Statistical Modeling for Some Real Applications in Reliability Analysis Using Non-Parametric Hypothesis Testing

Abstract

Probability life distributions usually describe the time to an event or survival time. Therefore, these life distributions play a crucial role in the analysis and projection of maximum life expectancy using the goodness of fit approach for nonparametric hypothesis testing. This study suggests a nonparametric technique to determine whether the data follow an exponential distribution or belong to the mathematical class of the moment generating function for used better than aged (UBAmgf). These tests can be applied to both censored and non-censored data. The upper percentile points of the test statistics are generated, and the suggested test’s asymptotic normality is established. Some well-known alternative asymmetric probability models are used to compute the Pitman asymptotic relative efficiency (PARE) and powers of the proposed test. To demonstrate the paper’s conclusions, some asymmetric real-world datasets are examined.

1. Introduction

The stochastic comparison of probability distributions, symmetry, and asymmetry are significant principles in probability, statistics, and various associated fields, such as economics, survival analysis, and reliability theory. In reliability analysis, both symmetry and asymmetry are crucial. In general, symmetrical data have an identifiable trend and are simpler to predict than asymmetrical data. Asymmetry assists in locating outliers or anomalies that should be taken into account in the predictive model, whereas symmetry helps uncover patterns within the data that may be used in predictive models. Over the past few decades, this technique has been used to test exponentiality. Patient survival times are noted after implementing the advised strategy. In this investigation, the behavior of the observed data was assessed, assuming the used is better than aged in the moment generating function ordered (UBAmgf) attribute or the constant failure rate (exponential scenario). The most well-known classes proposed over the past few decades include increasing failure rate and average (IFR and IFRA), new better than used and expectation (NBU and NBUE), used is better than aged (UBA), and used better than aged in moment generating function (UBAmgf). One might consult Deshpand et al. [1], Klefsjo [2], Ahmed [3,4], Bakr et al. [5], Barlow and Proschan [6], Rolski [7], Abu Youssef et al. [8], Etman et al. [9], EL-Sagheer et al. [10], Abu Youssef et al. [11], Bakr et al. [12], EL-Sagheer et al. [13], Mahmoud et al. [14], Gadallah et al. [15], and Bakr and Al Babtain [16].

The development of a systematic approach for the investigation of any event or activity taking place in the world was necessitated by the fundamental requirements of modern science and technology. Therefore, a technique like this would be required for examining the dependability of technological systems and products. In real life, there are times when the manufacturer’s warranty expires, and the system’s components gradually deteriorate over time, necessitating the replacement of spare parts in order to extend its functionality. In such cases, the process of renewal attempts to enhance the usability of the system without being able to return the system to its previous state.

We found that nonparametric tests of life distribution are not very effective and have a low power of test. As a result, we have developed a statistical nonparametric hypothesis test (goodness of fit) for the class of life distribution considered in this work, taking into account the efficiency and power of the test.

In reliability studies, specific classes of life distributions and their variants have been introduced. These classes of life distributions are used in engineering, social science, biological science, and maintenance. Statisticians and reliability analysts have demonstrated an increasing interest in modeling survival data using life distribution classifications that are based on aging-related characteristics. Concepts of aging explain how a population of units or systems gets better or worse as they get older. In the literature, statistical ordering is used to categorize and characterize several classes of life distributions.

If $X$ and $Y$ are two random variables with survivals $\overline{F}$ and $\overline{G}$, respectively, then we say that X is smaller than Y in the moment generating function order ($X{\le }_{mgf} Y$), if $M_X\left(s\right)\le M_Y\left(s\right), \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} s>0$, where

$$M_X\left(s\right)={\int }_0^{\infty }{e}^{sx}dF(x), and M_Y\left(s\right)={\int }_0^{\infty }{e}^{sy}dF(Y),$$

so that

$${\int }_0^{\infty }{e}^{sx}dF(x)\le {\int }_0^{\infty }{e}^{sy}dG\left(Y\right),s>0.$$

See Deshpande et al. [1], Kaur et al. [17], and Klar and MÄuller [18] for more details.

Definition 1.

The distribution function $F$ is said to be

i. 

increasing failure rate (IFR), if

$$\frac{\overline{F}(x+t)}{\overline{F}(t)}is increasing in t\ge 0 for all x\ge 0,$$

or

$${\overline{F}}^2\left(\frac{x+y}{2}\right)\ge \overline{F}\left(x\right)\overline{F}\left(y\right),for all x,y\ge 0.$$

ii. 

New better than used (NBU), if

$$\overline{F}\left(t\right)\overline{F}\left(x\right)\ge \overline{F}\left(t+x\right),\hspace{1em}for all t,x\ge 0.$$

iii. 

Used better than aged (UBA) for all$x,t\ge 0$, and if $0<\mu \left(\infty \right)<\infty$, (See Ahmed [4])

$$\overline{F}\left(t\right)e^{-x/\mu (\infty )}\le \overline{F}\left(x+t\right), x,t\ge 0$$

where

$${\overline{F}}_t\left(x\right)=\frac{\overline{F}(x+t)}{\overline{F}(t)},\hspace{1em}\overline{F}\left(t\right)>0,$$

and

$$\mu ={\int }_0^{\infty }\overline{F}\left(u\right)du,\mu \left(t\right)=E\left(X_t\right)=\frac{{\int }_t^{\infty }\overline{F}\left(u\right)du}{\overline{F}\left(t\right)},$$

$$\mu \left(\infty \right)=\underset{t\to \infty }{\mathit{lim}}\mu (t)=\frac{1}{h(\infty )}, h\left(t\right)=-\frac{d}{dt}\mathit{ln}\overline{F}\left(t\right)=\frac{f\left(t\right)}{\overline{F}\left(t\right)}, t\ge 0, \overline{F}\left(t\right)>0$$

iv. 

Used better than aged in expectation (UBAE), if

$$\mu \left(t\right)\ge \mu \left(\infty \right), x,t\ge 0.$$

For more details about these classes, see Bryson and Siddiqui [18], Marshall and Proschan [19], and Shaked and Shanthikumar [20].

The UBAmgf class is one of the most often utilized classes. It is described by Abu-Yossuf et al. [8] as a massive class of life distribution that incorporates all previously established classes, such as IFR, decreasing mean residual life (DMRL), and UBA. Klefsjo [2], Mahmoud and Diab [21], and Ghosh and Mitra [22] have since researched UBA from a variety of perspectives.

Definition 2.

The distribution function $F$ is said to be UBAmgf, if for all $x,t\ge 0$ and $0<\mu \left(\infty \right)<\infty$:

$${\int }_0^{\infty }{e}^{sx}\overline{F}(x+t)dx\ge \frac{\mu \left(\infty \right)}{1-s\mu \left(\infty \right)}\overline{F}\left(t\right),\hspace{1em}s>0.$$

It is equivalent to $X_t{\le }_{mgf}{X}_A$ for all t ≥ 0.

Now,

$$X{\le }_{st}{X}_A\Rightarrow X{\le }_{mgf}{X}_A.$$

It is obvious that $\mathrm{I}\mathrm{F}\mathrm{R}⟹\mathrm{U}\mathrm{B}\mathrm{A}⟹\mathrm{U}\mathrm{B}\mathrm{A}\mathrm{m}\mathrm{g}\mathrm{f}$.

Testing exponentiality concerns against several types of aging class life distributions have been explored by statisticians and reliability analysts from diverse perspectives; for example, see Mahmoud et al. [23], Navarro and Pellerey [24], Mahmoud El-Morshedy [25], and Navarro [26], among others. Several authors, including Mahmoud and Abdul Alim [27], Abu-Youssef and Gerges [28], and Abu-Youssef and El-Toony [29] have examined testing exponentiality using the goodness of fit approach.

The objective of this paper is to propose a nonparametric test statistic for both complete and censored data based on the goodness of fit approach for testing $H_0$: data is exponential against $H_1$: data is UBAmgf. The performance of the test statistic was evaluated based on the Pitman asymptotic relative efficiency (PARE) and empirical power of the test. Commonly used alternative distributions were also used.

The organization of this paper is as follows: The test statistic for complete data, along with PARE, the power of the test, and its real-life application, will be discussed in Section 2. The test statistic for censored data, along with the power of the test and its real-life application, will be proposed in Section 3. Finally, we provide a conclusion in Section 4.

2. Testing Exponentiality

We establish a measure of departure from exponentiality in the direction of the UBAmgf class.

2.1. Complete Data

Consider $F\left(x\right)=1-e^{-\beta x} \mathrm{f}\mathrm{o}\mathrm{r} \beta ,x>0$ as the class of exponential distribution function. We test $H_0:F$ is exponential against $H_1:F$ is UBAmgf not exponential.

The measure of deviation is derived from the following lemma.

Lemma 1.

$$\delta \left(s\right)=E\left[\left(1-s\mu \left(\infty \right)\right)\left(e^{sx}-1\right)-s(1+\mu \left(\infty \right))\left(1-e^{-x}\right)\right], s\ge 0$$

(1)

Proof. 

Let $\delta \left(s\right)$ be written as follows:

$$\delta \left(s\right)=\underset{0}{\overset{\infty }{\int }}\left[{\int }_0^{\infty }{e}^{sx}\overline{F}\left(x+t\right)dx-\frac{\mu \left(\infty \right)}{1-s\mu \left(\infty \right)}\overline{F}\left(t\right)\right]d{F}_0\left(t\right).$$

Since the measurement is scale invariant, we consider $\mu =1$ and take $F_0\left(x\right)=1-e^{-x}, x\ge 0, \phi \left(s\right)={\int }_0^{\infty }{e}^{sx}dF\left(x\right),$ which is affected by $e^{sx}$, and perform the integration as follows:

$$\begin{array}{c}\delta \left(s\right)={\int }_0^{\infty }{\int }_0^{\infty }{e}^{-t+su}\overline{F}\left(u+t\right)dudt-\frac{1}{1-s}\underset{0}{\overset{\infty }{\int }}\overline{F}\left(t\right)e^{-t}dt\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=I_1-I_2.\end{array}$$

where

$$\begin{array}{cc}{I}_1\hfill & ={\int }_0^{\infty }{\int }_0^{\infty }{e}^{su}\overline{F}\left(u+t\right)e^{-t}dudt\hfill \\ & ={\int }_0^{\infty }{\int }_t^{\infty }{e}^{-s(t-x)}\overline{F}\left(x\right)e^{-t}dxdt\hfill \\ & =\frac{1}{s}{\int }_0^{\infty }\left(e^{st}-1\right)\overline{F}\left(t\right)e^{-t}dt\hfill \\ & =\frac{1}{s(1+s)}\left[\left(\phi \left(s\right)-1\right)-s(1-\phi \left(-1\right))\right].\hfill \end{array}$$

and

$$\begin{array}{cc}{I}_2\hfill & =\frac{1}{1-s}\underset{0}{\overset{\infty }{\int }}\overline{F}\left(t\right)d{F}_0\left(t\right)\hfill \\ & =\frac{1}{1-s}(1-\phi \left(-1\right)).\hfill \end{array}$$

Therefore, the result was proved. □

The empirical estimator ${\widehat{\delta }}_n(s)$ based on $X_1,X_2,\dots ,X_n(Random variables)$ may be written as:

$${\widehat{\delta }}_n(s)=\frac{1}{\left(1+s\right)n}\sum _i\left\{\left(1-s\right)\left(e^{sx}-1\right)-2s\left(1-e^{-x}\right)\right\}.$$

So, ${\widehat{\delta }}_n(s)$ can be obtained as

$${\widehat{\delta }}_n(s)=\frac{1}{(1+s)n}\sum _j\varnothing \left(X_i\right),$$

where

$$\varnothing \left(X_i\right)=\left\{\left(1-s\right)\left(e^{s{x}_i}-1\right)-2s\left(1-e^{-x_i}\right)\right\}.$$

Set

$$\varnothing \left(X\right)=\left(1-s\right)\left(e^{sx}-1\right)-2s\left(1-e^{-x}\right).$$

The asymptotic normality is derived from the following theorem.

Theorem 1.

As $n\to \infty ,({\widehat{\delta }}_n(s)-\delta (s))$ is asymptotically normal with zero mean and ${\sigma }^2$ given in (2); under $H_0$, the variance reduces to (3).

Proof. 

Considering Lee [30], we calculate the following using the Wolfram Mathematica program, version 10:

$$\begin{array}{c}E\left[\varnothing \left(X\right)\right]=E\left(\left(1-s\right)\left(e^{sx}-1\right)-2s\left(1-e^{-x}\right)\right),\\ {\sigma }^2=E{\left(\left(1-s\right)\left(e^{sx}-1\right)-2s\left(1-e^{-x}\right)\right)}^2. \end{array}$$

(2)

One may demonstrate that the variance under $H_0$ is

$${\sigma }_0^2\left(s\right)=\frac{2s^2{(1+s)}^2}{3(-2+s)(-1+2s)},Re\left[s\right]<\frac{1}{2}.$$

(3)

□

2.2. Critical Values

In this subsection, we obtained the upper percentiles of ${\widehat{\delta }}_n\left(s\right)$, as in

Table 1. Critical values of ${\widehat{\delta }}_n\left(s\right)$ used to produce an exponential distribution-based 10,000-sample simulated sample.

${\widehat{\mathit{\delta }}}_{\mathit{n}}(0.1)$				${\widehat{\mathit{\delta }}}_{\mathit{n}}(0.01)$
n	90%	95%	99%	90%	95%	99%
5	0.0269568	0.0376037	0.0583109	0.00229131	0.003089	0.0044668
10	0.022075	0.0300524	0.0462416	0.00180829	0.00239556	0.0034263
15	0.0201512	0.0264841	0.0405875	0.00158865	0.00209656	0.00300109
20	0.0175885	0.0233625	0.0344135	0.00139708	0.00181139	0.0026182
25	0.0157562	0.0216115	0.0319283	0.0012824	0.00168031	00239751
30	0.0145093	0.0195368	0.0293987	0.00120351	0.00154123	0.00212703
35	0.0136182	0.0180498	0.0266419	0.00111429	0.00146195	0.00205294
40	0.0131065	0.017101	0.0248728	0.00103891	0.00135436	0.00188826
45	0.0127	0.0168735	0.0244385	0.00100398	0.00129905	0.00185749
50	0.0118313	0.0156481	0.0232832	0.000979758	0.00128491	0.00178655
55	0.0114589	0.0152271	0.021865	0.00092096	0.00116987	0.00166315
60	0.0111349	0.0145445	0.0210173	0.000897631	0.00114625	0.00160462
65	0.0104427	0.0139471	0.0202393	0.00086986	0.00111567	0.00154781
70	0.0105306	0.0137865	0.0195862	0.000829601	0.00108286	0.00153534
75	0.00972476	0.012878	0.0192712	0.000797139	0.00104552	0.00148272
80	0.00949736	0.0124992	0.0182769	0.000773635	0.000989083	0.0014461
85	0.00939424	0.012363	0.0179713	0.000759779	0.000981124	0.00137304
90	0.00909839	0.0120664	0.0174325	0.000724814	0.000938991	0.00134655
95	0.00893048	0.0117764	0.0170683	0.000723773	0.00092358	0.00129325
100	0.00878876	0.0113728	0.0167065	0.000701153	0.000916134	0.00129516

using the Wolfram Mathematica program, version 10.

Table 1 and Figure 1 show that the critical values behave well. Furthermore, the asymptotic normality of our test improves as s decreases.

2.3. Relative Efficiency

Pitman asymptotic efficiencies (PAEs) were investigated for $\delta (s)$ and examined with a variety of other tests for some distributions to evaluate the efficacy of this approach.

$$PAE\left(\delta \right)=\frac{{\left|\frac{\partial }{\partial \theta }\delta \right|}_{\theta \to {\theta }_0}}{{\sigma }_0}=\frac{1}{{\sigma }_0}\left|\frac{1-s}{1+s}{\int }_0^{\infty }{e}^{sx}{\overline{F}{}^{\prime }}_{{\theta }_0}\left(x\right)dx+\frac{2s}{1+s}{\int }_0^{\infty }{\overline{F}{}^{\prime }}_{{\theta }_0}\left(x\right)dx\right|,$$

where ${\overline{F}\prime }_{{\theta }_0}\left(x\right)={\left.\frac{d}{d\theta }{\overline{F}}_{\mathsf{\theta }}(u)\right|}_{\theta \to {\theta }_0}.$

Popular alternate distributions at $\theta =0,0,1$ respectively.

i.

LFR:

$${\overline{F}}_1\left(x\right)=e^{-x-\frac{x^2}{2}\theta },\hspace{1em}\theta ,x\ge 0.$$

ii.

Makeham distribution (MD):

$${\overline{F}}_3\left(x\right)=e^{-\theta (e^{-x}+x-1)-x},\hspace{1em}\theta ,x\ge 0.$$

iii.

Weibull distribution (WD):

$${\overline{F}}_2\left(x\right)=e^{-x^{\theta }},\hspace{1em}\theta \ge 1,\hspace{1em}x\ge 0.$$

A comparison of our test ${\widehat{\delta }}_n\left(s\right)$ to those of ${(\delta }_n^{(5)})$ from Ahmed et al. [31] and (${\delta }_{U_{2L}}$) from Abu-Youssef and El-Toony [29] is proposed in

Table 2. Selected values of s.

Distribution	${\mathit{\delta }}_{\mathit{n}}^{(5)}$	${\mathit{\delta }}_{{\mathit{U}}_{2\mathit{L}}}$	${\widehat{\mathit{\delta }}}_{\mathit{n}}(\mathit{s})$
			S = 0.01	S = 0.1
LFR	1.1456	1.3	1.35	1.30
Makeham	0.5455	0.58	0.803	0.86
Weibull	_____	0.791	0.99	0.97

for selected values of s.

From Table 2, we conclude that our statistic ${\widehat{\delta }}_n(s)$ performs well for LFR, Makeham, and Weibull distributions and is more efficient than the other statistics.

2.4. Empirical Power for Different Alternatives

For the test to be sensitive to a divergence from exponentiality towards the UBAmgf class, the power of the test statistics must be estimated. The test statistic’s ability to detect this variation is improved by larger power estimates.

Table 3. Empirical powers at $\alpha =0.05$ with 10,000 replications.

Distribution	n	$$\mathit{\theta }=1$$	$$\mathit{\theta }=2$$	$$\mathit{\theta }=3$$
LFR	10	0.9958	1	1
	20	1	1	1
	30	1	1	1
Gamma	10	1	1	1
	20	1	1	1
	30	1	1	1
Weibull	10	0.567	0.993	1
	20	0.568	1	1
	30	0.573	1	1

shows the test statistics’ power at $\alpha =0.05$ for the LFR, Gamma, and Weibull alternatives, using the Wolfram Mathematica program, version 10.

According to Table 3, our test yields optimal power for the LFR and Gamma families and good power for the Weibull family. The empirical powers increase as the sample size and the value of θ increase. The powers become stronger as the families deviate further from the exponential distribution.

2.5. Applications for Complete Data

In

Table 4. Some actual biological datasets.

Dataset No.	Observations	Source
1	0.019, 0.129, 0.159, 0.203, 0.485, 0.636, 0.748, 0.781, 0.869, 1.175, 1.206, 1.219, 1.219, 1.282, 1.356, 1.362, 1.458, 1.564, 1.586, 1.592, 1.781, 1.923, 1.959, 2.134, 2.413, 2.466, 2.548, 2.652, 2.951, 3.038, 3.6, 3.655, 3.745, 4.203, 4.690, 4.888, 5.143, 5.167, 5.603, 5.633, 6.192, 6.655, 6.874	Bakr et al. [5]
2	1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0	Jamal et al. [32]
3	3.1091, 3.3825, 3.1444, 3.2135, 2.4946, 3.5146, 4.9274, 3.3769, 6.8686, 3.0914, 4.9378, 3.1091, 3.2823, 3.8594, 4.0480, 4.1685, 3.6426, 3.2110, 2.8636, 3.2218, 2.9078, 3.6346, 2.7957, 4.2781, 4.2202, 1.5157, 2.6029, 3.3592, 2.8349, 3.1348, 2.5261, 1.5806, 2.7704, 2.1901, 2.4141, 1.9048	Almetwally et al. [33]
4	Censored observations: 0.57, 1.86, 3.00, 3.00, 0.14, 0.14, 0.29, 0.43, 0.57, 3.29, 3.29, 6.00 6.00, 6.14, 8.71, 10.57, 11.86, 15.57, 16.57, 27.57, 32.14, 33.14, 47.29, 17.29, 18.71, 21.29, 23.86, 26.00	Kamran Abbas et al. [34]
	Uncensored: 0.43, 2.86, 3.14, 3.14, 3.43, 3.43, 3.71, 3.86, 6.14, 6.86, 9.00, 9.43 10.71, 10.86, 11.14, 13.00, 14.43, 15.71, 18.43, 18.57, 20.71, 29.14, 29.71, 40.57, 48.57, 49.43, 53.86, 61.86, 66.57, 68.71, 68.96, 72.86, 72.86.

, we demonstrate the applicability of the findings of the study to specific real-world datasets.

• Data # 1: The test statistic from

  Table 5. Detailed statistics for datasets.

  	$\mathit{s}=0.1$	$\mathit{s}=0.01$
  Data # 1	${\widehat{\delta }}_n\left(s\right)=0.16$	${\widehat{\delta }}_n\left(s\right)=0.013$
  Data # 2	${\widehat{\delta }}_n\left(s\right)=0.026$	${\widehat{\delta }}_n\left(s\right)=0.0025$
  Data # 3	${\widehat{\delta }}_n\left(s\right)=0.17$	${\widehat{\delta }}_n\left(s\right)=0.014$

  was applied to the leukemia dataset and compared with the critical value from Table 1. A total of 40 patients with blood cancer (leukemia) from one of the Saudi Arabian Ministry of Health hospitals are represented by this statistic, which shows their ages (in years). The exponentiality of the null hypothesis is not rejected by our test for any value of s.
• Data # 2: The test statistic from Table 5 was applied to the lung cancer dataset and compared with the critical value from Table 1. The exponentiality of the null hypothesis is rejected by our test for any value of s.
• Data # 3: The test statistic from Table 5 was applied to the COVID-19 dataset and compared with the critical value from Table 1. The exponentiality of the null hypothesis is rejected by our test for any value of s.

3. Censored Data

Censored data or censored observations are results obtained from subjects who are no longer approachable after a research period. Particular patients in particular professions, such as the biological sciences, may still be alive or disease-free at the end of the study because it is unclear when survival or the end of the illness will occur.

3.1. Right-Censored Data Testing

Here, a test statistic is suggested to compare $H_0$ and $H_1$ using right-censored data. Suppose n components are put on test, and $X_1,X_2,\dots ,X_n$ denote their complete lifetime. (See the critical values in

Table 6. Critical values of the upper percentile for ${\widehat{\delta }}_c$.

${\widehat{\mathit{\delta }}}_{\mathit{c}};\mathit{s}=0.1$				${\widehat{\mathit{\delta }}}_{\mathit{c}};\mathit{s}=0.01$
n	90%	95%	99%	90%	95%	99%
5	0.498088	0.636364	0.636364	0.766628	0.960396	0.960396
10	0.336134	0.382473	0.517396	0.573876	0.666684	0.799073
15	0.265439	0.323231	0.430404	0.467642	0.55297	0.662725
20	0.208552	0.275179	0.356835	0.408777	0.471459	0.570148
25	0.165984	0.206825	0.290894	0.367748	0.414611	0.529359
30	0.152646	0.196752	0.277819	0.323765	0.369765	0.467179
35	0.124209	0.163155	0.240493	0.305978	0.35976	0.442183
40	0.101562	0.143398	0.214079	0.272452	0.319695	0.400209
45	0.0849952	0.123347	0.197554	0.263558	0.306832	0.391493
50	0.0887608	0.140263	0.202561	0.258478	0.312231	0.402083
55	0.0719956	0.109207	0.199241	0.243516	0.282715	0.345942
60	0.0642098	0.100345	0.168729	0.226144	0.261391	0.332767
65	0.0539372	0.0898383	0.162777	0.216027	0.245622	0.325733
70	0.0412581	0.0808866	0.149023	0.198931	0.234524	0.303373
75	0.0359189	0.0682674	0.127837	0.198921	0.236463	0.297495
80	0.0330002	0.0635518	0.119355	0.197349	0.223904	0.279493
85	0.0310688	0.0656815	0.120076	0.191855	0.227	0.300977
90	0.0243409	0.0542047	0.107965	0.182472	0.208744	0.288981
95	0.0182163	0.0497125	0.111521	0.17774	0.208666	0.262691
100	0.00599188	0.0371824	0.0824509	0.173808	0.205047	0.259568

calculated using the Wolfram Mathematica program, version 10).

Using the Kaplan and Meier estimator in the case of censored data, the measure of deviation ${\delta }_c\left(s\right)$ from $H_0$ in comparison to $H_1$ is provided as follows:

$${\delta }_c\left(s\right)=\frac{1}{n(1+s)}\left[\left(1-s\right)(\theta -1)-2s\left(\eta -1\right)\right]$$

(4)

where

$$\theta \left(s\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-sx}dF\left(x\right),$$

$$\widehat{\theta }\left(s\right)={\sum }_{m=1}^l{e}^{-s{Z}_{\left(m\right)}}({\prod }_{p=1}^{m-2}{C}_p^{{\delta }_p}-{\prod }_{p=1}^{m-1}{C}_p^{{\delta }_p}), \eta =\widehat{\theta }\left(-1\right),$$

$$d{F}_n\left(Z_i\right)=\prod _{q=1}^{j-2}{C}_i^{{\delta }_i}-\prod _{q=1}^{j-1}{C}_i^{{\delta }_i},$$

$${\overline{F}}_n\left(t\right)=\prod _{m<t}^{}{C}_m^{{\delta }_m},$$

$$\hspace{1em}{C}_m=\frac{n-m}{n-m+1} ,\hspace{1em} t\in \left[0, z_{\left(m\right)}\right].$$

Table 6 and Figure 2 show that the critical values behave well. Furthermore, the asymptotic normality of our test improves as s decreases.

3.2. Power Estimates

Table 7. Estimates of powers at $\alpha =0.05$ with 10,000 replications.

Distribution	n	$$\mathit{\theta }=1$$	$$\mathit{\theta }=2$$	$$\mathit{\theta }=3$$
LFR	10	0.956	0.96	0.971
	20	0.983	0.991	0.999
	30	0.991	0.992	1
Weibull	10	1	1	1
	20	1	1	1
	30	1	1	1

shows the power of our test once again for censored data, specifically for the LFR and Weibull distributions.

The simulation studies clearly indicate that the test exhibits high power for both distributions.

3.3. Applications

To apply our test statistic in (4), we considered the dataset in Table 4, which describes the survival periods, in weeks, of 61 patients with incurable lung cancer treated with cyclophosphamide. The patients whose treatment was discontinued due to a deteriorating condition are represented by 33 uncensored observations and 28 censored observations.

We calculated the statistic in (4) for both cases of $s=0.1 \mathrm{a}\mathrm{n}\mathrm{d} s=0.01$, and ${\delta }_c$ for the two cases are negative values as n = 61, which are both (0.37 and 0.27) greater than the critical value in Table 6. As a result, the exponentiality null hypothesis is rejected using our test for any value of s.

4. Conclusions

Exponential distribution is one of the most important continuous distributions in statistics. It has been used widely in reliability theory, life testing, and stochastic processes due to its memoryless property. Therefore, it is of great interest to determine whether a random sample follows an exponential distribution or not. In this paper, we proposed nonparametric tests for testing whether the data follow an exponential distribution against the data is (UBAmgf). Based on our studies, we were able to determine if the recommended treatments had a positive or negative impact on patients through their survival times (discussed in the Applications section). We considered both complete and censored data. The asymptotic normality of the proposed test was established and, using a Monte Carlo simulation study, the upper percentile points of the test statistics were obtained. The Pitman asymptotic relative efficiencies (PAEs) and power of the proposed test were calculated for LFR, Gamma, and Weibull distributions. Real-life datasets were analyzed to illustrate the findings of this paper.