A New Hjorth Distribution in Its Discrete Version

Abstract

The Hjorth distribution is more flexible in modeling various hazard rate shapes, including increasing, decreasing, and bathtub shapes. This makes it highly useful in reliability analysis and survival studies, where different failure rate behaviors must be captured effectively. In some practical experiments, the observed data may appear to be continuous, but their intrinsic discreteness requires the development of specialized techniques for constructing discrete counterparts to continuous distributions. This study extends this methodology by discretizing the Hjorth distribution using the survival function approach. The proposed discrete Hjorth distribution preserves the essential statistical characteristics of its continuous counterpart, such as percentiles and quantiles, making it a valuable tool for modeling lifetime data. The complexity of the transformation requires numerical techniques to ensure accurate estimations and analysis. A key feature of this study is the incorporation of Type-II censored samples. We also derive key statistical properties, including the quantile function and order statistics, and then employ maximum likelihood and Bayesian inference methods. A comparative analysis of these estimation techniques is conducted through simulation studies. Furthermore, the proposed model is validated using two real-world datasets, including electronic device failure times and ball-bearing failure analysis, by applying goodness-of-fit tests against alternative discrete models. The findings emphasize the versatility and applicability of the discrete Hjorth distribution in reliability studies, engineering, and survival analysis, offering a robust framework for modeling discrete data in practical scenarios. To our knowledge, no prior research has explored the use of censored data in analyzing discrete Hjorth-distributed data. This study fills this gap, providing new insights into discrete reliability modeling and broadening the application of the Hjorth distribution in real-world scenarios.

1. Introduction

In many lifetime experiments, data often seem continuous; however, they are actually discrete observations. This discrepancy drives the development of techniques that transform continuous distributions into discrete versions, offering a suitable match to the observed data. Discrete distributions in statistical modeling are widely encouraged for various compelling reasons. They address limitations in data collection methods, measurement intervals, or the inherent characteristics of observed phenomena, such as electronic device failures or maintenance cycles. Discrete distributions represent data with countable or finite values, such as the number of items tested, people in a queue, coin toss outcomes, or defective products in manufacturing. These distributions are intuitive and easy to interpret, corresponding to datasets with a defined range of possible values.

Moreover, many discrete distributions provide explicit expressions for their probability generating function and/or probability mass function (PMF), simplifying analytical operations and permitting effective moment and probability calculations without requiring integration. They are highly adaptable, making them suitable for modeling diverse real-world phenomena, including ecosystem species distribution, genetic diversity, and network traffic patterns. Their computational efficiency and versatility make them invaluable tools in statistical modeling.

Recently, a variety of discrete models have been developed, with applications in certain fields such as medicine, engineering, reliability studies, and survival analysis. For a deeper understanding and practical applications of discrete distributions, one can explore references Roy and Gupta [1] and Roy and Ghosh [2], among others. However, while these studies have advanced discrete reliability theory and other application areas, they primarily focus on continuous formulations or general discretization techniques that do not specifically address the nuances of the Hjorth distribution.

As a result, many scholars have significantly advanced discrete reliability theory, offering innovative approaches and perspectives. Transforming continuous models into their discrete counterparts involves various discretization techniques, which are extensively discussed and documented in academic research. These methods are designed to create discrete distributions that closely reflect their continuous origins. A range of approaches to discretization have been explored in the literature; one may refer to Bracquemond and Gaudoin [3], Lai [4], and Chakraborty [5].

A prominent approach in the development of discrete distributions involves the application of the survival function as a discretization technique. Notable contributions in this area include the derivation of discrete analogs for normal and Rayleigh distributions, presented by Roy [6,7], respectively, both employing the survival function methodology. Extending this framework, the discrete formulation of the Burr-II distribution was systematically examined by Al-Huniti and AL-Dayian [8]. Additionally, Bebbington et al. [9] provided a comprehensive study on the discrete additive counterpart of the Weibull distribution. Further advancements and examples of discrete transformations across various distributions are detailed by Haj Ahmad and Almetwally [10] and Chesneau et al. [11], among others.

This study employs the survival discretization method to transform the continuous Hjorth distribution into its discrete counterpart. Although the Hjorth distribution is highly flexible in modeling various hazard rate shapes in a continuous framework, practical scenarios often involve data recorded in discrete intervals or subject to censoring. Discretizing the Hjorth distribution using the survival function approach not only preserves key statistical features—like percentiles and quantiles—but also facilitates the analysis of data that are subject to constraints such as Type-II censoring. This discrete formulation directly addresses the practical challenges in reliability and survival analysis, where capturing the precise timing of failure events in a discrete manner is crucial for accurate modeling and inference. Unlike other discretization techniques, this method retains the intrinsic properties of the continuous model while providing a framework that is more robust to data irregularities.

Admittedly, the DH model’s computational intensity and reliance on numerical techniques are limitations that we address through rigorous simulation studies and methodological refinements. While the DH model has demonstrated superior performance in our comparative analysis, it may exhibit sensitivity to certain types of data distributions, particularly when the failure rate does not follow the bathtub or monotonic hazard rate structures that the model is designed to accommodate.

Consider a scenario where n items undergo a life-testing experiment, and only the first r failure times, denoted by $x_1<x_2<\cdots <x_r$, are observed. The set $\mathbf{x}=(x_1,x_2,\dots ,x_r)$ is called a Type-II censored sample. The remaining $n-r$ items are censored and known only to have more than $x_r$ failure times.

Various statistical characteristics, including the quantile function and order statistics, are derived, and statistical inference methods are explored. These methods utilize the maximum likelihood estimation approach and the Bayes framework. To assess the estimators, the two estimation techniques for the newly developed discrete distribution are compared. Simulation studies are conducted using numerical methodologies. To the best of our knowledge, the existing literature does not address the use of censored data to analyze discrete data following the Hjorth distribution. Therefore, this study aims to bridge this gap in distribution theory. Two real-world datasets are analyzed using goodness-of-fit tests to evaluate the effectiveness of the proposed model compared to other discrete alternatives. The first example discusses the failure times of some electronic devices, and the second considers ball-bearing failure analysis.

The rest of this paper is organized as follows: Section 2 introduces the discrete Hjorth distribution, while Section 3 presents some statistical functions. Section 4 and Section 5 calculate statistical inference, including maximum likelihood and Bayesian estimation, respectively. Monte Carlo results are presented and commented on in Section 6. Section 7 studies real-world data illustration examples. Section 8 explores the findings and concluding remarks.

2. Discrete Hjorth Distribution

The Hjorth distribution (by Hjorth [12]) is a continuous distribution that extends the Rayleigh, exponential, and linear failure rate distributions. Later, this distribution received some attention in reliability analysis and lifetime experiments. Hence, its statistical properties, inferential statistics, and reliability analysis have been studied by Guess et al. [13]. Yadav et al. [14] used progressive censoring and estimated the parameters of the Hjorth model in addition to the hazard rate and the reliability functions. Pushkarna et al. [15] studied some recurrence relations for the Hjorth moment model under progressive censoring. Pandey et al. [16] used Bayesian inference to estimate Hjorth parameters under a generalized Type-I progressive censoring sample. Korkmaz et al. [17] presented a new regression model using the Hjorth model. It is noted for its effectiveness in modeling datasets with smaller values rather than larger ones. Elshahhat and Nassar [18] examined survival analysis of the Hjorth model by adaptive Type-II progressive hybrid censoring.

We are motivated to use the Hjorth model because it has increasing, decreasing, constant, upside-down, unimodal, and bathtub hazard rates. It can also be considered a suitable model for fitting the bimodal, unimodal, U-shaped, and other-shaped data. Hence, it was observed in the literature that the Hjorth model outperforms several well-known lifetime distributions. Its continuous form, however, limits its use for datasets that are inherently discrete. By converting the Hjorth model into a discrete form, a new distribution is derived that accommodates count data while preserving Hjorth’s ability to model tail behavior.

The probability density function of the continuous Hjorth model is expressed as

$$f\left(x;\mathit{\theta }\right)=\left[\alpha x(1+\beta x)+\gamma \right]{\left(1+\beta x\right)}^{-\left({\displaystyle \frac{\gamma }{\beta }}+1\right)}{e}^{-\alpha {\displaystyle \frac{x^2}{2}}},x\ge 0,$$

(1)

where $\mathit{\theta }={(\alpha ,\beta ,\gamma )}^{\top }.$

The survival function is written as

$$S\left(x;\mathit{\theta }\right)={\left(1+\beta x\right)}^{-\left({\displaystyle \frac{\gamma }{\beta }}\right)}{e}^{-\alpha {\displaystyle \frac{x^2}{2}}},$$

(2)

in which $\beta$ and $\gamma$ are positive shape parameters, and $\alpha >0$ is a scale parameter.

Here, $\alpha$ controls the rate of decay in the exponential component. Hence, higher values of $\alpha$ result in a faster decay of the density. The parameter $\beta$ influences the shape of the distribution through the polynomial term $(1+\beta x)$. It affects the tail behavior and the curvature of the density function. A larger $\beta$ tends to increase the weight in the tail, which affects how quickly the probability mass of the distribution decreases as x increases. $\gamma$ adjusts the baseline level of the polynomial component. It plays a crucial role in determining the heaviness of the tail and the overall curvature near the origin, thereby influencing the skewness and the rate at which the distribution transitions from its peak to the tail. The hazard rate function for the Hjorth model is

$$h\left(x;\alpha ,\beta ,\gamma \right)=\left[\alpha x(1+\beta x)+\gamma \right].$$

(3)

This study aims to develop a novel discrete form of the Hjorth model and perform the statistical analysis under Type-II censored data for time and cost constraints. This counterpart, termed the discrete Hjorth (DH) model, is constructed using the survival discretization approach. In this section, the PMF, cumulative distribution function (CDF), and associated properties of DH model are outlined.

Roy [6,7] used the survival function to define the PMF for the new discrete distribution, which is presented in the following form:

$$P\left(X=k\right)=S\left(k\right)-S\left(k+1\right),k=0,1,2,\dots$$

(4)

where $S\left(x\right)$ is provided in Equation (2); hence, the DH PMF is provided by

$$P\left(X=k\right)=\Delta \left(\beta ,\gamma ,k\right)e^{-\alpha k_0}-\Delta \left(\beta ,\gamma ,k+1\right)e^{-\alpha k_1},$$

(5)

where $k_0={\displaystyle \frac{k^2}{2}}$, $k_1={\displaystyle \frac{{(k+1)}^2}{2}}$, and $\Delta \left(\beta ,\gamma ,i\right)={\left(1+\beta i\right)}^{-\left({\displaystyle \frac{\gamma }{\beta }}\right)}$.

The DH using the survival discretization method has the next CDF:

$$P\left(X<k\right)=1-\Delta \left(\beta ,\gamma ,k+1\right)e^{-\alpha k_1}.$$

(6)

The hazard rate function for the DH model is written as

$$h_{DH}\left(k\right)={\displaystyle \frac{\Delta \left(\beta ,\gamma ,k\right)}{\Delta \left(\beta ,\gamma ,k+1\right)}}{e}^{{\displaystyle \frac{\alpha (2k+1)}{2}}}-1.$$

(7)

Figure 1 illustrates various shapes of the PMF and HRF for the DH distribution, generated using different parameter selections. The PMF of the DH model is shown to exhibit right-skewed or symmetric patterns, while the HRF displays diverse forms, including decreasing, increasing, and bathtub-shaped patterns. Additionally, the shapes presented in the figure highlight that the DH model can naturally represent forward or backward recurrence time patterns within renewal processes.

In the next section, some statistical properties, functions, and characteristics of the DH model are derived, including quantiles, moments, and ordered statistics.

3. Statistical Functions

This section discusses statistical functions for the DH distribution, including the quantile function, moments, skewness, kurtosis, and order statistics.

3.1. Quantile Function and Moments

The quantile function (say, $Q\left(p\right)$) for a discrete distribution is the inverse of its CDF. It is used basically to generate random samples for simulation purposes. To determine the quantile function, we solve for k in terms of p:

$$p=1-\Delta \left(\beta ,\gamma ,k+1\right)e^{-\alpha {\displaystyle \frac{{(k+1)}^2}{2}}}.$$

(8)

Rearranging Equation (8), we obtain

$$\Delta \left(\beta ,\gamma ,k+1\right)e^{-\alpha {\displaystyle \frac{{(k+1)}^2}{2}}}=1-p.$$

(9)

Substituting $\Delta (\beta ,\gamma ,k+1)$ into (9), we have

$${\left(1+\beta (k+1)\right)}^{-\left({\displaystyle \frac{\gamma }{\beta }}\right)}{e}^{-\alpha {\displaystyle \frac{{(k+1)}^2}{2}}}=1-p,$$

(10)

and its natural logarithm becomes

$$-{\displaystyle \frac{\gamma }{\beta }}ln(1+\beta (k+1))-\alpha {\displaystyle \frac{{(k+1)}^2}{2}}=ln(1-p).$$

(11)

This equation must be solved numerically for k as an analytical closed-form expression is generally not feasible due to the combination of logarithmic and exponential terms. The complexity of the DH’s CDF makes direct algebraic inversion infeasible for deriving the quantile function, particularly due to the presence of exponential and power terms. Extracting k from the expression for $\Delta \left(\mathit{\theta },k\right)$ poses significant analytical difficulties and is unlikely to yield an exact solution because of the intricate structure and exponential decay term. As an alternative to an exact analytical form, approximations or numerical techniques can be employed for practical applications. Moments are a crucial statistical tool, offering detailed insights into the shape and properties of a probability distribution. They are widely applied in fields such as quality control, risk assessment, and environmental analysis.

To compute the moments for the DH model, consider a non-negative random variable l∼$DH\left(\mathit{\theta }\right)$. The sth moment, say ${\mu }_s^{\prime }$, can be expressed as follows:

$${\mu }_s^{\prime }=\sum _{l=0}^{\infty }{l}^s\left[\Delta \left(\beta ,\gamma ,l\right)e^{-\alpha {\displaystyle \frac{l^2}{2}}}-\Delta \left(\beta ,\gamma ,l+1\right)e^{-\alpha {\displaystyle \frac{{(l+1)}^2}{2}}}\right].$$

(12)

An exact expression for the $s\mathrm{th}$ moment is not obtainable; thus, numerical methods are necessary to evaluate the moment. Specifically, take $\alpha =(0.25,0.75,1.5)$ and several choices from $\beta$ and $\gamma$.

Table 1. Measurements of DH model using several choices regarding its parameter values.

$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\gamma }$	$\mathcal{M}$	$\mathcal{V}$	$\mathcal{ID}$	$\mathcal{CV}$	$\mathcal{S}$	$\mathcal{K}$
0.25	0.5	0.1	1.765	1.819	1.031	0.764	0.667	3.199
		0.5	1.082	1.442	1.333	1.110	1.075	4.093
		1.0	0.615	0.875	1.522	1.521	1.601	6.421
		1.5	0.364	0.507	1.652	1.954	2.153	9.699
		2.5	0.139	0.175	1.729	3.008	3.445	19.13
	2.5	0.1	1.877	1.858	0.990	0.726	0.616	3.147
		0.5	1.440	1.853	1.259	0.945	0.850	3.311
		1.0	1.041	1.584	1.287	1.209	1.226	4.164
		1.5	0.757	1.250	1.391	1.477	1.579	5.585
		2.5	0.408	0.705	1.422	2.059	2.337	10.21
0.75	0.5	0.1	0.859	0.636	0.741	0.929	0.671	3.074
		0.5	0.584	0.526	0.900	1.241	1.148	3.699
		1.0	0.367	0.367	1.001	1.652	1.768	5.261
		1.5	0.234	0.244	1.062	2.114	2.363	7.871
		2.5	0.098	0.104	1.149	3.291	3.590	17.51
	2.5	0.1	0.895	0.649	0.725	0.900	0.636	3.043
		0.5	0.715	0.615	0.860	1.097	0.887	3.297
		1.0	0.541	0.530	0.981	1.346	1.236	4.073
		1.5	0.410	0.436	1.045	1.609	1.645	5.276
		2.5	0.238	0.273	1.060	2.198	2.518	8.919
1.5	0.5	0.1	0.480	0.340	0.709	1.215	0.793	2.806
		0.5	0.340	0.276	0.812	1.544	1.218	3.582
		1.0	0.223	0.199	0.893	2.003	1.772	5.260
		1.5	0.146	0.138	0.941	2.536	2.379	7.626
		2.5	0.064	0.063	0.986	3.932	3.872	15.69
	2.5	0.1	0.497	0.347	0.699	1.186	0.757	2.769
		0.5	0.403	0.313	0.777	1.388	1.034	3.212
		1.0	0.311	0.265	0.852	1.655	1.384	4.026
		1.5	0.240	0.218	0.907	1.944	1.745	5.145
		2.5	0.143	0.140	0.976	2.609	2.530	8.638

illustrates the behavior with respect to the mean ($\mathcal{M}$), variance ($\mathcal{V}$), index of dispersion ($\mathcal{ID}$), coefficient of variation ($\mathcal{CV}$), skewness ($\mathcal{S}$), and kurtosis ($\mathcal{K}$) for the DH model. As a result, we summarize the following points:

• As $\alpha$ grows (for fixed $\beta$ and $\gamma$), the values of $\mathcal{M}$, $\mathcal{V}$, $\mathcal{ID}$, and $\mathcal{K}$ decrease while those for $\mathcal{CV}$ and $\mathcal{S}$ increase.
• As $\beta$ grows (for fixed $\alpha$ and $\gamma$), the values of $\mathcal{M}$ and $\mathcal{V}$ increase while those for $\mathcal{ID}$, $\mathcal{CV}$, $\mathcal{S}$, and $\mathcal{K}$ decrease.
• As $\gamma$ grows (for fixed $\alpha$ and $\beta$), the values of $\mathcal{M}$ and $\mathcal{V}$ decrease while those for $\mathcal{ID}$, $\mathcal{CV}$, $\mathcal{S}$, and $\mathcal{K}$ increase.
• The DH model is well suited for modeling both under- and over-dispersed data as its variance can be smaller than the mean for certain parameter values, while the variance is greater than the mean in other choices regarding parameter values.
• The DH model exhibits flexible dispersion characteristics depending on parameter choices.
• For reliability modeling, higher dispersion ($\mathcal{ID}$ > 1) suggests suitability for lifetime data with higher variability.
• The positive skewness values indicate that this distribution is right-skewed. Additionally, a skewness value approaching zero suggests the possibility of a symmetric curve for the PMF.
• Elevated kurtosis signifies greater tail risk and the likelihood of outliers relative to a normal distribution. Changes in the distribution can be observed by adjusting $\alpha$, $\beta$, and $\gamma$.

3.2. Ordered Statistics

Consider $W_1,W_2,\dots ,W_n$ as a random sample following the DH model, and assume that $W_{1:n},W_{2:n},\dots ,W_{n:n}$ represent the related order statistics; hence, the $i\mathrm{th}$ order statistics have a CDF at w, which is

$$F_{i:n}\left(w;\mathit{\theta }\right)=\sum _{i=1}^n\left(\begin{array}{c}n\\ m\end{array}\right){\left[F_i\left(w;\mathit{\theta }\right)\right]}^m{\left[1-F_i\left(w;\mathit{\theta }\right)\right]}^{n-m}.$$

(13)

The negative Binomial theorem can be used as a series representation for the $CDF$, which can be written as

$$F_{i:n}\left(w;\mathit{\theta }\right)=\sum _{i=1}^n\sum _{j=1}^{n-m}\left(\begin{array}{c}n\\ m\end{array}\right){\left(\begin{array}{c}n-m\\ j\end{array}\right)\left(-1\right)}^j{\left[F_i\left(w;\mathit{\theta }\right)\right]}^{m+j}.$$

(14)

Therefore,

$$F_{i:n}\left(w;\mathit{\theta }\right)=\sum _{i=1}^n\sum _{j=1}^{n-m}\left(\begin{array}{c}n\\ m\end{array}\right){\left(\begin{array}{c}n-m\\ j\end{array}\right)\left(-1\right)}^j\left[1-\Delta \left(\beta ,\gamma ,w+1\right)e^{-\alpha {\displaystyle \frac{{(w+1)}^2}{2}}}\right].$$

(15)

The ith order statistic under the DH model has PMF that can be expressed as follows

$$f_{i:n}\left(w;\mathit{\theta }\right)=\sum _{i=1}^n\sum _{j=1}^{n-m}\left(\begin{array}{c}n\\ m\end{array}\right){\left(\begin{array}{c}n-m\\ j\end{array}\right)\left(-1\right)}^j{[\Delta \left(\beta ,\gamma ,w\right)e^{-\alpha {\displaystyle \frac{w^2}{2}}}-\Delta \left(\beta ,\gamma ,w+1\right)e^{-\alpha {\displaystyle \frac{{(w+1)}^2}{2}}}]}^{m+j}.$$

Hence, the $r\mathrm{th}$ moments of $w_{i:n}$ can be written as follows:

$$E\left(w_{i:n}^r\right)=\sum _{w=0}^{\infty }\sum _{i=1}^n\sum _{j=1}^{n-m}\left(\begin{array}{c}n\\ m\end{array}\right){\left(\begin{array}{c}n-m\\ j\end{array}\right)\left(-1\right)}^j{w}^r{\left[\Delta \left(\beta ,\gamma ,w\right)e^{-\alpha {\displaystyle \frac{w^2}{2}}}-\Delta \left(\beta ,\gamma ,w+1\right)e^{-\alpha {\displaystyle \frac{{(w+1)}^2}{2}}}\right]}^{m+j}.$$

4. Maximum Likelihood Estimation

In this section, we estimate the undetermined parameters $\alpha$, $\beta$, and $\gamma$ for the DH model by applying the maximum likelihood estimation (MLE) approach. The MLE estimates are obtained based on the Type-II censoring scheme, and, by using the missing information principle, the variance–covariance matrix (VCM) of $\alpha$, $\beta$, and $\gamma$ is obtained. This matrix is then utilized to construct asymptotic confidence intervals for $\alpha$, $\beta$, and $\gamma$. Let $x_1,\dots ,x_r$ represent a Type-II censored sampling from DH model.

Referring to the PMF in Equation (5) and the CDF in Equation (7), the likelihood and the log-likelihood functions are explored, respectively, as

$$\begin{array}{cc}\hfill L(\mathit{\theta }\mid \mathbf{x})& =\left[\prod _{i=1}^r\left(\Delta (\beta ,\gamma ,x_i)e^{-\alpha {\displaystyle \frac{x_i^2}{2}}}-\Delta (\beta ,\gamma ,x_i+1)e^{-\alpha {\displaystyle \frac{{(x_i+1)}^2}{2}}}\right)\right]\hfill \\ \hfill & \times {\left[\Delta (\beta ,\gamma ,x_r)e^{-\alpha {\displaystyle \frac{x_r^2}{2}}}\right]}^{n-r},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \ell (\mathit{\theta }\mid \mathbf{x})& =\sum _{i=1}^{r}log\left(\Delta (\beta ,\gamma ,x_i)e^{-\alpha {\displaystyle \frac{x_i^2}{2}}}-\Delta (\beta ,\gamma ,x_i+1)e^{-\alpha {\displaystyle \frac{{(x_i+1)}^2}{2}}}\right)\hfill \\ \hfill & +(n-r)log\left(\Delta (\beta ,\gamma ,x_r)e^{-\alpha {\displaystyle \frac{x_r^2}{2}}}\right).\hfill \end{array}$$

(16)

The parameters’ MLEs are obtained by deriving the likelihood function (16) partially for each parameter such as

$${\displaystyle \frac{\partial \ell }{\partial \alpha }}=\sum _{i=1}^r{\displaystyle \frac{A_{i,\alpha }}{A_i(\alpha ,\beta ,\gamma )}}-{\displaystyle \frac{(n-r)x_r^2}{2}},$$

(17)

$${\displaystyle \frac{\partial \ell }{\partial \beta }}=\sum _{i=1}^r{\displaystyle \frac{A_{i,\beta }}{A_i(\alpha ,\beta ,\gamma )}}+(n-r)\psi (\beta ,x_r),$$

(18)

and

$${\displaystyle \frac{\partial \ell }{\partial \gamma }}=\sum _{i=1}^r{\displaystyle \frac{A_{i,\gamma }}{A_i(\alpha ,\beta ,\gamma )}}-{\displaystyle \frac{(n-r)}{\beta }}ln(1+\beta x_r),$$

(19)

where

• $A_i(\alpha ,\beta ,\gamma )=\Delta (\beta ,\gamma ,x_i)e^{-{\displaystyle \frac{\alpha x_i^2}{2}}}-\Delta (\beta ,\gamma ,x_i+1)e^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}},$
• $A_{i,\alpha }\equiv {\displaystyle \frac{\partial A_i}{\partial \alpha }}=-{\displaystyle \frac{x_i^2}{2}}\Delta (\beta ,\gamma ,x_i)e^{-{\displaystyle \frac{\alpha x_i^2}{2}}}+{\displaystyle \frac{{(x_i+1)}^2}{2}}\Delta (\beta ,\gamma ,x_i+1)e^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}},$
• $A_{i,\beta }\equiv {\displaystyle \frac{\partial A_i}{\partial \beta }}=e^{-{\displaystyle \frac{\alpha x_i^2}{2}}}\Delta (\beta ,\gamma ,x_i)\psi (\beta ,x_i)-e^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}}\Delta (\beta ,\gamma ,x_i+1)\psi (\beta ,x_i+1),$
• $A_{i,\gamma }\equiv {\displaystyle \frac{\partial A_i}{\partial \gamma }}=-{\displaystyle \frac{1}{\beta }}ln(1+\beta x_i)\Delta (\beta ,\gamma ,x_i)e^{-{\displaystyle \frac{\alpha x_i^2}{2}}}+{\displaystyle \frac{1}{\beta }}ln(1+\beta (x_i+1))\Delta (\beta ,\gamma ,x_i+1)e^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}},$

and

• $\psi (\beta ,x)={\displaystyle \frac{\gamma }{{\beta }^2}}ln(1+\beta x)-{\displaystyle \frac{\gamma x}{\beta (1+\beta x)}}.$

The obtained system of nonlinear Equations (17)–(19) is solved by using numerical techniques to determine the parameter estimates. These calculations can be complex and computationally intensive, especially when dealing with sums and exponentials. For actual applications, numerical methods or software packages capable of symbolic differentiation may be beneficial for accurately computing these derivatives, especially when optimizing parameters in practical scenarios. Various numerical techniques have been explored in the literature; in this study, the Newton–Raphson method is employed. This method was selected primarily due to its quadratic convergence rate that enables rapid refinement of estimates when the initial guess is reasonably close to the true solution. The results of the analysis are discussed in Section 6.

To construct the VCM of the DH model parameters $\mathit{\theta }$, we need to use the Fisher Information Matrix (FIM), which is derived from the second-order partial derivatives of the log-likelihood function $\ell \left(\mathit{\theta }\right)$. The FIM (say, $\mathcal{I}(·)$) is a $3\times 3$ symmetric matrix where each entry corresponds to a second-order partial derivative. The inverse of the FIM provides the VCM (denoted by $\Sigma (·)\cong {\mathcal{I}}^{-1}(·)$), which is then used to build asymptotic confidence intervals for the parameters.

The FIM and VCM (at $\mathit{\theta }=\widehat{\mathit{\theta }}$) are expressed, respectively, as

$$\mathcal{I}\left(\mathit{\theta }\right)=\left(\begin{array}{ccc}{\mathcal{I}}_{\alpha \alpha }& {\mathcal{I}}_{\alpha \beta }& {\mathcal{I}}_{\alpha \gamma }\\ {\mathcal{I}}_{\beta \alpha }& {\mathcal{I}}_{\beta \beta }& {\mathcal{I}}_{\beta \gamma }\\ {\mathcal{I}}_{\gamma \alpha }& {\mathcal{I}}_{\gamma \beta }& {\mathcal{I}}_{\gamma \gamma }\end{array}\right)$$

and

$$\Sigma \left(\mathit{\theta }\right)=\left(\begin{array}{ccc}\mathrm{Var}\left(\alpha \right)& \mathrm{Cov}(\alpha ,\beta )& \mathrm{Cov}(\alpha ,\gamma )\\ \mathrm{Cov}(\beta ,\alpha )& \mathrm{Var}\left(\beta \right)& \mathrm{Cov}(\beta ,\gamma )\\ \mathrm{Cov}(\gamma ,\alpha )& \mathrm{Cov}(\gamma ,\beta )& \mathrm{Var}\left(\gamma \right)\end{array}\right).$$

The second-order partial derivatives of the log-likelihood function ℓ are developed as follows:

$${\displaystyle \frac{{\partial }^2\ell }{\partial {\alpha }^2}}=\sum _{i=1}^r\left[{\displaystyle \frac{A_{i,\alpha \alpha }}{A_i}}-{\left({\displaystyle \frac{A_{i,\alpha }}{A_i}}\right)}^2\right],$$

(20)

$${\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }^2}}=\sum _{i=1}^r\left[{\displaystyle \frac{A_{i,\beta \beta }}{A_i}}-{\left({\displaystyle \frac{A_{i,\beta }}{A_i}}\right)}^2\right]+(n-r){\psi }^{\prime }(\beta ,x_r),$$

(21)

$${\displaystyle \frac{{\partial }^2\ell }{\partial {\gamma }^2}}=\sum _{i=1}^r\left[{\displaystyle \frac{A_{i,\gamma \gamma }}{A_i}}-{\left({\displaystyle \frac{A_{i,\gamma }}{A_i}}\right)}^2\right],$$

(22)

$${\displaystyle \frac{{\partial }^2\ell }{\partial \alpha \partial \beta }}=\sum _{i=1}^r\left[{\displaystyle \frac{A_{i,\alpha \beta }}{A_i}}-{\displaystyle \frac{A_{i,\alpha }{A}_{i,\beta }}{A_i^2}}\right],$$

(23)

$${\displaystyle \frac{{\partial }^2\ell }{\partial \alpha \partial \gamma }}=\sum _{i=1}^r\left[{\displaystyle \frac{A_{i,\alpha \gamma }}{A_i}}-{\displaystyle \frac{A_{i,\alpha }{A}_{i,\gamma }}{A_i^2}}\right],$$

(24)

and

$${\displaystyle \frac{{\partial }^2\ell }{\partial \beta \partial \gamma }}=\sum _{i=1}^r\left[{\displaystyle \frac{A_{i,\beta \gamma }}{A_i}}-{\displaystyle \frac{A_{i,\beta }{A}_{i,\gamma }}{A_i^2}}\right]+(n-r)\left[{\displaystyle \frac{1}{{\beta }^2}}ln(1+\beta x_r)-{\displaystyle \frac{x_r}{\beta (1+\beta x_r)}}\right],$$

(25)

where ${\Delta }^{\circ }=\Delta (\beta ,\gamma ,x_i)$ and ${\Delta }^{•}=\Delta (\beta ,\gamma ,x_i+1)$

• $A_{i,\alpha \alpha }={\displaystyle \frac{x_i^4}{4}}{\Delta }^{\circ }{e}^{-{\displaystyle \frac{\alpha x_i^2}{2}}}-{\displaystyle \frac{{(x_i+1)}^4}{4}}{\Delta }^{•}{e}^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}},$
• $A_{i,\beta \beta }=e^{-{\displaystyle \frac{\alpha x_i^2}{2}}}{\Delta }^{\circ }\left[\psi {(\beta ,x_i)}^2+{\psi }^{\prime }(\beta ,x_i)\right]-e^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}}{\Delta }^{•}\left[\psi {(\beta ,x_i+1)}^2+{\psi }^{\prime }(\beta ,x_i+1)\right],$
• $A_{i,\gamma \gamma }={\displaystyle \frac{1}{{\beta }^2}}ln{(1+\beta x_i)}^2{\Delta }^{\circ }{e}^{-{\displaystyle \frac{\alpha x_i^2}{2}}}-{\displaystyle \frac{1}{{\beta }^2}}ln{(1+\beta (x_i+1))}^2{\Delta }^{•}{e}^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}},$
• ${\psi }^{\prime }(\beta ,x)=-{\displaystyle \frac{2\gamma }{{\beta }^3}}ln(1+\beta x)+{\displaystyle \frac{\gamma x}{{\beta }^2(1+\beta x)}}+{\displaystyle \frac{\gamma x(1+2\beta x)}{{\left[\beta (1+\beta x)\right]}^2}},$
• $A_{i,\alpha \beta }=-{\displaystyle \frac{x_i^2}{2}}{e}^{-{\displaystyle \frac{\alpha x_i^2}{2}}}{\Delta }^{\circ }\psi (\beta ,x_i)+{\displaystyle \frac{{(x_i+1)}^2}{2}}{e}^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}}{\Delta }^{•}\psi (\beta ,x_i+1),$
• $A_{i,\alpha \gamma }={\displaystyle \frac{x_i^2}{2\beta }}ln(1+\beta x_i)e^{-{\displaystyle \frac{\alpha x_i^2}{2}}}{\Delta }^{\circ }-{\displaystyle \frac{{(x_i+1)}^2}{2\beta }}ln(1+\beta (x_i+1))e^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}}{\Delta }^{•}$

and

• $A_{i,\beta \gamma }=-{\displaystyle \frac{1}{\beta }}{e}^{-{\displaystyle \frac{\alpha x_i^2}{2}}}{\Delta }^{\circ }\psi (\beta ,x_i)ln(1+\beta x_i)+{\displaystyle \frac{1}{\beta }}{e}^{-{\displaystyle \frac{\alpha {(x_i+1)}^2}{2}}}{\Delta }^{•}\psi (\beta ,x_i+1)ln(1+\beta (x_i+1)).$

Hence, the asymptotic confidence interval for $\theta =\left(\mathit{\theta }\right)$ is provided by $\theta \pm z_{\frac{\lambda }{2}}\sqrt{\mathrm{Var}\left(\theta \right)}$, where $z_{\frac{\lambda }{2}}$ is the critical value from the standard normal distribution for a given $\lambda$th confidence level.

5. Bayesian Inference

This section deals with the Bayesian estimation, which is applied to estimate the unknown parameters of the DH model. This method treats the parameters as random variables following a certain model, referred to as the prior distribution. Since prior information is often unavailable, it becomes necessary to choose a suitable prior.

A joint conjugate prior distribution is selected for parameters $\alpha$, $\beta$, and $\gamma$, with each parameter assumed to follow a gamma distribution. Consequently, $\alpha \sim Gamma(a_1,b_1)$, $\beta \sim Gamma(a_2,b_2)$, and $\gamma \sim Gamma(a_3,b_3)$, where $a_i$ and $b_i$ (for $i=1,2,3$) represent non-negative hyper-parameters of the specified distributions. Therefore, the prior distributions for $\alpha$, $\beta$, and $\gamma$ are defined as

$$\begin{array}{cc}\hfill & {\pi }_1\left(\alpha \right)={\displaystyle \frac{{b_1}^{a_1}}{\Gamma \left(a_1\right)}}{\alpha }^{a_1-1}{e}^{-b_1\alpha },\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & {\pi }_2\left(\beta \right)={\displaystyle \frac{{b_2}^{a_2}}{\Gamma \left(a_2\right)}}{\beta }^{a_2-1}{e}^{-b_2\beta },\hfill \end{array}$$

(27)

and

$${\pi }_3\left(\gamma \right)={\displaystyle \frac{{b_3}^{a_3}}{\Gamma \left(a_3\right)}}{\gamma }^{a_3-1}{e}^{-b_3\gamma }.$$

(28)

Therefore, the joint prior function for $\alpha$, $\beta$, and $\gamma$ is

$$\pi \left(\mathit{\theta }\right)\propto {\alpha }^{a_1-1}{\beta }^{a_2-1}{\gamma }^{a_3-1}{e}^{-b_1\alpha -b_2\beta -b_3\gamma }.$$

(29)

The joint posterior (say, $P(·)$) of $\alpha$, $\beta$, and $\gamma$ with data availability is

$$P(\mathit{\theta }\mid \mathbf{x})={\displaystyle \frac{1}{K}}L(\mathit{\theta }\mid \mathbf{x})\pi \left(\mathit{\theta }\right),$$

(30)

where $K=\int \int \int L(\mathit{\theta }\mid \mathbf{x})\pi \left(\mathit{\theta }\right)d\alpha d\beta d\gamma$ is the normalizing factor.

The parameter estimation for the DH model has been analyzed using the squared error (SE) loss function. Later, to evaluate the efficiency of the estimation methods and to examine the impact of parameter values on these techniques, a simulation analysis is conducted to assess the efficiency of the estimators based on several metrics, namely average point estimate, mean absolute bias, mean square error, average interval length, and coverage probability.

Under the SE loss function, Bayesian estimation of a parameter $\theta$ (for example) is defined as the expected value for the joint posterior distribution as

$${\widehat{\theta }}_{SE}={\displaystyle \frac{1}{K}}\int \theta L(\theta \mid \mathbf{x})\pi \left(\theta \right)d\theta .$$

(31)

It is necessary to employ numerical approaches for calculating the triple integration mentioned in Equation (31). To achieve this, we adopted the Markov Chain Monte Carlo (MCMC) approach. A suitable $\mathsf{R}$ code was developed to execute this process.

Applying the survival discretization method leads to the development of the DH model, whose PMF is defined in Equation (5). The joint posterior density under a Type-II censored sample is expressed as follows:

$$\begin{array}{cc}\hfill P(\mathit{\theta }\mid \mathbf{x})& ={\displaystyle \frac{1}{K}}\left[\prod _{i=1}^r\left(\Delta (\beta ,\gamma ,x_i)e^{-\alpha {\displaystyle \frac{x_i^2}{2}}}-\Delta (\beta ,\gamma ,x_i+1)e^{-\alpha {\displaystyle \frac{{(x_i+1)}^2}{2}}}\right)\right]{\left[\Delta (\beta ,\gamma ,x_r)e^{-\alpha {\displaystyle \frac{x_r^2}{2}}}\right]}^{n-r}\hfill \\ \hfill & \times {\alpha }^{a_1-1}{\beta }^{a_2-1}{\gamma }^{a_3-1}{e}^{-b_1\alpha -b_2\beta -b_3\gamma }.\hfill \end{array}$$

(32)

Bayesian analysis for the parameters $\alpha$, $\beta$, and $\gamma$ under the SE loss function is beyond developing their conditional posterior functions, respectively, as

$$\begin{array}{c}\hfill P_1(\alpha \mid \beta ,\gamma ,\mathbf{x})=L(\mathit{\theta }\mid \mathbf{x}){\alpha }^{a_1}{e}^{-b_1\alpha },\end{array}$$

(33)

$$\begin{array}{c}\hfill P_2(\beta \mid \alpha ,\gamma ,\mathbf{x})=L(\mathit{\theta }\mid \mathbf{x}){\beta }^{a_2}{e}^{-b_2\beta },\end{array}$$

(34)

and

$$\begin{array}{c}\hfill P_3(\gamma \mid \alpha ,\beta \mathbf{x})=L(\mathit{\theta }\mid \mathbf{x}){\gamma }^{a_3}{e}^{-b_3\gamma }.\end{array}$$

(35)

The Bayes estimators of $\alpha$, $\beta$, and $\gamma$, based on Equations (33)–(35), respectively, cannot follow any known statistical distribution. To handle this problem, the Metropolis–Hastings (M–H) sampler, the most important term of the MCMC family of Bayes calculation techniques, is recommended to evaluate the Bayes point estimates and their Bayesian credible intervals (BCIs). It is useful to remember that the BCI represents a range of values where an unknown parameter is likely to reside with a specified probability based on the obtained data and prior beliefs regarding the parameters.

Now, the implementation of the MCMC approach through the M–H sampler is outlined as listed below:

Step 1:

Start by assuming a basic value $({\alpha }^{\left(0\right)},{\beta }^{\left(0\right)},{\gamma }^{\left(0\right)})=(\widehat{\alpha },\widehat{\beta },\widehat{\gamma }).$

Step 2:

Set $m=1.$

Step 3:

From (33), create ${\alpha }^{\star }$ from $N({\alpha }^{(m-1)},\mathrm{Var}\left(\alpha \right))$, and then follow the next steps (a)–(d):

(a)

Find ${\mathcal{C}}_1={\displaystyle \frac{P_1({\alpha }^{\star }\mid {\beta }^{(m-1)},{\gamma }^{(m-1)},\mathbf{x})}{P1({\alpha }^{(m-1)}\mid {\beta }^{(m-1)},{\gamma }^{(m-1)},\mathbf{x})}}$.

(b)

Obtain ${\mathcal{Q}}_1=min\{1,{\mathcal{C}}_1\}$.

(c)

Generate $u_1$ from $\mathcal{U}(0,1)$.

(d)

If $u_1<{\mathcal{Q}}_1$, adopt the proposal and set ${\alpha }^{\left(m\right)}={\alpha }^{\star }$; otherwise, set ${\alpha }^{\left(m\right)}={\alpha }^{(m-1)}$.

Step 4:

Redo Step 3 for $\beta$ and $\gamma$.

Step 5:

Set $m=m+1$.

Step 6:

Obtain (${\alpha }^{\left(l\right)},{\beta }^{\left(l\right)},{\gamma }^{\left(l\right)}$) for $l=1,2,\dots N$ by redoing Steps 2 to 4 N times.

Step 7:

Discard the first iterations (say, $N^{*}$) as burn-in, and obtain the Bayes estimates of $\alpha$, $\beta$, and $\gamma$ (say, ${\psi }_j,j=1,2,3$) as

$$\begin{array}{c}\hfill {\tilde{\psi }}_j={\displaystyle \frac{1}{N-N^{*}}}\sum _{m=N^{*}+1}^N{\psi }^{\left(m\right)},\end{array}$$

where $\left({\psi }_1,{\psi }_2,{\psi }_3\right)=\left(\alpha ,\beta ,\gamma \right).$

Step 8:

Sort ${\psi }_j^{\left(m\right)},j=1,2,3,m=N^{*}+1,N^{*}+2,\dots ,N,$ in ascending order and obtain the $100(1-\lambda )\%$ BCIs for ${\psi }_j$ as

$$\begin{array}{c}\hfill \left({\psi }_{j\left((N-N^{*}){\displaystyle \frac{\lambda }{2}}\right)},{\psi }_{j\left((N-N^{*})\left(1-{\displaystyle \frac{\lambda }{2}}\right)\right)}\right).\end{array}$$

6. Numerical Comparisons

In this part, to test the efficiency of the acquired estimators of DH’s parameters $\alpha$, $\beta$, and $\gamma$, different comparisons via Monte Carlo experiments are performed. After that, some comments on the outcomes of the simulation are provided. The numerical part was performed using the R (64) software package.

6.1. Simulation Scenarios

We now suggest the following steps to gather a Type-II censored dataset from the proposed model:

Step 1:

Fix the number of replications as 2000.

Step 2:

Fix the values of $\mathrm{DH}(\alpha ,\beta ,\gamma )$; that is, (a) Set-1:$\mathrm{DH}$(0.5,0.8,1) and (b) Set-2:$\mathrm{DH}$(1,1.5,2).

Step 3:

Set sample size n as $(30,60,100,150,200)$.

Step 4:

Obtain $u_i$ for $i=1,2,\dots ,n$ as an independent observation from a uniform distribution $U(0,1)$.

Step 5:

Obtain pseudo-random values of size n from the DH distribution as

$$x_i=F^{-1}(u_i;\alpha ,\beta ,\gamma ),i=1,2,\dots ,n.$$

Step 6:

Sort the outputs in Step 5; then, for a failure percentage (FP%), determine the value of r such as ${\displaystyle \frac{r}{n}}\times 100=40$, 80, and 100%. It is important to note that, when FP%$\to 100\%$, censored (incomplete) sampling reverts to completed sampling.

Step 7:

Fix the informative prior sets of $a_i$ and $b_i$ for $i=1,2,3$ to develop the Bayes point and interval estimations, such as

(i)

For Set-1:

• Prior-A:$(a_1,a_2,a_3)=(2.5,0.16,5)$ and $b_i=5$ for $i=1,2,3$;
• Prior-B:$(a_1,a_2,a_3)=(5,0.08,10)$ and $b_i=10$ for $i=1,2,3$.

(ii)

For Set-2:

• Prior-A:$(a_1,a_2,a_3)=(5,0.3,10)$ and $b_i=5$ for $i=1,2,3$;
• Prior-B:$(a_1,a_2,a_3)=(10,0.15,20)$ and $b_i=10$ for $i=1,2,3$.

Step 8:

Compute the average point estimate (APE) of $\alpha$ (as an example):

$$\begin{array}{c}\hfill \mathrm{APE}\left(\breve{\alpha }\right)={\displaystyle \frac{1}{2000}}{\sum }_{i=1}^{2000}{\breve{\alpha }}^{\left[i\right]},\end{array}$$

where $\breve{\alpha }$ is the offered estimate of $\alpha$ at ith sample.

Step 9:

Compute the mean squared error (MSE) and mean absolute bias (MAB) of $\alpha$:

$$\begin{array}{c}\hfill \mathrm{MSE}\left(\breve{\alpha }\right)={\displaystyle \frac{1}{2000}}{\sum }_{i=1}^{2000}{\left({\breve{\alpha }}^{\left[i\right]}-\alpha \right)}^2,\end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{MAB}\left(\breve{\alpha }\right)={\displaystyle \frac{1}{2000}}{\sum }_{i=1}^{2000}\left|{\breve{\alpha }}^{\left[i\right]}-\alpha \right|.\end{array}$$

Step 10:

Compute the average interval length (AIL) and coverage provability (CP) of $\alpha$:

$$\begin{array}{cc}\hfill {\mathrm{ACL}}_{95\%}\left(\alpha \right)& ={\displaystyle \frac{1}{2000}}{\sum }_{i=1}^{2000}\left({\mathcal{U}}_{{\breve{\alpha }}^{\left[i\right]}}\left(\alpha \right)-{\mathcal{L}}_{{\breve{\alpha }}^{\left[i\right]}}\left(\alpha \right)\right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\mathrm{CP}}_{95\%}\left(\alpha \right)={\displaystyle \frac{1}{2000}}{\sum }_{i=1}^{2000}{\mathbb{I}}_{\left({\mathcal{L}}_{{\breve{\alpha }}^{\left[i\right]}};{\mathcal{U}}_{{\breve{\alpha }}^{\left[i\right]}}\right)}\left(\alpha \right),\end{array}$$

where ($\mathcal{L}(·)$,$\mathcal{U}(·)$) refers to the estimated interval limits and $\mathbb{I}(·)$ denotes the indicator operator.

The chosen prior distributions reflect reasonable assumptions about parameters $\alpha$, $\beta$, and $\gamma$ based on prior research and empirical observations in reliability and survival studies. Prior-A represents a more informative prior with moderate variance, assuming prior knowledge about the expected parameter range, whereas Prior-B has greater uncertainty by enabling higher variance, making it suitable for scenarios where less prior knowledge is available.

6.2. Simulation Results and Discussion

In

Table 2. The point results of $\alpha$, $\beta$, and $\gamma$ from Set-1.

n	FP%	Par.	MLE			Bayes
Prior →						A			B
30	40%	$\alpha$	0.4595	0.3357	0.5327	0.4399	0.0892	0.2437	0.6927	0.0691	0.2125
		$\beta$	1.2955	1.1791	1.0403	0.7193	0.6275	0.4825	1.0522	0.1867	0.3960
		$\gamma$	1.1438	0.5379	0.6113	0.9556	0.3334	0.5583	0.8510	0.1499	0.3706
	80%	$\alpha$	0.4691	0.3085	0.4952	0.4802	0.0714	0.2283	0.4713	0.0629	0.2090
		$\beta$	0.9478	0.9663	0.8685	0.7054	0.5176	0.4365	1.1733	0.1691	0.3778
		$\gamma$	1.0751	0.5133	0.5963	1.0021	0.2452	0.4660	1.0803	0.1170	0.2934
	100%	$\alpha$	0.9581	0.2854	0.4581	0.4886	0.0661	0.2112	0.6242	0.0582	0.1933
		$\beta$	0.9965	0.8939	0.8033	0.7267	0.4788	0.4038	1.1701	0.1516	0.3495
		$\gamma$	1.0180	0.4748	0.5452	1.0763	0.2168	0.4311	1.1086	0.1024	0.2714
60	40%	$\alpha$	0.4219	0.2749	0.4831	0.4904	0.0680	0.1995	0.4689	0.0576	0.1895
		$\beta$	1.2951	0.6072	0.6936	1.8196	0.4526	0.4048	1.4113	0.1578	0.3384
		$\gamma$	0.9944	0.4727	0.5763	0.8103	0.2444	0.4587	0.9580	0.0931	0.2838
	80%	$\alpha$	0.5119	0.2613	0.4792	0.5481	0.0660	0.1952	0.5068	0.0520	0.1827
		$\beta$	0.9122	0.6063	0.5426	1.6449	0.1853	0.3708	1.2936	0.1378	0.2825
		$\gamma$	1.2847	0.4236	0.5363	1.2421	0.2334	0.4364	1.2864	0.0900	0.2750
	100%	$\alpha$	0.8758	0.2417	0.4433	0.5070	0.0611	0.1805	0.4688	0.0481	0.1690
		$\beta$	0.8925	0.5609	0.5019	1.5815	0.1714	0.3430	1.1966	0.1275	0.2613
		$\gamma$	1.2541	0.3918	0.4961	1.1489	0.2159	0.4037	0.8994	0.0833	0.2544
100	40%	$\alpha$	0.4891	0.2569	0.4455	0.4676	0.0628	0.1914	0.4523	0.0517	0.1678
		$\beta$	1.2951	0.1612	0.4051	1.4877	0.1383	0.3322	0.7992	0.1179	0.2665
		$\gamma$	1.1676	0.3724	0.5123	0.8201	0.1300	0.2919	0.9716	0.0620	0.2213
	80%	$\alpha$	0.5378	0.2450	0.2192	0.5405	0.0611	0.1885	0.6602	0.0503	0.1603
		$\beta$	0.9293	0.1407	0.3679	1.4782	0.0978	0.2756	0.8528	0.0907	0.1902
		$\gamma$	1.1275	0.3165	0.4853	1.2888	0.1222	0.2854	1.2627	0.0517	0.1909
	100%	$\alpha$	0.8758	0.2267	0.2028	0.4999	0.0565	0.1743	0.6507	0.0465	0.1482
		$\beta$	0.8925	0.1265	0.3240	1.3674	0.0910	0.2489	0.8881	0.0839	0.1759
		$\gamma$	1.2541	0.2793	0.4249	1.1921	0.1160	0.2395	1.2680	0.0448	0.1707
150	40%	$\alpha$	0.4803	0.2251	0.1942	0.6836	0.0501	0.1817	0.5732	0.0383	0.1525
		$\beta$	1.2951	0.1385	0.3390	0.8417	0.0822	0.1833	1.0697	0.0337	0.1548
		$\gamma$	0.9868	0.2757	0.4275	0.9444	0.1205	0.2725	0.7920	0.0417	0.1708
	80%	$\alpha$	0.5174	0.1751	0.1879	0.5085	0.0495	0.1710	0.5407	0.0353	0.1436
		$\beta$	0.9403	0.1307	0.3130	1.0509	0.0741	0.1741	0.9142	0.0292	0.1484
		$\gamma$	1.3767	0.2259	0.3968	1.0379	0.1003	0.2597	1.0862	0.0277	0.1462
	100%	$\alpha$	0.5174	0.1593	0.1704	0.4704	0.0458	0.1581	0.5002	0.0326	0.1328
		$\beta$	0.9403	0.1169	0.2896	0.9721	0.0685	0.1561	0.8457	0.0271	0.1354
		$\gamma$	1.3767	0.2050	0.3567	0.9600	0.0918	0.2402	1.0047	0.0226	0.1305
200	40%	$\alpha$	0.5254	0.1451	0.1778	0.4864	0.0454	0.1390	0.4949	0.0229	0.1241
		$\beta$	1.2951	0.1249	0.2758	1.0504	0.0617	0.1646	0.8633	0.0185	0.1299
		$\gamma$	1.0568	0.1973	0.3713	0.9843	0.0905	0.1636	0.9404	0.0201	0.1256
	80%	$\alpha$	0.5208	0.1351	0.1668	0.5389	0.0397	0.1217	0.4800	0.0214	0.1099
		$\beta$	0.9369	0.1187	0.2494	1.0374	0.0565	0.1319	1.0322	0.0139	0.1182
		$\gamma$	1.3792	0.1822	0.3581	1.0428	0.0775	0.1405	1.1908	0.0179	0.1148
	100%	$\alpha$	0.4965	0.1269	0.1543	0.4984	0.0368	0.1125	0.4440	0.0192	0.1016
		$\beta$	0.8965	0.1076	0.2307	0.9562	0.0522	0.1172	0.9548	0.0128	0.1120
		$\gamma$	1.1125	0.1569	0.3237	0.9646	0.0716	0.1259	1.1015	0.0165	0.1082

and

Table 3. The point results of $\alpha$, $\beta$, and $\gamma$ from Set-2.

n	FP%	Par.	MLE			Bayes
Prior →						A			B
30	40%	$\alpha$	1.1737	1.0966	0.9495	0.9770	0.6260	0.7776	1.0820	0.5066	0.6748
		$\beta$	2.1669	1.4447	1.2663	1.8700	0.7929	0.7170	1.3988	0.4136	0.5917
		$\gamma$	1.8538	0.9218	1.2516	1.8769	0.8264	0.9441	1.8726	0.4612	0.7424
	80%	$\alpha$	0.9265	0.8285	0.9006	0.9564	0.6096	0.7260	0.9832	0.4650	0.6317
		$\beta$	1.8313	1.2496	1.1377	1.5113	0.6358	0.5897	1.8926	0.3733	0.5402
		$\gamma$	2.1315	0.8859	1.1467	1.9839	0.7323	0.7222	2.1495	0.4286	0.6854
	100%	$\alpha$	0.9087	0.7567	0.8244	0.8966	0.5271	0.6806	0.9176	0.4326	0.5923
		$\beta$	1.6717	1.1671	1.0566	1.4168	0.5796	0.5653	1.5774	0.3450	0.5065
		$\gamma$	1.8317	0.8231	1.0450	1.8599	0.6965	0.6877	1.9870	0.3984	0.6343
60	40%	$\alpha$	0.9654	0.7529	0.8538	0.9551	0.4540	0.5908	0.8934	0.3848	0.5543
		$\beta$	1.6969	1.0875	0.9139	1.6998	0.5755	0.5570	2.1598	0.3604	0.4984
		$\gamma$	1.7585	0.8333	1.0948	1.9571	0.6826	0.6194	1.8390	0.3676	0.5637
	80%	$\alpha$	1.0674	0.4009	0.5696	1.0545	0.3752	0.5317	0.9412	0.3502	0.4902
		$\beta$	2.1337	0.9372	0.8490	1.8577	0.4576	0.5197	1.9307	0.3154	0.4759
		$\gamma$	2.1798	0.7992	0.8250	2.1503	0.6112	0.5445	2.5056	0.3228	0.5062
	100%	$\alpha$	1.1265	0.3758	0.5234	0.9886	0.3518	0.4985	0.8982	0.3308	0.4596
		$\beta$	1.7550	0.8679	0.7796	1.7416	0.4290	0.4872	1.9781	0.3030	0.4462
		$\gamma$	1.9808	0.7349	0.7673	1.8665	0.5730	0.5105	2.1490	0.2964	0.4675
100	40%	$\alpha$	1.1365	0.3637	0.5191	1.1271	0.3403	0.4763	1.0304	0.3155	0.4386
		$\beta$	1.5970	0.8156	0.8346	1.7846	0.4136	0.4969	1.7985	0.2996	0.4597
		$\gamma$	2.1245	0.7064	0.7892	1.9665	0.5843	0.4941	1.9662	0.2271	0.4634
	80%	$\alpha$	0.9541	0.3366	0.4964	1.0945	0.3168	0.4542	1.1486	0.2911	0.4155
		$\beta$	1.8494	0.7897	0.7556	1.3489	0.3485	0.4526	1.4606	0.1999	0.3945
		$\gamma$	2.2307	0.5262	0.5429	2.1190	0.3600	0.4599	2.3979	0.2115	0.4166
	100%	$\alpha$	1.0448	0.3155	0.4654	1.0261	0.2970	0.4358	1.0769	0.2729	0.3895
		$\beta$	1.9734	0.7540	0.7084	1.2646	0.3287	0.4143	1.3693	0.1874	0.3698
		$\gamma$	2.0913	0.4933	0.5134	1.9866	0.3175	0.4231	2.2480	0.1983	0.3905
150	40%	$\alpha$	1.1643	0.2987	0.4520	1.1920	0.2493	0.3800	1.1528	0.1394	0.3367
		$\beta$	1.7970	0.6886	0.7264	1.6941	0.2998	0.4167	2.0060	0.1849	0.3676
		$\gamma$	1.9938	0.4749	0.5035	1.7605	0.3181	0.4152	1.9308	0.1753	0.3690
	80%	$\alpha$	0.8833	0.2784	0.4221	1.1387	0.2424	0.3674	1.1861	0.1243	0.3116
		$\beta$	1.8238	0.3184	0.4862	1.4928	0.2754	0.3709	1.5934	0.1658	0.3313
		$\gamma$	2.1682	0.4368	0.4095	2.5074	0.1682	0.3576	2.1656	0.1541	0.3284
	100%	$\alpha$	0.8281	0.2610	0.3957	1.0675	0.2273	0.3445	1.1119	0.1165	0.2922
		$\beta$	1.7099	0.2985	0.4558	1.3995	0.2582	0.3477	1.6813	0.1595	0.3155
		$\gamma$	2.0327	0.4095	0.3839	2.3507	0.1577	0.3352	2.0655	0.1445	0.3108
200	40%	$\alpha$	1.1909	0.2637	0.3785	1.0137	0.2002	0.3081	0.9611	0.1173	0.2631
		$\beta$	1.5970	0.3024	0.4747	1.5268	0.2465	0.3314	1.8722	0.1473	0.3138
		$\gamma$	2.1364	0.4039	0.3760	1.7792	0.1450	0.3124	1.9750	0.1259	0.2689
	80%	$\alpha$	1.0551	0.2451	0.2793	1.1547	0.1754	0.2645	1.1641	0.1063	0.2259
		$\beta$	1.8140	0.2531	0.3711	1.7848	0.2150	0.3140	1.8269	0.1287	0.2724
		$\gamma$	2.1482	0.3846	0.3571	2.1201	0.1265	0.2719	2.2701	0.1138	0.2158
	100%	$\alpha$	0.9892	0.2298	0.2562	1.0825	0.1645	0.2379	1.0914	0.0987	0.1976
		$\beta$	1.7807	0.2373	0.3408	1.6732	0.2016	0.2944	1.7127	0.1169	0.2355
		$\gamma$	1.9045	0.3605	0.3280	1.9876	0.1186	0.2549	2.1283	0.0986	0.1961

, the APEs, MSEs, and MABs of $\alpha$, $\beta$, and $\gamma$ can be found in the first, second, and third columns, respectively. Additionally, in

Table 4. The interval results of $\alpha$, $\beta$, and $\gamma$ from Set-1.

n	FP%	Par.	95% ACI		95% BCI
Prior →					A		B
30	40%	$\alpha$	2.583	0.845	1.117	0.902	0.750	0.918
		$\beta$	1.691	0.899	1.407	0.904	0.336	0.921
		$\gamma$	1.987	0.876	1.275	0.894	0.616	0.911
	80%	$\alpha$	2.302	0.852	0.909	0.907	0.676	0.922
		$\beta$	1.454	0.902	1.313	0.907	0.310	0.922
		$\gamma$	1.695	0.882	1.029	0.900	0.577	0.913
	100%	$\alpha$	2.190	0.858	0.879	0.909	0.644	0.926
		$\beta$	1.274	0.908	1.274	0.908	0.288	0.927
		$\gamma$	1.565	0.900	0.979	0.901	0.524	0.915
60	40%	$\alpha$	2.237	0.857	0.897	0.909	0.637	0.924
		$\beta$	1.339	0.905	1.237	0.910	0.283	0.924
		$\gamma$	1.412	0.885	0.981	0.902	0.551	0.914
	80%	$\alpha$	2.200	0.860	0.814	0.912	0.536	0.928
		$\beta$	1.272	0.908	0.893	0.914	0.256	0.928
		$\gamma$	1.297	0.889	0.977	0.904	0.530	0.915
	100%	$\alpha$	1.960	0.873	0.787	0.915	0.516	0.930
		$\beta$	1.140	0.912	0.818	0.916	0.226	0.931
		$\gamma$	1.140	0.902	0.876	0.911	0.469	0.920
100	40%	$\alpha$	2.080	0.865	0.768	0.915	0.491	0.931
		$\beta$	1.233	0.910	0.754	0.918	0.219	0.931
		$\gamma$	1.094	0.902	0.927	0.905	0.486	0.917
	80%	$\alpha$	1.946	0.870	0.720	0.917	0.471	0.933
		$\beta$	1.201	0.912	0.686	0.923	0.190	0.932
		$\gamma$	0.933	0.904	0.764	0.910	0.456	0.918
	100%	$\alpha$	1.839	0.874	0.680	0.919	0.445	0.937
		$\beta$	1.135	0.914	0.619	0.926	0.168	0.934
		$\gamma$	0.882	0.909	0.702	0.913	0.431	0.920
150	40%	$\alpha$	1.688	0.878	0.685	0.919	0.440	0.936
		$\beta$	1.150	0.914	0.628	0.925	0.168	0.934
		$\gamma$	0.788	0.907	0.591	0.914	0.447	0.918
	80%	$\alpha$	1.467	0.883	0.609	0.922	0.417	0.937
		$\beta$	1.068	0.915	0.585	0.929	0.147	0.935
		$\gamma$	0.610	0.909	0.558	0.915	0.427	0.919
	100%	$\alpha$	1.372	0.887	0.569	0.925	0.390	0.939
		$\beta$	0.930	0.918	0.547	0.931	0.124	0.938
		$\gamma$	0.527	0.912	0.522	0.918	0.359	0.922
200	40%	$\alpha$	1.275	0.887	0.525	0.925	0.371	0.940
		$\beta$	1.003	0.916	0.548	0.932	0.122	0.936
		$\gamma$	0.583	0.912	0.517	0.916	0.412	0.919
	80%	$\alpha$	1.064	0.893	0.483	0.928	0.318	0.942
		$\beta$	0.904	0.919	0.518	0.934	0.101	0.938
		$\gamma$	0.543	0.913	0.467	0.919	0.387	0.920
	100%	$\alpha$	0.995	0.896	0.452	0.930	0.297	0.943
		$\beta$	0.845	0.923	0.484	0.937	0.094	0.940
		$\gamma$	0.479	0.915	0.436	0.922	0.362	0.921

and

Table 5. The interval results of $\alpha$, $\beta$, and $\gamma$ from Set-2.

n	FP%	Par.	95% ACI		95% BCI
Prior →					A		B
30	40%	$\alpha$	3.315	0.832	1.775	0.893	1.167	0.912
		$\beta$	2.382	0.883	1.348	0.902	0.711	0.916
		$\gamma$	2.221	0.867	1.487	0.886	0.969	0.893
	80%	$\alpha$	3.157	0.837	1.473	0.897	1.018	0.915
		$\beta$	2.154	0.886	1.279	0.905	0.661	0.919
		$\gamma$	1.882	0.874	1.433	0.887	0.930	0.895
	100%	$\alpha$	2.876	0.842	1.367	0.901	0.982	0.917
		$\beta$	1.994	0.889	1.212	0.907	0.581	0.921
		$\gamma$	1.764	0.876	1.317	0.889	0.894	0.893
60	40%	$\alpha$	2.967	0.841	1.341	0.899	0.850	0.918
		$\beta$	1.818	0.890	1.149	0.908	0.587	0.921
		$\gamma$	1.643	0.877	1.388	0.890	0.814	0.900
	80%	$\alpha$	2.559	0.846	1.190	0.903	0.828	0.920
		$\beta$	1.756	0.892	1.082	0.910	0.527	0.922
		$\gamma$	1.533	0.880	1.278	0.892	0.790	0.902
	100%	$\alpha$	2.230	0.851	0.895	0.892	0.739	0.923
		$\beta$	1.365	0.899	0.930	0.893	0.457	0.925
		$\gamma$	1.297	0.901	0.997	0.888	0.653	0.907
100	40%	$\alpha$	2.233	0.852	0.934	0.906	0.760	0.923
		$\beta$	1.642	0.895	0.918	0.912	0.467	0.924
		$\gamma$	1.496	0.882	1.204	0.893	0.762	0.906
	80%	$\alpha$	1.993	0.857	0.884	0.909	0.715	0.926
		$\beta$	1.548	0.897	0.791	0.915	0.425	0.925
		$\gamma$	1.233	0.885	1.022	0.896	0.735	0.907
	100%	$\alpha$	1.883	0.860	0.815	0.912	0.676	0.930
		$\beta$	1.463	0.901	0.708	0.917	0.401	0.928
		$\gamma$	1.165	0.888	0.927	0.901	0.664	0.910
150	40%	$\alpha$	1.781	0.863	0.850	0.911	0.690	0.928
		$\beta$	1.493	0.900	0.769	0.917	0.386	0.928
		$\gamma$	1.006	0.888	0.859	0.901	0.691	0.909
	80%	$\alpha$	1.559	0.868	0.812	0.913	0.628	0.930
		$\beta$	1.451	0.901	0.732	0.918	0.344	0.929
		$\gamma$	0.986	0.890	0.814	0.903	0.667	0.910
	100%	$\alpha$	1.457	0.872	0.759	0.916	0.587	0.933
		$\beta$	1.246	0.904	0.685	0.920	0.322	0.931
		$\gamma$	0.922	0.893	0.761	0.905	0.624	0.913
200	40%	$\alpha$	1.498	0.871	0.796	0.915	0.590	0.932
		$\beta$	1.390	0.903	0.672	0.920	0.317	0.931
		$\gamma$	0.925	0.892	0.780	0.907	0.631	0.912
	80%	$\alpha$	1.262	0.875	0.755	0.917	0.495	0.935
		$\beta$	1.323	0.904	0.620	0.921	0.298	0.932
		$\gamma$	0.887	0.895	0.674	0.910	0.604	0.914
	100%	$\alpha$	1.180	0.877	0.706	0.919	0.463	0.937
		$\beta$	1.237	0.906	0.579	0.923	0.278	0.935
		$\gamma$	0.829	0.898	0.622	0.912	0.565	0.916

, the AILs and CPs of $\alpha$, $\beta$, and $\gamma$ can be found in the first and second columns, respectively.

From the facts presented in Table 2 and Table 5 and the plots in Figure 2, the statistical behavior of the proposed estimators for parameters $\alpha$, $\beta$, and $\gamma$ under various experimental conditions is summarized as follows:

• As the sample size n increases, the performance of all the estimators improves significantly. Specifically, lower MSE, MAB, and AIL values are observed, along with higher CP values. These results demonstrate the asymptotic consistency of the proposed estimation methods, reinforcing their reliability for larger datasets.
• Increasing r (or, equivalently, increasing FP%) enhances the precision of the calculated estimators. This improvement is reflected in reduced MSE, MAB, and AIL values, while CP values exhibit an increasing trend, indicating stronger inferential accuracy.
• As we anticipate, when FP%$\to 100\%$, the precision of all simulation results of $\alpha$, $\beta$, or $\gamma$ behaves better under a complete sampling situation than others.
• When comparing point estimates of $\alpha$, $\beta$, and $\gamma$, the Bayesian estimation approach consistently outperforms the likelihood approach in terms of lower simulated MSE and MAB values. This suggests that the Bayesian framework provides more stable and efficient estimators in finite samples.
• Regarding interval estimation, the BCI method demonstrates superior performance over the ACI method. Specifically, BCI-derived intervals exhibit shorter AIL values while maintaining higher CP values, emphasizing their greater informativeness and accuracy.
• The choice regarding the prior distribution significantly influences the Bayesian estimation results. For all three parameters, $\alpha$, $\beta$, and $\gamma$, estimates obtained using Prior-B outperform those derived from Prior-A. This can be attributed to the smaller variance of Prior-B, which leads to more precise and concentrated posterior distributions.
• The overall estimation accuracy varies depending on the data structure. The results from Set-1 yield more precise estimates of $\alpha$, $\beta$, and $\gamma$ compared to Set-2. Moreover, as the values of these parameters increase, a deterioration in estimation precision is observed, characterized by higher MSE, MAB, and AIL values and a corresponding decline in CP values.
• Finally, for analyzing discrete Hjorth data in the presence of censored observations, the Bayesian framework is strongly recommended due to its robust inferential properties, particularly in handling incomplete data efficiently.

Using the simulated values of MSEs and AILs (for example) corresponding to DH parameters $\alpha$, $\beta$, and $\gamma$, Figure 2 provides clearer insights into the simulation results and confirms all the data presented in Table 2, Table 3, Table 4 and Table 5.

7. Real-World Data Analysis

This section analyzes two applications using two separate actual datasets to (i) assess the offered model’s adaptation and effectiveness to real problems; (ii) show how the inferential outcomes can be applied to a real-world scenario; and (iii) assess if the suggested model is more appropriate than eight other discrete models in the literature. Now, we consider the following applications:

• The first application (say, App [A]) examines the failure times of eighteen electronic devices. These data were first introduced by Wang [19]; see

  Table 6. Failure time points in the two applications.

  Application	Time
  [A]	0.5	1.1	2.1	3.1	4.6	7.5	9.8	12.2	14.5	16.5	19.6
  	22.4	24.5	29.3	32.1	33.0	35.0	42.0
  [B]	1.788	2.892	3.300	4.152	4.212	4.560	4.880	5.184	5.196	5.412	5.556
  	6.780	6.864	6.864	6.888	8.412	9.312	9.864	10.512	12.792	12.804	17.340

  .
• The second application (say, App [B]) analyzes how many millions of revolutions each of 22 ball bearings will make before they fail; see Caroni [20]. For computational purposes, we divide each revolution by ten and list the newly transformed ball-bearing data in Table 6.

Briefly, in

Table 7. Failure time statistics in the two applications.

Application	Minimum	Maximum	Quartiles			Mode	Mean	St.D.	Skewness	Kurtosis
			1st	2nd	3rd
[A]	0.500	42.00	5.325	15.50	28.10	0.500	17.21	13.15	0.314	1.850
[B]	1.788	17.34	4.640	6.168	9.087	6.864	7.071	3.762	1.066	3.742

, several statistics for applications [A] and [B], namely minimum, maximum, quartiles (1st, 2nd, and 3rd), mean, mode, standard deviation (St.D.), skewness, and kurtosis, are evaluated.

Aside from three well-known discrete models, namely negative binomial (NB), geometric (Geom), and Poisson (Pois), based on the above two applications, the DH model will be compared with eight other comparable models from the literature to demonstrate the reliability and advantage of the newly developed model. The competitive distributions include the following:

• Discrete exponentiated-Chen (DEC$(\alpha ,\beta ,\gamma )$) by Alotaibi et al. [21];
• Exponentiated discrete Weibull (EDW$(\alpha ,\beta ,\gamma )$) by Nekoukhou and Bidram [22];
• Discrete modified Weibull (DMW$(\alpha ,\beta ,\gamma )$) by Almalki and Nadarajah [23];
• Discrete Burr Type-XII (DB$(\alpha ,\beta )$) by Krishna and Pundir [24];
• Discrete Perks (DP$(\alpha ,\beta )$) by Tyagi et al. [25];
• Discrete generalized-exponential (DGE$(\alpha ,\beta )$) by Nekoukhou et al. [26];
• Discrete gamma (DG$(\alpha ,\beta )$) by Chakraborty and Chakravarty [27];
• Discrete Burr–Hatke (DBH$\left(\alpha \right)$) by El-Morshedy et al. [28].

To judge the best model, several metrics are utilized, namely negative log-likelihood (N-LogL), Akaike (Ak.), consistent-Akaike (C-Ak.), Bayesian (Bayes), Hannan–Quinn (H–Q), and Kolmogorov–Smirnov (K–S) statistic (with its p-value). From both applications [A] and [B], the fitted values corresponding to these criteria as well as the MLEs (along with their standard errors (Std.Ers)) of $\alpha$, $\beta$, and $\gamma$ are obtained; see

Table 8. Goodness of fit for the DH and its competitive distributions from the two applications.

App.	Model	MLE (Std.Er)			N-LogL	Ak.	Bayes	C-Ak.	H–Q	K–S (p-Value)
		α	β	γ
[A]	DH	0.0028(0.0009)	0.3521(0.6590)	0.0683(0.0648)	68.138	142.275	143.990	144.947	142.644	0.106(0.975)
	DEC	0.0241(0.0486)	0.4237(0.1062)	0.8808(0.7033)	68.582	143.164	144.878	145.835	143.532	0.113(0.956)
	EDW	0.9968(0.0019)	1.7621(0.1801)	0.5609(0.1580)	68.648	143.296	145.010	145.967	143.664	0.113(0.957)
	DMW	1.0186(0.0497)	5.0673(11.339)	1.0427(0.0165)	68.407	142.814	144.529	145.485	143.183	0.109(0.973)
	DW	1.2421(0.2470)	18.880(3.7583)	-	69.186	142.372	144.172	145.153	142.677	0.122(0.924)
	DB	13.730(32.516)	0.0298(0.0708)	-	78.846	161.693	162.493	163.473	161.938	0.284(0.089)
	DP	0.0869(0.0286)	0.5727(0.6924)	-	69.025	142.492	144.085	144.983	142.695	0.109(0.971)
	DG	0.0723(0.0265)	1.2811(0.3875)	-	69.419	142.838	143.999	144.969	143.083	0.109(0.966)
	NB	1.2227(0.4391)	17.113(3.7763)	-	69.478	142.955	144.736	145.755	143.078	0.136(0.849)
	DBH	0.9979(0.0111)	-	-	92.832	187.664	187.914	188.555	187.787	0.647(0.002)
	Geom	0.0552(0.0126)	-	-	69.631	143.262	144.152	144.951	143.385	0.141(0.819)
	Pois	17.111(0.9750)	-	-	135.56	273.129	274.015	273.380	273.250	0.342(0.022)
[B]	DH	0.0609(0.0167)	0.0454(0.0068)	0.1119(0.0407)	55.819	117.639	118.972	120.912	118.410	0.147(0.728)
	DEC	0.8106(0.7237)	0.2929(0.1010)	32.879(60.685)	56.971	119.941	121.275	123.214	120.712	0.171(0.541)
	EDW	0.5577(0.6397)	0.8308(0.6029)	11.825(27.309)	56.982	119.964	121.297	123.237	120.735	0.149(0.726)
	DMW	0.9917(0.0086)	7.9939(9.0815)	1.1866(0.0416)	62.367	130.734	132.067	134.007	131.505	0.237(0.167)
	DW	2.1995(0.3484)	8.5805(0.8846)	-	58.258	120.516	121.148	122.698	121.030	0.222(0.231)
	DB	8.8347(32.276)	0.0594(0.2172)	-	78.218	160.436	161.067	162.618	160.950	0.390(0.003)
	DP	0.4233(0.1557)	0.0627(0.0906)	-	58.861	121.723	122.354	123.905	122.237	0.180(0.473)
	DG	0.6151(0.1913)	4.6565(1.3750)	-	57.253	118.506	119.137	120.988	119.020	0.199(0.345)
	NB	8.8040(5.8286)	7.0904(0.7627)	-	57.741	119.483	121.665	120.914	119.997	0.150(0.722)
	DBH	0.9868(0.0261)	-	-	90.521	183.042	183.242	184.133	183.299	0.760(0.004)
	Geom	0.1236(0.0247)	-	-	66.577	135.154	136.245	135.354	135.411	0.366(0.006)
	Pois	7.0909(0.5677)	-	-	60.298	122.596	123.687	122.796	122.853	0.165(0.585)

. It is clear from Table 8 for both analyzed applications that the fitted DH distribution produces the lowest values for all fitted metrics except the highest p-value among all other fitted competing models. As a result, the DH probability model is better than others.

Several goodness-of-fit visualization tools are employed, including (i) histogram of data with fitted probability lines, (ii) fitted reliability lines, (iii) probability–probability (PP), and (iv) total-time-test (TTT) plots for the DH and its competing distributions, as displayed in Figure 3. Clearly, in Figure 3, we only compared the DH with models with p-values greater than 5%. It exhibits that the newly proposed DH distribution provides the best fit for both applications in terms of estimated PMF, RF, and PP curves.

The estimated TTT lines shown in Figure 3 indicate that the given datasets employed in applications [A] and [B] provide bathtub and increasing failure rates, respectively. These failure rate shapes support the same shapes of the DH model represented in Figure 1. Additionally, all facts shown in Figure 3 support the same numerical findings reported in Table 8. We can therefore infer from the results presented in Figure 3 (or Table 8) that, compared to other traditional (or modern) statistical models, the new DH distribution provides a significantly superior fit.

Type-II censoring is a powerful tool in engineering reliability studies. It enables cost-effective, efficient, and statistically robust failure data analysis, ensuring better product design, maintenance, and quality control. As a result, it provides reliable failure estimates faster and cheaper than complete sampling.

To assess the obtained frequentist and Bayes estimators of $\alpha$, $\beta$, and $\gamma$, three artificial Type-II censored samples are obtained for each dataset based on varying the values of r; refer to

Table 9. Artificial Type-II censored samples from the two applications.

App.	r	Data
[A]	8	0.5	1.1	2.1	3.1	4.6	7.5	9.8	12.2
	12	0.5	1.1	2.1	3.1	4.6	7.5	9.8	12.2	14.5	16.5	19.6	22.4
	18	0.5	1.1	2.1	3.1	4.6	7.5	9.8	12.2	14.5	16.5	19.6	22.4
		24.5	29.3	32.1	33	35	42
[B]	10	1.788	2.892	3.300	4.152	4.212	4.560	4.880	5.184	5.196	5.412
	15	1.788	2.892	3.300	4.152	4.212	4.560	4.880	5.184	5.196	5.412	5.556
		6.780	6.864	6.864	6.888
	22	1.788	2.892	3.300	4.152	4.212	4.560	4.880	5.184	5.196	5.412	5.556
		6.780	6.864	6.864	6.888	8.412	9.312	9.864	10.512	12.792	12.804	17.340

. In every sample, the MLEs (along with the 95% ACIs) as well as the Bayes estimation (along with the 95% BCIs) are obtained. All Bayes evaluations are conducted by running the MCMC sampler 50,000 times and discarding the first 10,000 iterations as burn-in. Due to the lack of prior information about the DH’s parameters, we set $a_i=b_i=0.001$ for $i=1,2,3.$ In

Table 10. Estimates of $\alpha$, $\beta$, and $\gamma$ from the two applications.

Data	r	Par.	MLE		Bayes		95% ACI			95% BCI
			Est.	Std.Er	Est.	Std.Er	Low.	Upp.	IL	Low.	Upp.	IL
[A]	8	$\alpha$	0.00135	0.00475	0.00130	0.00026	0.00000	0.01065	0.01065	0.00081	0.00180	0.00099
		$\beta$	0.12382	0.47007	0.12382	0.00025	0.00000	1.04515	1.04515	0.12333	0.12431	0.00098
		$\gamma$	0.05994	0.05176	0.05993	0.00025	0.00000	0.16138	0.16138	0.05944	0.06043	0.00099
	12	$\alpha$	0.00187	0.00178	0.00184	0.00025	0.00000	0.00536	0.00536	0.00136	0.00233	0.00097
		$\beta$	0.18077	0.44236	0.18077	0.00025	0.00000	1.04778	1.04778	0.18028	0.18127	0.00099
		$\gamma$	0.06241	0.05428	0.06241	0.00025	0.00000	0.16880	0.16880	0.06192	0.06291	0.00099
	18	$\alpha$	0.00274	0.00103	0.00272	0.00024	0.00072	0.00475	0.00403	0.00225	0.00320	0.00095
		$\beta$	0.31689	0.57888	0.31689	0.00025	0.00000	1.45147	1.45147	0.31641	0.31739	0.00098
		$\gamma$	0.06686	0.06152	0.06686	0.00025	0.00000	0.18744	0.18744	0.06637	0.06734	0.00098
[B]	10	$\alpha$	0.50352	0.14345	0.50317	0.00106	0.22236	0.78467	0.56231	0.50118	0.50514	0.00396
		$\beta$	0.14670	0.00005	0.14669	0.00101	0.14660	0.14679	0.00019	0.14474	0.14868	0.00394
		$\gamma$	0.12099	0.04284	0.12088	0.00101	0.03702	0.20496	0.16794	0.11891	0.12284	0.00394
	15	$\alpha$	0.28233	0.10461	0.28189	0.00109	0.07729	0.48737	0.41008	0.27993	0.28388	0.00395
		$\beta$	0.09847	0.00389	0.09855	0.00101	0.09085	0.10608	0.01524	0.09659	0.10053	0.00394
		$\gamma$	0.49199	0.18261	0.49189	0.00101	0.13409	0.84990	0.71582	0.48991	0.49387	0.00397
	22	$\alpha$	0.06090	0.01667	0.06038	0.00114	0.02823	0.09357	0.06534	0.05833	0.06235	0.00402
		$\beta$	0.04554	0.00729	0.04554	0.00101	0.03124	0.05983	0.02859	0.04358	0.04749	0.00391
		$\gamma$	0.11187	0.03987	0.11175	0.00102	0.03373	0.19002	0.15629	0.10979	0.11373	0.00394

, the maximum likelihood and Bayes estimates (with their Std.Ers) as well as 95% ACI/BCI bounds (with their interval widths (ILs)) of $\alpha$, $\beta$, and $\gamma$ are listed. It indicates that the point and interval estimation results developed from the Bayes method outperformed those developed from the likelihood method. When r is increased, all estimates of $\alpha$, $\beta$, or $\gamma$ behave well because their Std.Er and IL values decreased.

Based on the last 40,000 outputs for $\alpha$, $\beta$, and $\gamma$ obtained from the two applications when $r=8$ and 10 (for example), respectively, Figure 4 shows the trace and density plots along with their sample averages (in solid lines) and 95% two-bound BCIs (in dashed lines). It shows that the MCMC sampler converges highly effectively and that the calculated posteriors for all DH parameters are reasonably symmetric.

Additionally, to explore the convergence and blending of Markovian chains, the acceptance rates of the M–H proposals are calculated, such as

• For App. [A]: The acceptance rates from $r(=8,10,12)$ are 90.12, 93.40, and 95.38%;
• For App. [B]: The acceptance rates from $r(=10,15,22)$ are 89.63, 92.78, and 97.47%.

As a consequence, the estimated acceptance rates in App. [A] (or App. [B]) confirm the same facts revealed in Figure 4; that is, the percentage of iterations in which proposals were approved is significantly higher.

The numerical findings from the two applications employed, among other things, demonstrate that the suggested model outperformed the other models. As a result, the methods for applying the newly discrete Hjorth probability model to electrical devices or ball-bearing datasets provide an adequate explanation.

8. Conclusions

Discrete distributions are well suited for representing data confined to a finite or countably infinite set of data. The straightforward structure, availability of closed-form expressions, and capacity to capture real-world scenarios make them an ideal option. Additionally, they are computationally efficient and effectively handle categorical data modeling. This paper introduced a discrete version of the Hjorth distribution using the survival discretization method, addressing the gap in the literature regarding the use of censored data for analyzing discrete models. The proposed distribution preserved the essential statistical features of the original Hjorth distribution, such as its median, percentiles, and general structure, while providing a flexible framework for modeling real-world discrete data. The statistical properties of the DH model, including the quantile function and order statistics, were thoroughly derived. Maximum likelihood inference and Bayesian methods were employed for parameter estimation, and their performance was evaluated through simulation studies. Type-II censored samples were utilized to demonstrate the practical applicability of the model, with variance–covariance matrices enabling the construction of asymptotic confidence intervals for the parameters. The effectiveness of the DH model was further validated using two real-world datasets: one concerning failure times of electronic devices and another focusing on ball-bearing failures. The goodness-of-fit measures highlighted the superior performance of the proposed model compared to the existing discrete alternatives. Overall, the DH model proved to be a versatile and robust tool for statistical modeling, particularly in reliability and survival analysis. Its ability to adapt to different data patterns, such as decreasing, increasing, and bathtub-shaped hazard rates, underscores its potential for broader applications in various fields, including engineering, medicine, and beyond.

While the proposed DH model demonstrated strong flexibility and effectiveness in capturing various hazard rate shapes, certain limitations can be acknowledged. The model’s performance may be less reliable with very small sample sizes due to increased variability in parameter estimates, necessitating techniques such as Bayesian estimation with informative priors. Additionally, while the DH model demonstrated flexibility across various hazard rate trends, its fit may be affected when dealing with datasets exhibiting extreme variance or strong skewness.

Future research could explore refined model selection criteria to guide practitioners in choosing between the DH model and alternative approaches based on dataset characteristics. Future work may explore extending this framework to other distributions, different censoring schemes, and refining the computational methods for enhanced efficiency.