Search Results (113)

The Unit Inverse Maxwell–Boltzmann (UIMB) distribution is introduced as a novel single-parameter model for data constrained within the unit interval ($0,1$), derived through an exponential transformation of the Inverse Maxwell–Boltzmann distribution. Designed to address the limitations of traditional unit-interval distributions, the UIMB model exhibits flexible density shapes and hazard rate behaviors, including right-skewed, left-skewed, unimodal, and bathtub-shaped patterns, making it suitable for applications in reliability engineering, environmental science, and health studies. This study derives the statistical properties of the UIMB distribution, including moments, quantiles, survival, and hazard functions, as well as stochastic ordering, entropy measures, and the moment-generating function, and evaluates its performance through simulation studies and real-data applications. Various estimation methods, including maximum likelihood, Anderson–Darling, maximum product spacing, least-squares, and Cramér–von Mises, are assessed, with maximum likelihood demonstrating superior accuracy. Simulation studies confirm the model’s robustness under normal and outlier-contaminated scenarios, with MLE showing resilience across varying skewness levels. Applications to manufacturing and environmental datasets reveal the UIMB distribution’s exceptional fit compared to competing models, as evidenced by lower information criteria and goodness-of-fit statistics. The UIMB distribution’s computational efficiency and adaptability position it as a robust tool for modeling complex unit-interval data, with potential for further extensions in diverse domains. Full article

The accelerated lifetime test is a widely used and effective approach in reliability analysis because of its shorter testing duration. In this study, product lifetimes are assumed to follow the Gompertz distribution. This article primarily focuses on performance comparisons between the linear model and the inverse power-law model, both of which are utilized to characterize the relationship between the shape parameter and stress levels. To test model robustness, we also generate data from the Sine-Modified Power Gompertz distribution, a more flexible alternative. We conduct Monte Carlo simulations using four estimation methods: the maximum likelihood method, the least squares method, the maximum product of spacing method, and the Cramér-von Mises method, for small, medium, and large sample sizes. The comparison of mean squared error serves as a critical indicator for evaluating the performance of different methods and models. Additionally, the shape parameter and reliability function are obtained based on the estimation results. Finally, a real dataset is analyzed to demonstrate the most suitable accelerated life model, and the Akaike Information Criterion is used to further assess model fit. Furthermore, we employ leave-one-out cross-validation (LOOCV) to prove this model’s generalizability. Full article

This study focuses on analyzing progressive Type-II right censoring competing risks datasets. The latent causes of failures are assumed to follow independent generalized linear exponential distributions. The maximum likelihood and maximum product of spacing methods are employed to estimate the unknown parameters and survival indices. Furthermore, approximate confidence intervals are derived using the asymptotic normality of the maximum likelihood and the maximum product of spacing estimators. Additionally, bootstrap methods are employed to construct confidence intervals. A comprehensive simulation study is carried out to evaluate the effectiveness of these estimation approaches. Finally, real-world datasets are analyzed to illustrate the practical applicability of the proposed model. Full article

We present Bayesian Binary Search (BBS), a novel framework that bridges statistical learning theory/probabilistic machine learning and binary search. BBS utilizes probabilistic methods to learn the underlying probability density of the search space. This learned distribution then informs a modified bisection strategy, where the split point is determined by probability density rather than the conventional midpoint. This learning process for search space density estimation can be achieved through various supervised probabilistic machine learning techniques (e.g., Gaussian Process Regression, Bayesian Neural Networks, and Quantile Regression) or unsupervised statistical learning algorithms (e.g., Gaussian Mixture Models, Kernel Density Estimation (KDE), and Maximum Likelihood Estimation (MLE)). Our results demonstrate substantial efficiency improvements using BBS on both synthetic data with diverse distributions and in a real-world scenario involving Bitcoin Lightning Network channel balance probing (3–6% efficiency gain), where BBS is currently in production. Full article

This paper introduces a novel and flexible class of continuous probability distributions, termed the Alpha Power Survival Ratio-X (APSR-X) family. Unlike many existing transformation-based families, the APSR-X class integrates an alpha power transformation with a survival ratio structure, offering a new mechanism for enhancing shape flexibility while maintaining mathematical tractability. This construction enables fine control over both the tail behavior and the symmetry properties, distinguishing it from traditional alpha power or survival-based extensions. We focus on a key member of this family, the two-parameter Alpha Power Survival Ratio Exponential (APSR-Exp) distribution, deriving essential mathematical properties including moments, quantile functions and hazard rate structures. We estimate the model parameters using eight frequentist methods: the maximum likelihood (MLE), maximum product of spacings (MPSE), least squares (LSE), weighted least squares (WLSE), Anderson–Darling (ADE), right-tailed Anderson–Darling (RADE), Cramér–von Mises (CVME) and percentile (PCE) estimation. Through comprehensive Monte Carlo simulations, we evaluate the estimator performance using bias, mean squared error and mean relative error metrics. The proposed APSR-X framework uniquely enables preservation or controlled modification of the symmetry in probability density and hazard rate functions via its shape parameter. This capability is particularly valuable in reliability and survival analyses, where symmetric patterns represent balanced risk profiles while asymmetric shapes capture skewed failure behaviors. We demonstrate the practical utility of the APSR-Exp model through three real-world applications: economic (tax revenue durations), engineering (mechanical repair times) and medical (infection durations) datasets. In all cases, the proposed model achieves a superior fit over that of the conventional alternatives, supported by goodness-of-fit statistics and visual diagnostics. These findings establish the APSR-X family as a unique, symmetry-aware modeling framework for complex lifetime data. Full article

This study introduces a new and flexible class of probability distributions known as the novel alpha-power X (NAP-X) family. A key development within this framework is the novel alpha-power half-logistic (NAP-HL) distribution, which extends the classical half-logistic model through an alpha-power transformation, allowing for greater adaptability to various data shapes. The paper explores several theoretical aspects of the proposed model, including its moments, quantile function and hazard rate. To assess the effectiveness of parameter estimation, a detailed simulation study is conducted using seven estimation techniques: Maximum likelihood estimation (MLE), Cramér–von Mises estimation (CVME), maximum product of spacings estimation (MPSE), least squares estimation (LSE), weighted least squares estimation (WLSE), Anderson–Darling estimation (ADE) and a right-tailed version of Anderson–Darling estimation (RTADE). The results offer comparative insights into the performance of each method across different sample sizes. The practical value of the NAP-HL distribution is demonstrated using two real datasets from the metrology and engineering domains. In both cases, the proposed model provides a better fit than the traditional half-logistic and related distributions, as shown by lower values of standard model selection criteria. Graphical tools such as fitted density curves, Q–Q and P–P plots, survival functions and box plots further support the suitability of the model for real-world data analysis. Full article

Probability models are instrumental in a wide range of applications by being able to accurately model real-world data. Over time, numerous probability models have been developed and applied in practical scenarios. This study introduces the AAP-X family of distributions—a novel, flexible framework for continuous data analysis named after authors Aadil Ajaz and Parvaiz. The proposed family effectively accommodates both symmetric and asymmetric characteristics through its shape-controlling parameter, an essential feature for capturing diverse data patterns. A specific subclass of this family, termed the “AAP Exponential” (AAPEx) model is designed to address the inflexibility of classical exponential distributions by accommodating versatile hazard rate patterns, including increasing, decreasing and bathtub-shaped patterns. Several fundamental mathematical characteristics of the introduced family are derived. The model parameters are estimated using six frequentist estimation approaches, including maximum likelihood, Cramer–von Mises, maximum product of spacing, ordinary least squares, weighted least squares and Anderson–Darling estimation. Monte Carlo simulations demonstrate the finite-sample performance of these estimators, revealing that maximum likelihood estimation and maximum product of spacing estimation exhibit superior accuracy, with bias and mean squared error decreasing systematically as the sample sizes increases. The practical utility and symmetric–asymmetric adaptability of the AAPEx model are validated through five real-world applications, with special emphasis on cancer survival times, COVID-19 mortality rates and reliability data. The findings indicate that the AAPEx model outperforms established competitors based on goodness-of-fit metrics such as the Akaike Information Criteria (AIC), Schwartz Information Criteria (SIC), Akaike Information Criteria Corrected (AICC), Hannan–Quinn Information Criteria (HQIC), Anderson–Darling (A*) test statistic, Cramer–von Mises (W*) test statistic and the Kolmogorov–Smirnov (KS) test statistic and its associated p-value. These results highlight the relevance of symmetry in real-life data modeling and establish the AAPEx family as a powerful tool for analyzing complex data structures in public health, engineering and epidemiology. Full article

This paper investigates various estimation methods for the parameters of the unit Lindley distribution (U-LD) under both ranked set sampling (RSS) and simple random sampling (SRS) designs. The distribution parameters are estimated using the maximum likelihood estimation, ordinary least squares, weighted least squares, maximum product of spacings, minimum spacing absolute distance, minimum spacing absolute log-distance, minimum spacing square distance, minimum spacing square log-distance, linear-exponential, Anderson–Darling (AD), right-tail AD, left-tail AD, left-tail second-order, Cramér–von Mises, and Kolmogorov–Smirnov. A comprehensive simulation study is conducted to assess the performance of these estimators, ensuring an equal number of measuring units across both designs. Additionally, two real datasets of items failure time and COVID-19 are analyzed to illustrate the practical applicability of the proposed estimation methods. The findings reveal that RSS-based estimators consistently outperform their SRS counterparts in terms of mean squared error, bias, and efficiency across all estimation techniques considered. These results highlight the advantages of using RSS in parameter estimation for the U-LD distribution, making it a preferable choice for improved statistical inference. Full article

This study presents a model for estimating forest productivity based on a sample of 2048 permanent field plots covering a wide range of growing sites in Mexico. Our state-space approach assumes that the growth behavior of any stand over time can be estimated on the basis of its current state, defined by the dominant height (H), number of trees per hectare (N), and stand basal area (BA). We used transition functions to estimate the change in states as a function of the current state. We also present transition functions for the change in stand volume (V) and total above-ground biomass (AGB). The first transition function relates dominant height to dominant diameter by using the guide-curve method to estimate site form. The transition function for N consists of two models, one for estimating natural mortality and the other for estimating recruitment. These models were developed in two steps: in the first step, the logistic regression and maximum likelihood approach were used to estimate the probability of the occurrence of mortality or recruitment, and in the second step, the rate of change associated with each event was modeled when mortality or recruitment was assumed to have occurred as a result of the first step. The remaining three transition functions (BA, V, and AGB) were fitted simultaneously to account for possible correlations between errors. The model estimating total above-ground biomass (AGB), which can be considered a state variable that summarizes the performance of the whole model, explained more than 97% of the observed variability, with a root mean square error value of 10.57 Mg/ha. Full article

In this study, we explore the use of C. kluyveri in synthetic biofilms for the production of 1-butyrate and 1-hexanoate, investigating the impact of inoculation temperature during biofilm formation and the presence of yeast extract. Therefore, a novel synthetic biofilm reactor has been designed and constructed. Prior to investigating synthetic biofilms in this reactor, we carried out preliminary batch experiments in anaerobic flasks containing an inoculated agar hydrogel fixed at the bottom and overlaid medium. For the operation of the novel synthetic biofilm reactor, specific volumes of inoculated agar hydrogel were dispensed into a cylindrical mold with a diameter of 102 mm, forming the synthetic biofilm with a height of 4 mm, which was then transferred into the biofilm reaction chamber onto the support grid. The biofilm support grid separates the gas phase (CO₂, N₂) above the synthetic biofilm from the aqueous phase (medium) below. Our results show that C. kluyveri remains metabolically active at biofilm preparation temperatures of up to 45 °C, with extended lag phases observed at 70 °C. The synthetic biofilm demonstrated efficient chain elongation in batch processes, converting ethanol and acetate into 1-butyrate and 1-hexanoate, with final concentrations of 2.7 g L⁻¹ and 10.1 g L⁻¹, respectively, with yeast extract in the circulating liquid medium of the synthetic biofilm reactor setup. The maximum estimated space-time yields for 1-butyrate and 1-hexanoate, referenced to the biofilm volume, were 1.331 g L⁻¹ h⁻¹ and 4.947 g L⁻¹ h⁻¹, respectively. Experiments without yeast extract lead to final concentrations of 2.0 g L⁻¹ 1-butyrate, and 7.3 g L⁻¹ 1-hexanoate and maximum estimated space-time yields, referenced to the biofilm volume, were 0.332 g L⁻¹ h⁻¹ and 1.123 g L⁻¹ h⁻¹, respectively. The use of synthetic biofilms, even without yeast extract, eliminates the need for significant cell growth during chain elongation. However, product concentrations were lower without yeast extract. Full article

This review article is concerned to provide a global context to several works on the fitting of continuous time nonhomogeneous Markov chains with finite state space and also to point out some selected aspects of two techniques previously introduced—estimation and calibration—relevant for applications and used to fit a continuous time Markov chain model to data by the adequate selection of parameters. The denomination estimation suits the procedure better when statistical techniques—e.g., maximum likelihood estimators—are employed, while calibration covers the case where, for instance, some optimisation technique finds a best approximation parameter to ensure good model fitting. For completeness, we provide a short summary of well-known important notions and results formulated for nonhomogeneous Markov chains that, in general, can be transferred to the homogeneous case. Then, as an illustration for the homogeneous case, we present a selected Billingsley’s result on parameter estimation for irreducible chains with finite state space. In the nonhomogeneous case, we quote two recent results, one of the calibration type and the other with more of a statistical flavour. We provide an ample set of bibliographic references so that the reader wanting to pursue her/his studies will be able to do so more easily and productively. Full article

This study introduces a novel trigonometric-based family of distributions for modeling continuous data through a newly proposed framework known as the ASP family, where ‘ASP’ represents the initials of the authors Aadil, Shamshad, and Parvaiz. A specific subclass of this family, termed the “ASP Rayleigh distribution” (ASPRD), is introduced that features two parameters. We conducted a comprehensive statistical analysis of the ASPRD, exploring its key properties and demonstrating its superior adaptability. The model parameters are estimated using four classical estimation methods: maximum likelihood estimation (MLE), least squares estimation (LSE), weighted least squares estimation (WLSE), and maximum product of spaces estimation (MPSE). Extensive simulation studies confirm these estimation techniques’ robustness, showing that biases, mean squared errors, and root mean squared errors consistently decrease as sample sizes increase. To further validate its applicability, we employ ASPRD on three real-world engineering datasets, showcasing its effectiveness in modeling complex data structures. This work not only strengthens the theoretical framework of probability distributions but also provides valuable tools for practical applications, paving the way for future advancements in statistical modeling. Full article

This article introduces a new continuous lifetime distribution within the Gamma family, called the induced Xgamma distribution, and explores its various statistical properties. The proposed distribution’s estimation and prediction are investigated using Bayesian and non-Bayesian approaches under progressively Type-II censored data. The maximum likelihood and maximum product spacing methods are applied for the non-Bayesian approach, and some of their performances are evaluated. In the Bayesian framework, the numerical approximation technique utilizing the Metropolis–Hastings algorithm within the Markov chain Monte Carlo is employed under different loss functions, including the squared error loss, general entropy, and LINEX loss. Interval estimation methods, such as asymptotic confidence intervals, log-normal asymptotic confidence intervals, and highest posterior density intervals, are also developed. A comprehensive numerical study using Monte Carlo simulations is conducted to evaluate the performance of the proposed point and interval estimation methods through progressive Type-II censored data. Furthermore, the applicability and effectiveness of the proposed distribution are demonstrated through three real-world datasets from the fields of medicine and engineering. Full article

Crystallography, a cornerstone of materials science, provides critical insights into material structures through techniques such as X-ray diffraction (XRD). Among the metrics derived from XRD, intensity serves as a key parameter, reflecting the electron density distribution and offering information about atomic arrangements and sample quality. Due to its inherent variability and susceptibility to extreme values, intensity is best modeled using heavy-tailed, location-scale probability distributions. This paper investigates the model parameter estimation problem for three such distributions—log-Cauchy, half-Cauchy, and Cauchy Birnbaum–Saunders—using several methods, including maximum likelihood and the maximum product of spacings estimation methods. Monte Carlo simulations are conducted to assess the performance of these methods across various scenarios. Additionally, two real XRD intensity datasets are analyzed to compare the applicability and effectiveness of the proposed models. The results demonstrate the potential of heavy-tailed distributions for modeling XRD intensity data, providing a robust framework for future research and practical applications in material characterization. Full article

This article defines a new distribution using a novel alpha power-transformed method extension. The model obtained has three parameters and is quite effective in modeling skewed, complex, symmetric, and asymmetric datasets. The new approach has one additional parameter for the model. Certain distributional and mathematical properties are investigated, notably reliability, quartile, moments, skewness, kurtosis, and order statistics, and several approaches of estimation, notably the maximum likelihood, least square, weighted least square, maximum product spacing, Cramer-Von Mises, and Anderson Darling estimators of the model parameters were obtained. A Monte Carlo simulation study was conducted to evaluate the performance of the proposed techniques of estimation of the model parameters. The actuarial measures are computed for our recommended model. At the end of the paper, two insurance applications are illustrated to check the potential and utility of the suggested distribution. Evaluation using four selection criteria indicates that our recommended model is the most appropriate probability model for modeling insurance datasets. Full article