Improved EDF-Based Tests for Weibull Distribution Using Ranked Set Sampling

Abstract

It is well known that ranked set sampling (RSS) is superior to conventional simple random sampling (SRS) in that it frequently results in more effective inference techniques. One of the most popular and broadly applicable models for lifetime data is the Weibull distribution. This article proposes different modified goodness-of-fit tests based on the empirical distribution function (EDF) for the Weibull distribution. The recommended RSS tests are compared to their SRS counterparts. For each scheme, the critical values of the relevant test statistics are computed. A comparison of the power of the suggested goodness-of-fit tests based on a number of alternatives is investigated. RSS tests are more effective than their SRS equivalents, according to simulated data.

1. Introduction

The actual measurement of sample observations may be difficult in various statistical contexts, such as agriculture, environment, ecology, sociology, and others, since measures may be damaging, intrusive, costly, or time-consuming. Despite the fact that data collection might be difficult, ranking possible sample observations can be simple. In such situations, the ranked set sample (RSS) methodology proposed in [1] may be more appropriate than traditional random sampling methods. The selection of the RSS can be summarized in the following steps:

• Randomly select $m^2$ sample units denoted by $X_{(i:j)},i=1,2,\dots ,m$ and $j=1,2,\dots ,m.$
• Allocate the selected $m^2$ units as randomly as possible into m sets, each of size m.
• Rank units inside each row depending on a criterion specified by the researcher, which will be one cycle, without collecting any measurements.
• Select the sample for proper analysis by selecting the least ranked unit for RSS from the first set (row), then the second smallest ranked units from the second set, and so on until the largest ranked unit is picked from the last set. The RSS associated with this cycle will be $(X_{11},X_{22},\dots ,X_{mm})$, which are independent but not identically distributed. Note that $X_{ii}$ is distributed as the ith-order statistic $X_i$ of a sample of size m.
• Repeat steps 1 through 4 at r cycles until the desired sample size, $n=mr$, is obtained for analysis.
• Use the following matrix notation to express the RSS design:

  $$\begin{array}{c}BeforeRank\hfill \\ \hfill \left(\begin{array}{c}\hfill X_{(1:1)}{X}_{(1:2)}\dots X_{(1:m)}\hfill \\ \hfill X_{(2:1)}{X}_{(2:2)}\dots X_{(2:m)}\hfill \\ \hfill ⋮⋮\ddots ⋮\hfill \\ \hfill X_{(m:1)}{X}_{(m:2)}.X_{(m:m)}\hfill \end{array}\right)\Rightarrow \hfill \end{array}\begin{array}{c}AfterRank\hfill \\ \hfill \left(\begin{array}{c}\hfill X_{11}{X}_{21}\dots X_{m1}\hfill \\ \hfill X_{12}{X}_{22}\dots X_{m2}\hfill \\ \hfill ⋮⋮\ddots ⋮\hfill \\ \hfill X_{1m}{X}_{2m}.X_{mm}\hfill \end{array}\right)\hfill \end{array}$$

The mathematical theory of RSS was established in [2]. The authors demonstrated that, when compared to the simple random sample (SRS) mean, the RSS mean is a more unbiased estimator of the population mean with lower variance. Reference [3] demonstrated that whether the ranking is perfect or not, the RSS mean remains unbiased. In reality, two elements influence RSS efficiency: set size and ranking errors. The greater the set size, the more efficient RSS becomes [4]. As a result, the larger the set, the more complex visual ranking and ranking errors become [5]. Several authors adjusted RSS to decrease ranking error and make visual ranking tractable by an experimenter for this purpose.

It is always interesting to determine whether a set of data may be considered to originate from a population managed by a single family. The distance between the empirical distribution function (EDF) and the hypothesized distribution function is one type of goodness-of-fit (GOF) test that may be employed. When there are no unknown parameters in the predicted distribution, these tests are valid. However, if these tests are employed in situations where unknown parameters must be inferred from sample data or data that are not SRS, one should use a high degree of caution. Many authors, including those of [6,7,8,9,10,11,12,13,14], have explored modified GOF tests under SRS.

Over the previous two decades, authors have paid little attention to GOF tests based on data acquired using the RSS approach and its variants. The characterization of RSS was examined in [15]. The authors also provided an unbiased estimate of the population distribution function based on the RSS empirical distribution function. Then, based on the EDF, they presented a Kolmogorov–Smirnov (KS) GOF test. They calculated the null distribution for the presented test. Reference [16] compared the power of a series of empirical distribution function GOF tests for the logistic distribution under SRS and RSS. Ref. [17] suggested a strategy for improving the power of GOF tests for logistic distribution under extreme RSS (ERSS). They also performed a simulated study to examine the power of each test when using the ERSS and SRS. The power of many GOF tests under SRS and RSS was investigated in [18]. Under selective RSS, the authors of [19] created critical value tables for the exponentiated Pareto distribution. Ref. [20] examined GOF tests for the inverse Gaussian distribution based on novel entropy estimates using RSS and double RSS approaches. For a variety of alternative distributions, the power of modified test statistics was evaluated using ERSS and SRS. Reference [21] used a moving ERSS to present modified GOF tests for the Weibull distribution. Different GOF tests for Rayleigh distribution were examined in [22]. They discovered that Anderson–Darling is the most effective of all the tests. Reference [23] investigated GOF tests based on the sample entropy and EDF for the Laplace distribution using RSS.

In the study of survival data, an appropriate parametric model is typically important because it gives insight into aspects of failure periods and hazard functions that nonparametric approaches may not provide. One of the most often used distributions for modeling systems with monotone failure rates is the Weibull distribution. It also includes specific models for the exponential and Rayleigh distributions. Then, it may be sufficient for fitting a variety of kinds of data. The Weibull distribution is a powerful tool for modeling failure rates with the virtue of being monotonic since it can adapt negatively and positively skewed density shapes. The cumulative distribution function (CDF) of the Weibull distribution is given by:

$$F(x;\lambda ,\alpha )=1-exp-{\left(\frac{x}{\lambda }\right)}^{\alpha },x>0,\alpha ,\lambda >0,$$

(1)

where $\lambda$ is a scale parameter and $\alpha$ is referred to as the shape parameter. The probability density function (PDF) corresponding to (1) is given by

$$f(x;\lambda ,\alpha )=\frac{\alpha }{{\lambda }^{\alpha }}{x}^{\alpha -1}exp-{\left(\frac{x}{\lambda }\right)}^{\alpha },x>0,\alpha ,\lambda >0.$$

(2)

The Weibull distribution has been widely used in several areas, such as survival analysis [24], reliability engineering [25], and weather forecasting [26]. A number of authors have performed Weibull distribution studies. Reference [27] presented a modified profile likelihood method for estimating the Weibull shape parameter. The estimation of the Weibull distribution in a complete and censored sample was provided in [28]. In the medical, biological, and Earth sciences, the Weibull distribution was used (see [29]). The parameter estimator of Weibull distribution in a partially accelerated test model under hybrid censored samples was discussed in [30]. Reference [31] proposed different estimation methods for the Weibull distribution. The estimation of the PDF and the CDF of the Weibull distribution was studied in [32], and the estimation in a stress–strength model using different methods of estimation was discussed in [33]. Reference [34] considered the reliability estimators of a multicomponent stress strength Weibull model under record values. The sum of Weibull random variables is naturally of utmost significance in wireless communications applications, including optical, mobile, radar, signal detection, phase jitter, intersymbol interference, and related fields. Determining the performance of many wireless communication systems requires sums of Weibull random variables (see [35,36]). According to reference [37], the sums of Weibull random variables have never yielded any conclusions (not even approximations). However, due to the intricate task of evaluating Weibull sums, only a few works deal with exact sum statistics. Some works have resorted to different approaches in order to obtain simpler formulations for the PDF and the CDF of sums of Weibull variates. Reference [38] was able to derive exact statistical expressions for the sum of identically independent distributed Weibull variates in terms of nested infinite sum products, demonstrating a high computational burden and a high mathematical complexity that tends to grow as the number of Weibull variates in the sum increases. More recently, in [39], the authors found closed-form expressions for the PDF and CDF of the harmonic sum of two independent Weibull variates, which were derived in terms of the bivariate Fox H-function. Reference [40] derived novel exact expressions for the PDF and the CDF of the sum of independent and identically distributed Weibull random variables. They obtained simpler, faster, and more manageable solutions based on the residue theory (Cauchy’s residue theorem) and the Laplace transform.

Complex systems, such as a car, can fail through many different mechanisms, often requiring a sequence or combination of events for a component to fail. In the context of human diseases, cancer in particular, the same is possible. One of the risk models was originally proposed in [41], namely, multistage modeling. The multistage model explains that the condition state changes from one state to other states only in one step. Multistage models explain how systems may fail via one or more possible routes. They are sometimes referred to as “multistep” or “multihit” models, since each route typically involves the failure of one or more sequential or nonsequential steps. Other applications that address these concerns can be found in [42,43,44,45,46,47].

Due to the importance of Weibull distribution, we devote special attention to the modified GOF tests in the case of SRS and RSS. We produce large tables of critical values for the Weibull distribution using the SRS and RSS sampling procedures. Critical values for test statistics are calculated using the SRS and RSS tables for Weibull distributions with unknown parameters. Furthermore, power comparisons between KS, Cramer–von Mises (CvM), Anderson–Darling (AD), Watson (W), Kuiper (K), and Liao and Shimokawa (LS) test statistics for numbers of alternative distributions are studied. Furthermore, the power efficiency of the suggested test statistics under RSS in relation to SRS is investigated.

The structure of this article is as follows. The unknown parameters from the Weibull distribution are estimated using maximum likelihood estimation in Section 2. The collection of modified EDF goodness-of-fit tests under SRS and RSS is presented in Section 3. A simulation study is performed in Section 4 to obtain the critical values for the modified test statistics via SRS and RSS. The power efficiency of these test statistics under RSS compared to SRS is given in Section 5. In Section 6, there are a few concluding notes.

2. Parameter Estimation

This section provides the maximum likelihood estimators (MLEs) of the Weibull distribution parameters $\alpha$ and $\lambda$ under SRS and RSS designs.

2.1. MLEs under SRS

Let $X_1,X_2,\dots ,X_r$ be an SRS from a Weibull distribution with unknown parameters $\alpha$ and $\lambda$. The likelihood function is given by

$$l(\alpha ,\lambda )={\alpha }^r{\lambda }^{-\alpha r}{\displaystyle \prod _{i=1}^r}{x}_i^{\alpha -1}exp\left[-{\left(\frac{x_i}{\lambda }\right)}^{\alpha }\right].$$

The log-likelihood function $\ell \equiv l(\alpha ,\lambda )$ is given by

$$\ell =rlog\alpha -\alpha rlog\lambda +(\alpha -1){\displaystyle \sum _{i=1}^r}log{x}_i-{\displaystyle \sum _{i=1}^r}{\left(\frac{x_i}{\lambda }\right)}^{\alpha }.$$

(3)

Differentiate (3) with respect to $\alpha$ and $\lambda$, and the following normal equations are obtained

$$\frac{\partial \ell }{\partial \alpha }=\frac{r}{\widehat{\alpha }}-rlog\widehat{\lambda }+{\displaystyle \sum _{i=1}^r}log{x}_i-{\displaystyle \sum _{i=1}^r}{\left(\frac{x_i}{\widehat{\lambda }}\right)}^{\widehat{\alpha }}log\left(\frac{x_i}{\widehat{\lambda }}\right)=0,$$

(4)

and,

$$\frac{\partial \ell }{\partial \lambda }=-\frac{r\widehat{\alpha }}{\widehat{\lambda }}+{\displaystyle \sum _{i=1}^r}\frac{\widehat{\alpha }{x}_i^{\widehat{\alpha }}}{{\left(\widehat{\lambda }\right)}^{\widehat{\alpha }+1}}=0.$$

(5)

From Equation (5), an estimate of $\lambda$, say $\widehat{\lambda }$, can be obtained as follows:

$$\widehat{\lambda }={\left(\frac{1}{r}{\displaystyle \sum _{i=1}^r}{x_i}^{\widehat{\alpha }}\right)}^{1/\widehat{\alpha }}.$$

(6)

The estimate of $\alpha$ can be obtained by substituting (6) in (4), as follows:

$$\frac{r}{\widehat{\alpha }}-rlog{\left(\frac{1}{r}{\displaystyle \sum _{i=1}^r}{x}_i^{\widehat{\alpha }}\right)}^{1/\phantom{1\widehat{\alpha }}\widehat{\alpha }}+{\displaystyle \sum _{i=1}^r}log{x}_i-r{\displaystyle \sum _{i=1}^r}\frac{x_i^{\widehat{\alpha }}}{{\displaystyle {\displaystyle \sum _{i=1}^r}}{x}_i^{\widehat{\alpha }}}log\left(\frac{x_i}{{\displaystyle {\displaystyle \sum _{i=1}^r}}{\left(x_i^{\widehat{\alpha }}/r\right)}^{1/\widehat{\alpha }}}\right)=0.$$

(7)

The solution to (7) requires a numerical solution. Once the value of $\widehat{\alpha }$ is obtained, the value of $\widehat{\lambda }$ can be obtained by substituting (7) in (6).

2.2. MLEs under RSS

Let $Y_1,Y_2,\dots ,Y_r$ be a random sample of size r selected via the ith-order statistics from Weibull distribution with unknown parameters $\alpha$ and $\lambda$. Therefore, the PDF of the ith-order statistic of odd set size from Weibull distribution is given by using CDF (1) and PDF (2) as the following:

$$\begin{array}{ll}f\left(y_{\left(i\right)}\right)\hfill & =C{\left[F_0(y_{\left(i\right)};\alpha ,\lambda )\right]}^{i-1}{[1-F_0(y_{\left(i\right)};\alpha ,\lambda )]}^{2m-1-i}{f}_0(y_{\left(i\right)};\alpha ,\lambda )\\ & =C\frac{\alpha }{{\lambda }^{\alpha }}{y_{\left(i\right)}}^{\alpha -1}exp\left[-(2m-i){\left(\frac{y_{\left(i\right)}}{\lambda }\right)}^{\alpha }\right]{\left[1-exp\left[-{\left(\frac{y_{\left(i\right)}}{\lambda }\right)}^{\alpha }\right]\right]}^{i-1},\hfill \end{array}$$

where $C=\frac{(2m-1)!}{(i-1)!(2m-1-i)!}$. The likelihood function of the sample $Y_1,Y_2,\dots ,Y_r$ is given by

$$l_1(\alpha ,\lambda )={\displaystyle \prod _{i=1}^r}\frac{C\alpha }{{\lambda }^{\alpha }}{\left(y_{\left(i\right)}\right)}_{}^{\alpha -1}{\left[1-exp-{\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}\lambda }\lambda \right)}^{\alpha }\right]}^{i-1}exp\left[-(2m-i){\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}\lambda }\lambda \right)}^{\alpha }\right].$$

(8)

The log-likelihood function ${\ell }_1\equiv l_1(\alpha ,\lambda )$ is given by

$$\begin{array}{c}\hfill {\ell }_1={\displaystyle \sum _{i=1}^r}\left(logC+(i-1)log\left(1-exp-{\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}\lambda }\lambda \right)}^{\alpha }\right)-(2m-i){\left(\frac{y_{\left(i\right)}}{\lambda }\right)}^{\alpha }\right.\hfill \\ \hfill \left.+log\alpha -\alpha log\lambda +(\alpha -1)log{y}_{\left(i\right)}\right).\hfill \end{array}$$

(9)

The MLEs ${\widehat{\alpha }}_1$ and ${\widehat{\lambda }}_1$ for the parameters $\alpha$ and $\lambda$ are the values which maximize the logarithm of the likelihood function ${\ell }_1$ obtained in (9). The first partial derivatives of ${\ell }_1$ with respect to $\alpha$ and $\lambda$ are obtained as follows:

$$\begin{array}{ll}\frac{\partial {\ell }_1}{\partial \alpha }\hfill & ={\displaystyle \sum _{i=1}^r}(i-1){\left(exp\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}{\widehat{\lambda }}_1}{\widehat{\lambda }}_1\right)-1\right)}^{-1}{\left(\frac{y_{\left(i\right)}}{{\widehat{\lambda }}_1}\right)}^{{\widehat{\alpha }}_1}log\left(\frac{y_{\left(i\right)}}{{\widehat{\lambda }}_1}\right)-(2m-i){\left(\frac{y_{\left(i\right)}}{{\widehat{\lambda }}_1}\right)}^{{\widehat{\alpha }}_1}log\left(\frac{y_{\left(i\right)}}{{\widehat{\lambda }}_1}\right)\\ & +{\displaystyle \sum _{i=1}^r}\left(\frac{1}{{\widehat{\alpha }}_1}-log{\widehat{\lambda }}_1+log{y}_{\left(i\right)}\right)=0,\hfill \end{array}$$

(10)

and

$$\frac{\partial {\ell }_1}{\partial \lambda }={\displaystyle \sum _{i=1}^r}\frac{(2m-i)\alpha y_{\left(i\right)}^{{\widehat{\alpha }}_1}}{{{\widehat{\lambda }}_1}^{{\widehat{\alpha }}_1+1}}-{\displaystyle \sum _{j=1}^r}(i-1){\left(exp\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}{\widehat{\lambda }}_1}{\widehat{\lambda }}_1\right)-1\right)}^{-1}{\widehat{\alpha }}_1\left(\frac{{y_{\left(i\right)}}^{{\widehat{\alpha }}_1}}{{{\widehat{\lambda }}_1}^{{\widehat{\alpha }}_1+1}}\right)-\frac{r{\widehat{\alpha }}_1}{{\widehat{\lambda }}_1}=0.$$

(11)

Nonlinear Equations (10) and (11) are clearly difficult to solve in a closed form. These equations can be solved numerically using a fairly straightforward iterative approach.

3. Goodness-of-Fit Tests

The distance between a continuous distribution function and the EDF is measured in several ways using empirical distribution function GOF tests. Continuous underlying distributions with known parameters are required for these tests. However, if these tests are applied in situations where unknown parameters must be calculated from sample data or the data must be ranked, they become exceedingly conservative. Modified EDF tests are used when the parameters are estimated or the data are not SRS. Under SRS and RSS, a modified EDF goodness-of-fit test is considered.

3.1. Modified GOF Tests via SRS

A goodness-of-fit test is a test of the hypothesis

$$H_0:F\left(x\right)=F_0\left(x\right)\forall x,vs.H_1:F\left(x\right)\ne F_0\left(x\right)forsomex,$$

(12)

where $F_0\left(x\right)$ is a known distribution function based on $X_1,X_2,\dots ,X_r$, a random sample from the Weibull CDF with two unknown parameters defined in (1).

The following set of the modified EDF goodness-of-fit tests under SRS is defined as follows:

• The KS test statistic, represented by D, is

  $$D=\underset{1⩽i⩽r}{max}\left\{\underset{}{\underset{1⩽i⩽r}{max}\left(\frac{i}{r}-F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda }\right),\underset{1⩽i⩽r}{max}\left(F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })-\frac{i-1}{r}\right)}\right\}.$$

  (13)
• The CvM statistic, say $W^2$, is

  $$W^2={\displaystyle \sum _{i=1}^r}[F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })-(2i-1)/2r{]}^2+(1/12r).$$

  (14)
• The K statistic, denoted by V, is

  $$V=\underset{}{\underset{1⩽i⩽r}{max}\left(\frac{i}{r}-F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })\right)+\underset{1⩽i⩽r}{max}\left(F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })-\frac{i-1}{r}\right)}.$$

  (15)
• The W statistic, say $U^2$, is

  $$U^2=W^2-r{\left[\frac{1}{r}{\displaystyle \sum _{i=1}^r}{F}_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })-\frac{1}{2}\right]}^2.$$

  (16)
• The AD statistic, say $A^2$, is

  $$A^2=-\frac{1}{r}\left\{{\displaystyle \sum _{i=1}^r}(2i-1)[log(F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })]+log(1-F_0(x_{(r+1-i)},\widehat{\alpha },\widehat{\lambda })\left)\right]\right\}-r.$$

  (17)
• The LS statistic, say L, is

  $$L=\frac{1}{\sqrt{r}}{\displaystyle \sum _{i=1}^r}\frac{max\left[(\frac{i}{r}-F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })),(F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })-\frac{i-1}{r})\right]}{\sqrt{F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })(1-F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda }))}}.$$

  (18)

Let us denote test statistics (13)–(18) by T, under SRS.

3.2. Modified GOF Tests via RSS

To test the hypothesis based on RSS, let $Y_1,Y_2,\dots ,Y_r$ be a random sample of size r selected via the ith-order statistic. According to [16], testing the hypothesis $H_0:F\left(y\right)=F_0\left(y\right)$$\forall y$ vs. $H_1:F\left(y\right)\ne F_0\left(y\right),$ for some y, is equivalent to testing the hypothesis

$$H_0^{*}:G_i\left(\mathrm{y}\right)=G_{0i}\left(y\right),\forall \mathrm{y}vs.H_1^{*}:G_i\left(y\right)\ne G_{0i}\left(y\right)forsomei,$$

(19)

where $G_i\left(y\right)$, $G_{0i}\left(y\right)$, are the CDFs of the ith-order statistics of random samples of an odd size ($2m-1$) chosen from $F_0\left(y\right)$, where $F_0\left(y\right)$ is the Weibull distribution.

Thus, the GOF test for hypothesis (19) in RSS, denoted by $T^{*}$, can be performed as follows:

• The KS statistic

  $$D=\underset{1⩽i⩽n}{max}\left\{\underset{1⩽i⩽n}{max}\left[\frac{i}{n}-F(y_{(i:n)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)\right],\underset{1⩽i⩽n}{max}\left[F_0(y_{(i:n)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)-\frac{i-1}{n}\right]\right\},$$

  (20)

  where,

  $$F\left(y_{(i:n)}\right)=\frac{1}{m}\left(F_{(1:m)}\left(y_{(i:n)}\right)+F_{(2:m)}\left(y_{(i:n)}\right)+\dots +F_{(m:m)}\left(y_{(i:n)}\right)\right),n=rm.$$
• The CvM statistic

  $$W^2={\displaystyle \sum _{i=1}^r}[F\left(y_{(i:n)}\right)-(2i-1)/2n{]}^2+(1/12n).$$

  (21)
• The K statistic

  $$V=\underset{}{\underset{1⩽i⩽n}{max}\left(\frac{i}{n}-F_0(y_{(i:n)},\widehat{\alpha },\widehat{\lambda })\right)+\underset{1⩽i⩽n}{max}\left(F_0(y_{(i:n)},\widehat{\alpha },\widehat{\lambda })-\frac{i-1}{n}\right)}.$$

  (22)
• The W statistic

  $$U^2=W^2-n{\left[\frac{F\left(y_{(i:n)}\right)}{n}-\frac{1}{2}\right]}^2.$$

  (23)
• The AD statistic

  $$A^2=-\frac{1}{n}\left[{\displaystyle \sum _{i=1}^n}(2i-1)\left(logF\left(y_{i:n)}\right)+log(1-F\left(y_{(i:n)}\right)\right)\right]-n.$$

  (24)
• The LS statistic

  $$L=\frac{1}{\sqrt{n}}{\displaystyle \sum _{i=1}^n}\frac{max\left[(\frac{i}{n}-F_0(y_{(i:n)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)),(F_0(y_{(i:n)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)-\frac{i-1}{n})\right]}{\sqrt{F_0(y_{(i:n)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)(1-F_0(y_{(i:n)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)}}.$$

  (25)

4. Calculation of Critical Values

The major goal of this section is to find critical values for the test statistics, T and $T^{*}$, for the Weibull distribution with unknown parameters when using SRS and RSS schemes. Monte Carlo simulation was carried out to create tables of critical values for KS, CvM, AD, K, W, and LS test statistics under two different sampling techniques, SRS and RSS.

4.1. Critical Values via SRS

Here, we construct tables of critical values for six suggested Weibull distribution test statistics. For the Weibull distribution, these critical values of test statistics are provided with unknown parameters. The critical values of the modified test statistics KS, CvM, K, W, AD, and LS are calculated using the techniques below.

• Selected SRS of size $r=10\left(10\right)30$. An SRS $X_1,X{}_2,\dots ,X_r$ from a Weibull distribution with chosen parameters $\alpha =3$ and $\lambda =3.5$ is generated.
• The unknown parameters $\alpha$ and $\lambda$ are estimated using the maximum likelihood technique from this random sample. To estimate the shape parameter $\alpha$, the nonlinear Equation (7) is numerically solved. Once the value of $\alpha$ is determined, the estimate may be calculated using (6).
• The resulting MLEs of $\alpha$ and $\lambda$ are used to calculate the Weibull distribution’s hypothesized CDF as follows:

  $$F_0(x_{\left(i\right)},\widehat{\alpha },\widehat{\lambda })=1-exp-{\left(\frac{x_{\left(i\right)}}{\widehat{\lambda }}\right)}^{-\widehat{\alpha }}.$$

  (26)
• Selected SRS of size $r=10\left(10\right)30$. The modified KS, CvM, AD, K, W, and LS test statistics are calculated for a given value of r.
• This technique is performed 5000 times, resulting in 5000 independent values for the test statistics. The values of these test statistics at seven significance levels, i.e., $\gamma =0.25,0.20,0.15,0.10,0.05,0.025$, and $0.01$, are determined after these 5000 values are ranked. These are the critical values for each sample size utilized for that particular test.
• Table 1. Simulated critical values for test statistics for Weibull distribution based on SRS.

  r	Tests	Significant Level $\mathit{\gamma }$
  		0.25	0.2	0.15	0.1	0.05	0.025	0.01
  10	$A^2$	0.459	0.496	0.544	0.612	0.721	0.854	1.02
  	$W^2$	0.072	0.079	0.088	0.099	0.12	0.143	0.174
  	$U^2$	0.069	0.075	0.084	0.094	0.114	0.134	0.164
  	D	0.209	0.217	0.227	0.241	0.26	0.28	0.304
  	V	0.361	0.376	0.392	0.414	0.445	0.482	0.52
  	L	1.016	1.048	1.088	1.143	1.241	1.346	1.483
  20	$A^2$	0.464	0.497	0.544	0.61	0.732	0.844	0.987
  	$W^2$	0.072	0.078	0.086	0.098	0.119	0.141	0.168
  	$U^2$	0.069	0.075	0.083	0.093	0.113	0.133	0.158
  	D	0.151	0.157	0.164	0.174	0.189	0.204	0.219
  	V	0.267	0.277	0.29	0.305	0.334	0.363	0.39
  	L	0.897	0.921	0.958	1.009	1.095	1.174	1.28
  30	$A^2$	0.471	0.509	0.561	0.631	0.749	0.865	1.018
  	$W^2$	0.072	0.08	0.089	0.101	0.122	0.142	0.175
  	$U^2$	0.069	0.076	0.085	0.095	0.116	0.135	0.166
  	D	0.125	0.13	0.136	0.144	0.157	0.168	0.182
  	V	0.225	0.233	0.245	0.26	0.248	0.306	0.331
  	L	0.841	0.874	0.908	0.958	1.035	1.109	1.227

  shows the critical values for the modified test statistics KS, CvM, AD, K, W, and LS. The null hypothesis of a Weibull distribution should be rejected at the given significance level if the computed value of the test statistics exceeds the tabulated value.

4.2. Critical Values via RSS

For the Weibull distribution, tables of critical values are constructed for the suggested test statistics for RSS determined by order statistic, minimum, median, and maximum. For the Weibull distribution, these critical values of test statistics are presented. The critical values of RSS for the modified test statistics KS, CvM, K, AD, W, and LS are calculated using the techniques below.

• Create an RSS of size $r=10\left(10\right)30$ using the Weibull distribution with $\alpha =3$ and $\lambda =3.5$ for odd set sizes ($2m-1),m=1,2,3,4$, where ($m=1$) means the SRS case. In particular, for set size $m=2$, $Y_1,Y{}_2,\dots ,Y_r$ is a random sample generated from $G_{i0}\left(y\right),$$i=1,2,3$ (i.e., random sample of size r selected via smallest, median, and largest order statistics).
• The RSS is used to estimate $\alpha$ and $\lambda$ numerically by solving Equations (10) and (11).
• The MLEs of $\alpha$ and $\lambda$ are used to determine the hypothesized CDF $G_{i0}(y_{\left(i\right)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)$.
• The hypothesized CDF for the minimum order statistics for Weibull distribution is obtained from $G_{i0}(y_{\left(i\right)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)=1-exp-\left(2m-1\right){\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}}{\widehat{\lambda }}_1\right)}^{{\widehat{\alpha }}_1},$$m=2,3,4$.
• Additionally, the hypothesized CDF for the maximum order statistics is obtained from $G_{i0}(y_{\left(i\right)},{\widehat{\alpha }}_1,{\widehat{\lambda }}_1)={\left[1-exp-{\left(y_{\left(i\right)}/\phantom{y_{\left(i\right)}{\widehat{\lambda }}_1}{\widehat{\lambda }}_1\right)}^{{\widehat{\alpha }}_1}\right]}^{\left(2m-1\right)},$$m=2,3,4$.
• Obtain the EDF from RSS as follows:

  $${\widehat{F}}_{RSS}=\frac{1}{r}{\displaystyle \sum _{j=1}^r}I(Y_{ij}⩽x),I\left(Y_{ij}\right)=\left\{\begin{array}{cc}1& Y_{ij}⩽x\\ 0& otherwise\end{array}\right..$$
• Calculate the modified test statistics (KS, CvM, AD, K, W, LS).
• This technique is performed 5000 times, resulting in 5000 independent values for the test statistics. The values of these test statistics at seven significance levels, i.e., $\gamma =0.25,0.20,0.15,0.10,0.05,0.025$, and $0.01$, are determined after these 5000 values are ranked. These are the critical values for that specific test for each sample size and set size that are employed.
• Table 2. Simulated critical values for test statistics for Weibull distribution based on the maximum-order statistic.

  r	m	Tests	Significant Level $\mathit{\gamma }$
  			0.25	0.2	0.15	0.1	0.05	0.025	0.01
  10	2	$A^2$	0.465	0.499	0.551	0.621	0.719	0.825	0.969
  		$W^2$	0.074	0.08	0.089	0.101	0.12	0.14	0.166
  		$U^2$	0.07	0.076	0.085	0.096	0.114	0.131	0.153
  		D	0.211	0.219	0.229	0.243	0.263	0.279	0.301
  		V	0.363	0.375	0.39	0.413	0.443	0.475	0.512
  		L	1.016	1.046	1.085	1.139	1.224	1.316	1.437
  	3	$A^2$	0.459	0.493	0.544	0.6	0.707	0.804	0.951
  		$W^2$	0.073	0.08	0.088	0.1	0.118	0.137	0.164
  		$U^2$	0.069	0.076	0.083	0.094	0.111	0.128	0.154
  		D	0.21	0.217	0.228	0.242	0.261	0.278	0.303
  		V	0.36	0.372	0.389	0.41	0.44	0.475	0.51
  		L	1.014	1.045	1.08	1.129	1.215	1.296	1.393
  	4	$A^2$	0.464	0.498	0.545	0.606	0.718	0.822	0.955
  		$W^2$	0.074	0.08	0.089	0.1	0.121	0.141	0.165
  		$U^2$	0.07	0.076	0.084	0.094	0.112	0.131	0.153
  		D	0.209	0.218	0.227	0.24	0.262	0.28	0.302
  		V	0.36	0.374	0.39	0.411	0.441	0.475	0.514
  		L	1.015	1.044	1.08	1.134	1.212	1.29	1.419
  20	2	$A^2$	0.465	0.504	0.552	0.661	0.732	0.852	0.993
  		$W^2$	0.073	0.08	0.089	0.1	0.123	0.145	0.17
  		$U^2$	0.069	0.076	0.084	0.096	0.115	0.138	0.161
  		D	0.152	0.159	0.165	0.175	0.19	0.205	0.221
  		V	0.269	0.279	0.293	0.309	0.335	0.362	0.394
  		L	0.896	0.927	0.959	1.008	1.087	1.164	1.256
  	3	$A^2$	0.459	0.493	0.537	0.601	0.716	0.839	0.982
  		$W^2$	0.073	0.079	0.087	0.098	0.119	0.141	0.17
  		$U^2$	0.069	0.075	0.082	0.093	0.112	0.133	0.158
  		D	0.152	0.159	0.166	0.175	0.189	0.203	0.221
  		V	0.268	0.279	0.291	0.307	0.335	0.359	0.391
  		L	0.887	0.917	0.952	1.001	1.081	1.163	1.278
  	4	$A^2$	0.466	0.504	0.56	0.632	0.737	0.859	0.987
  		$W^2$	0.074	0.081	0.09	0.103	0.122	0.147	0.172
  		$U^2$	0.07	0.076	0.084	0.097	0.115	0.135	0.161
  		D	0.153	0.159	0.167	0.176	0.192	0.206	0.223
  		V	0.268	0.28	0.293	0.311	0.34	0.366	0.397
  		L	0.898	0.927	0.966	1.016	1.095	1.172	1.266
  30	2	$A^2$	0.463	0.504	0.547	0.619	0.74	0.859	1.019
  		$W^2$	0.072	0.08	0.088	0.1	0.121	0.143	0.171
  		$U^2$	0.096	0.075	0.084	0.095	0.114	0.135	0.161
  		D	0.125	0.13	0.136	0.143	0.156	0.169	0.185
  		V	0.223	0.232	0.244	0.258	0.282	0.305	0.337
  		L	0.839	0.867	0.903	0.948	1.032	1.103	1.225
  	3	$A^2$	0.469	0.505	0.556	0.628	0.736	0.858	1.011
  		$W^2$	0.073	0.08	0.09	0.103	0.125	0.146	0.174
  		$U^2$	0.069	0.076	0.085	0.097	0.116	0.138	0.159
  		D	0.125	0.13	0.137	0.146	0.159	0.171	0.184
  		V	0.224	0.234	0.246	0.263	0.287	0.31	0.335
  		L	0.838	0.867	0.901	0.953	1.026	1.101	1.178
  	4	$A^2$	0.456	0.496	0.544	0.609	0.728	0.833	0.96
  		$W^2$	0.072	0.078	0.088	0.1	0.121	0.143	0.164
  		$U^2$	0.068	0.075	0.082	0.094	0.097	0.132	0.156
  		D	0.125	0.13	0.136	0.144	0.156	0.17	0.185
  		V	0.363	0.375	0.39	0.413	0.443	0.475	0.512
  		L	0.835	0.861	0.894	0.941	1.016	1.079	1.184

  ,

  Table 3. Simulated critical values for test statistics for Weibull distribution based on the median-order statistic.

  r	m	Tests	Significant Level $\mathit{\gamma }$
  			0.25	0.2	0.15	0.1	0.05	0.025	0.01
  10	2	$A^2$	0.46	0.496	0.54	0.598	0.714	0.827	0.977
  		$W^2$	0.073	0.079	0.088	0.098	0.117	0.138	0.164
  		$U^2$	0.069	0.076	0.083	0.093	0.109	0.129	0.156
  		D	0.209	0.218	0.228	0.241	0.261	0.278	0.301
  		V	0.36	0.373	0.389	0.411	0.44	0.472	0.519
  		L	1.012	1.044	1.08	1.134	1.223	1.328	1.519
  	3	$A^2$	0.465	0.5	0.547	0.617	0.717	0.842	1.023
  		$W^2$	0.074	0.08	0.09	0.101	0.121	0.142	0.176
  		$U^2$	0.07	0.076	0.085	0.096	0.114	0.133	0.164
  		D	0.212	0.22	0.231	0.244	0.264	0.285	0.309
  		V	0.363	0.377	0.394	0.415	0.449	0.482	0.527
  		L	1.014	1.046	1.082	1.133	1.221	1.337	1.519
  	4	$A^2$	0.468	0.508	0.556	0.619	0.746	0.9	1.235
  		$W^2$	0.075	0.081	0.09	0.102	0.126	0.153	0.221
  		$U^2$	0.071	0.077	0.085	0.097	0.118	0.141	0.199
  		D	0.212	0.221	0.231	0.245	0.267	0.291	0.345
  		V	0.365	0.377	0.391	0.416	0.454	0.492	0.592
  		L	1.02	1.051	1.09	1.145	1.24	1.358	1.827
  20	2	$A^2$	0.46	0.495	0.546	0.614	0.732	0.837	1.016
  		$W^2$	0.071	0.079	0.087	0.099	0.118	0.141	0.171
  		$U^2$	0.068	0.075	0.082	0.093	0.112	0.132	0.158
  		D	0.151	0.158	0.165	0.175	0.189	0.201	0.218
  		V	0.266	0.277	0.29	0.307	0.334	0.359	0.387
  		L	0.892	0.92	0.957	1.01	1.096	1.173	1.277
  	3	$A^2$	0.478	0.517	0.575	0.664	0.818	1.084	1.982
  		$W^2$	0.074	0.081	0.09	0.103	0.123	0.143	0.169
  		$U^2$	0.07	0.076	0.085	0.096	0.116	0.134	0.158
  		D	0.152	0.159	0.166	0.177	0.192	0.204	0.217
  		V	0.268	0.279	0.294	0.312	0.338	0.363	0.387
  		L	0.9	0.926	0.964	1.014	1.087	1.171	1.277
  	4	$A^2$	0.469	0.511	0.563	0.63	0.739	0.843	0.987
  		$W^2$	0.074	0.082	0.091	0.103	0.122	0.143	0.167
  		$U^2$	0.07	0.077	0.086	0.097	0.115	0.134	0.158
  		D	0.154	0.159	0.167	0.177	0.191	0.204	0.223
  		V	0.27	0.281	0.294	0.311	0.34	0.362	0.398
  		L	0.9	0.93	0.966	1.015	1.093	1.181	1.314
  30	2	$A^2$	0.47	0.505	0.559	0.625	0.748	0.851	0.882
  		$W^2$	0.074	0.081	0.09	0.104	0.124	0.144	0.172
  		$U^2$	0.071	0.076	0.085	0.097	0.118	0.135	0.158
  		D	0.126	0.131	0.138	0.146	0.159	0.17	0.183
  		V	0.225	0.236	0.247	0.262	0.288	0.307	0.333
  		L	0.842	0.869	0.904	0.952	1.027	1.103	1.218
  	3	$A^2$	0.466	0.503	0.55	0.62	0.747	0.873	1.028
  		$W^2$	0.073	0.08	0.089	0.101	0.124	0.15	0.177
  		$U^2$	0.069	0.075	0.084	0.096	0.117	0.137	0.165
  		D	0.127	0.132	0.138	0.146	0.159	0.172	0.185
  		V	0.226	0.236	0.246	0.261	0.287	0.311	0.34
  		L	0.839	0.866	0.9	0.947	1.032	1.117	1.224
  	4	$A^2$	0.471	0.515	0.564	0.635	0.755	0.889	1.034
  		$W^2$	0.074	0.082	0.091	0.105	0.125	0.148	0.171
  		$U^2$	0.07	0.077	0.086	0.098	0.118	0.138	0.161
  		D	0.126	0.131	0.138	0.147	0.159	0.171	0.187
  		V	0.225	0.236	0.248	0.265	0.287	0.31	0.341
  		L	0.841	0.873	0.912	0.957	1.04	1.122	1.207

  and

  Table 4. Simulated critical values for test statistics for Weibull distribution based on minimum-order statistic.

  r	m	Tests	Significant Level $\mathit{\gamma }$
  			0.25	0.2	0.15	0.1	0.05	0.025	0.01
  10	2	$A^2$	0.464	0.495	0.540	0.607	0.724	0.826	0.997
  		$W^2$	0.073	0.079	0.087	0.099	0.119	0.139	0.166
  		$U^2$	0.069	0.076	0.083	0.095	0.113	0.132	0.154
  		D	0.209	0.217	0.226	0.238	0.259	0.277	0.302
  		V	0.362	0.374	0.39	0.41	0.442	0.474	0.519
  		L	1.015	1.043	1.082	1.135	1.234	1.341	1.523
  	3	$A^2$	0.469	0.504	0.554	0.617	0.723	0.817	0.956
  		$W^2$	0.073	0.08	0.088	0.1	0.12	0.137	0.161
  		$U^2$	0.07	0.076	0.084	0.095	0.114	0.13	0.152
  		D	0.21	0.218	0.237	0.24	0.26	0.276	0.297
  		V	0.364	0.377	0.392	0.41	0.441	0.474	0.509
  		L	1.023	1.056	1.097	1.148	1.229	1.344	1.506
  	4	$A^2$	0.469	0.504	0.547	0.617	0.739	0.848	0.985
  		$W^2$	0.074	0.08	0.088	0.099	0.122	0.141	0.166
  		$U^2$	0.071	0.077	0.085	0.095	0.114	0.132	0.157
  		D	0.21	0.218	0.228	0.241	0.261	0.277	0.295
  		V	0.364	0.377	0.395	0.414	0.445	0.475	0.513
  		L	1.02	1.05	1.089	1.146	1.254	1.37	1.517
  20	2	$A^2$	0.476	0.515	0.568	0.636	0.742	0.853	1.009
  		$W^2$	0.074	0.081	0.09	0.103	0.122	0.143	0.171
  		$U^2$	0.07	0.077	0.086	0.098	0.116	0.135	0.16
  		D	0.153	0.159	0.165	0.176	0.19	0.204	0.219
  		V	0.27	0.28	0.294	0.309	0.337	0.363	0.391
  		L	0.905	0.934	0.972	1.022	1.099	1.185	1.29
  	3	$A^2$	0.466	0.506	0.556	0.623	0.744	0.866	1.027
  		$W^2$	0.073	0.08	0.089	0.101	0.123	0.143	0.171
  		$U^2$	0.07	0.076	0.085	0.096	0.116	0.136	0.158
  		D	0.151	0.158	0.165	0.175	0.189	0.203	0.22
  		V	0.269	0.279	0.292	0.309	0.333	0.36	0.39
  		L	0.898	0.928	0.963	1.01	1.097	1.185	1.34
  	4	$A^2$	0.474	0.515	0.563	0.632	0.75	0.85	1.015
  		$W^2$	0.074	0.081	0.09	0.102	0.122	0.141	0.166
  		$U^2$	0.071	0.078	0.086	0.096	0.114	0.135	0.158
  		D	0.152	0.158	0.165	0.175	0.19	0.202	0.215
  		V	0.269	0.279	0.292	0.31	0.336	0.357	0.389
  		L	0.903	0.93	0.967	1.019	1.106	1.198	1.324
  30	2	$A^2$	0.468	0.503	0.552	0.621	0.735	0.825	0.986
  		$W^2$	0.072	0.079	0.088	0.1	0.121	0.138	0.166
  		$U^2$	0.096	0.076	0.083	0.095	0.114	0.132	0.156
  		D	0.125	0.131	0.136	0.144	0.155	0.168	0.182
  		V	0.226	0.234	0.245	0.258	0.28	0.303	0.334
  		L	0.84	0.865	0.903	0.955	1.028	1.105	1.225
  	3	$A^2$	0.47	0.511	0.557	0.625	0.737	0.872	1.018
  		$W^2$	0.072	0.08	0.089	0.102	0.122	0.144	0.174
  		$U^2$	0.069	0.077	0.085	0.097	0.116	0.136	0.162
  		D	0.126	0.131	0.137	0.144	0.157	0.168	0.183
  		V	0.225	0.235	0.246	0.26	0.283	0.304	0.333
  		L	0.84	0.87	0.902	0.951	1.028	1.106	1.214
  	4	$A^2$	0.465	0.505	0.555	0.619	0.746	0.876	1.015
  		$W^2$	0.072	0.079	0.088	0.101	0.122	0.141	0.176
  		$U^2$	0.068	0.075	0.084	0.096	0.116	0.134	0.163
  		D	0.124	0.129	0.136	0.144	0.156	0.167	0.178
  		V	0.223	0.232	0.244	0.259	0.283	0.303	0.325
  		L	0.838	0.867	0.902	0.951	1.024	1.121	1.218

  show the critical values for test statistics KS, CvM, AD, K, W, and LS under RSS (using minimum, median, and maximum order statistics).
• If the calculated values of the test statistics exceed the tabulated value, the null hypothesis of maximum, median, and minimum Weibull distribution should be rejected at the chosen significance level.

The following points are clear from Table 1.

• When r increases, the critical values of $D,V$, and L decrease when $\gamma$ increases. The critical values of all tests decrease when $\gamma$ decreases.
• For different values of r, the critical values of $W^2$, $A^2$, and $U^2$ vary but increase when $\gamma$ decreases.
• The critical values of the L test statistic are the highest, but critical values of the $W^2$ test statistic are the smallest.

The following points are clear from Table 2, Table 3 and Table 4:

• For a fixed sample size, the critical values of the ith-order statistic for all test statistics vary for different values of a set size but they increase when $\gamma$ decreases.
• For a fixed set size, the critical values of $D,V$, and L decrease while r increases, and they increase when $\gamma$ decreases. Additionally, the critical values of $W^2$, $A^2$, and $U^2$ vary for different values of r, but they increase when $\gamma$ decreases.
• The critical values for the median-order statistic lie between those of the maximum- and minimum-order statistics.
• The critical values of the L test statistic are the highest, but the critical values of the $W^2$ test statistic are the smallest.

5. Power Study

In this section, a power study is conducted to determine the null hypothesis power of the six test statistics using two sampling techniques: SRS and RSS. A power comparison is carried out between the KS, CvM, AD, K, W, and LS test statistics for the Weibull distribution. In addition, efficiency will be determined as the ratio of the two approaches’ powers.

The probability that a statistic will lead to the rejection of the null hypothesis $H_0$ when it is untrue is defined as the power of a GOF. The power of a GOF at the significance level is indicated by ($1-\beta$), where $\beta$ is the probability of making a type II error and rejecting a false null hypothesis. The power function may be used to determine how good a test is. A test’s power function is good when its value is close to one.

A power comparison is made among KS, CvM, AD, K, W, and LS test statistics for the Weibull distribution with an unknown shape parameter $\alpha$ and an unknown scale parameter $\lambda$ via SRS and RSS. First, the null hypothesis $H_0$ that SRS $X_1,X{}_2,\dots ,X_r$ arises from a Weibull distribution with undetermined shape and scale parameters is tested through a simulated study. The alternative hypothesis, $H_1$ is that the sample follows a different distribution.

Second, the null hypothesis $H_0$ is that RSS $Y_1,Y{}_2,\dots ,Y_r$ comes from minimum, median, or maximum of Weibull distribution with unspecified shape and scale parameters. For the alternative hypothesis $H_1$, that the sample follows the minimum, median, or maximum of other distributions. The alternative distributions investigated in this study, are as follows:

• Uniform distribution, denoted by U(1,3).
• Normal distribution, denoted by N(3,1).
• Log-normal distribution, denoted by LN(3,1).
• Logistic distribution, denoted by Log(3,1).
• Chi-square distribution, denoted by Chi(5).

The steps below outline the technique for establishing tests under five different scenarios:

• From the different distributions, an SRS of size r = 10(10)30 is created. Additionally, a random sample $Y_1,Y{}_2,\dots ,Y_r$ of size r = 10(10)30 with set sizes ($2m-1$), where $m=1,2,3,4$, is generated from the selected alternatives distributions.
• Using the critical values in Table 1 via SRS, the test statistics are computed, and the fraction of rejection is presented as the power for that condition. The test statistics are calculated using the critical values given in Table 2, Table 3 and Table 4, and the proportion of rejection is reported as the power for that situation.
• If the estimated value of test statistics exceeds the corresponding tabulated critical values for a particular distribution and significance level $\gamma =0.01,0.05,$ and $0.1$, the hypothesis $H_0$ is rejected.
• To obtain separate sets of test statistics, repeat the above steps 5000 times.
• The power of each test is calculated by dividing the number of null hypothesis rejections by 5000.
• Table 5. Power of tests for Weibull distribution based on SRS.

  Alternatives	Tests	Sample Size (r)
  		10			20			30
  		Significant Level $\mathit{\gamma }$			Significant Level $\mathit{\gamma }$			Significant Level $\mathit{\gamma }$
  		0.01	0.05	0.1	0.01	0.05	0.1	0.01	0.05	0.1
  U(1,3)	$A^2$	0.351	0.52	0.615	0.349	0.522	0.626	0.359	0.552	0.654
  	$W^2$	0.324	0.489	0.585	0.323	0.493	0.592	0.34	0.518	0.621
  	$U^2$	0.076	0.204	0.303	0.083	0.21	0.315	0.069	0.215	0.327
  	D	0.243	0.413	0.512	0.129	0.257	0.368	0.268	0.444	0.551
  	V	0.127	0.251	0.343	0.238	0.411	0.519	0.155	0.296	0.387
  	L	0.287	0.47	0.584	0.296	0.467	0.565	0.271	0.451	0.552
  N(3,1)	$A^2$	0.349	0.52	0.616	0.353	0.519	0.625	0.338	0.538	0.633
  	$W^2$	0.322	0.492	0.584	0.331	0.488	0.588	0.322	0.499	0.601
  	$U^2$	0.079	0.212	0.32	0.08	0.21	0.309	0.076	0.207	0.309
  	D	0.241	0.404	0.521	0.135	0.261	0.365	0.252	0.43	0.533
  	V	0.127	0.247	0.359	0.243	0.412	0.507	0.144	0.285	0.373
  	L	0.277	0.477	0.572	0.286	0.461	0.563	0.266	0.44	0.532
  LN(3,1)	$A^2$	0.333	0.505	0.598	0.371	0.536	0.65	0.339	0.542	0.638
  	$W^2$	0.311	0.466	0.564	0.345	0.51	0.609	0.324	0.51	0.61
  	$U^2$	0.072	0.184	0.279	0.091	0.223	0.327	0.075	0.217	0.318
  	D	0.228	0.39	0.504	0.148	0.28	0.383	0.263	0.433	0.535
  	V	0.127	0.236	0.333	0.263	0.433	0.527	0.148	0.3	0.383
  	L	0.267	0.451	0.553	0.304	0.482	0.582	0.266	0.441	0.541
  Chi(5)	$A^2$	0.341	0.513	0.612	0.369	0.54	0.639	0.34	0.537	0.629
  	$W^2$	0.316	0.484	0.578	0.341	0.505	0.603	0.323	0.498	0.601
  	$U^2$	0.069	0.197	0.296	0.089	0.218	0.319	0.072	0.208	0.317
  	D	0.237	0.403	0.508	0.134	0.266	0.373	0.25	0.428	0.53
  	V	0.119	0.24	0.336	0.251	0.427	0.526	0.142	0.278	0.372
  	L	0.286	0.473	0.565	0.303	0.469	0.574	0.27	0.448	0.539
  Log(3,1)	$A^2$	0.347	0.514	0.611	0.378	0.534	0.638	0.337	0.533	0.63
  	$W^2$	0.324	0.486	0.582	0.355	0.513	0.606	0.321	0.491	0.586
  	$U^2$	0.077	0.2	0.303	0.092	0.223	0.322	0.074	0.208	0.304
  	D	0.24	0.406	0.513	0.146	0.277	0.381	0.25	0.43	0.523
  	V	0.13	0.252	0.343	0.258	0.427	0.53	0.133	0.277	0.362
  	L	0.285	0.468	0.566	0.309	0.479	0.578	0.266	0.446	0.541

  shows the power results for tests via SRS at the significant level $\gamma =0.01,0.05$, and $0.1$.

  Table 6. Power of tests for Weibull distribution based on maximum-order statistic at $\gamma =0.05$.

  Alternatives	Tests	Sample Size (r)
  		30			20			10
  		m			m			m
  		4	3	2	4	3	2	4	3	2
  U(1,3)	$A^2$	0.534	0.524	0.509	0.531	0.546	0.54	0.542	0.533	0.539
  	$W^2$	0.449	0.469	0.477	0.504	0.502	0.488	0.502	0.496	0.503
  	$U^2$	0.293	0.197	0.2	0.22	0.219	0.207	0.214	0.225	0.213
  	D	0.424	0.389	0.397	0.412	0.42	0.411	0.424	0.419	0.424
  	V	0.42	0.383	0.393	0.407	0.404	0.405	0.41	0.404	0.409
  	L	0.499	0.484	0.463	0.476	0.484	0.49	0.473	0.449	0.46
  N(3,1)	$A^2$	0.523	0.523	0.528	0.529	0.541	0.522	0.541	0.542	0.543
  	$W^2$	0.479	0.478	0.495	0.493	0.502	0.477	0.493	0.508	0.505
  	$U^2$	0.288	0.205	0.218	0.198	0.213	0.201	0.217	0.215	0.208
  	D	0.41	0.394	0.422	0.399	0.418	0.407	0.434	0.427	0.424
  	V	0.4	0.391	0.411	0.392	0.404	0.398	0.412	0.411	0.406
  	L	0.484	0.473	0.478	0.471	0.484	0.47	0.474	0.463	0.457
  LN(3,1)	$A^2$	0.544	0.537	0.535	0.534	0.533	0.522	0.553	0.547	0.54
  	$W^2$	0.449	0.486	0.509	0.496	0.496	0.483	0.509	0.513	0.507
  	$U^2$	0.3	0.19	0.217	0.205	0.217	0.208	0.231	0.22	0.215
  	D	0.42	0.41	0.434	0.402	0.412	0.416	0.436	0.433	0.432
  	V	0.412	0.404	0.425	0.392	0.398	0.409	0.418	0.415	0.41
  	L	0.505	0.498	0.484	0.482	0.474	0.476	0.483	0.468	0.461
  Chi(5)	$A^2$	0.536	0.526	0.523	0.534	0.541	0.541	0.541	0.562	0.542
  	$W^2$	0.492	0.474	0.496	0.502	0.494	0.497	0.498	0.527	0.509
  	$U^2$	0.301	0.191	0.21	0.214	0.218	0.208	0.218	0.235	0.211
  	D	0.418	0.386	0.417	0.413	0.412	0.428	0.43	0.445	0.426
  	V	0.412	0.382	0.411	0.403	0.397	0.421	0.411	0.431	0.408
  	L	0.489	0.484	0.474	0.481	0.48	0.492	0.469	0.486	0.463
  Log(3,1)	$A^2$	0.544	0.526	0.513	0.527	0.551	0.53	0.544	0.554	0.541
  	$W^2$	0.51	0.477	0.483	0.488	0.522	0.482	0.501	0.52	0.512
  	$U^2$	0.303	0.203	0.2	0.199	0.225	0.201	0.228	0.228	0.222
  	D	0.433	0.387	0.411	0.4	0.432	0.404	0.426	0.434	0.437
  	V	0.426	0.381	0.405	0.388	0.42	0.399	0.412	0.415	0.422
  	L	0.513	0.481	0.468	0.47	0.492	0.48	0.479	0.467	0.468

  ,

  Table 7. Power of tests for Weibull distribution based on median-order statistic at $\gamma =0.05$.

  Alternatives	Tests	Sample Size (r)
  		30			20			10
  		m			m			m
  		4	3	2	4	3	2	4	3	2
  U(1,3)	$A^2$	0.502	0.534	0.527	0.517	0.459	0.536	0.518	0.539	0.545
  	$W^2$	0.468	0.494	0.468	0.478	0.472	0.503	0.472	0.497	0.515
  	$U^2$	0.177	0.199	0.202	0.206	0.199	0.209	0.187	0.211	0.235
  	D	0.38	0.406	0.398	0.402	0.399	0.42	0.394	0.408	0.421
  	V	0.374	0.401	0.392	0.384	0.399	0.409	0.369	0.383	0.404
  	L	0.457	0.484	0.492	0.467	0.469	0.476	0.446	0.46	0.466
  N(3,1)	$A^2$	0.511	0.523	0.518	0.527	0.458	0.528	0.52	0.54	0.543
  	$W^2$	0.467	0.486	0.481	0.488	0.485	0.504	0.482	0.5	0.517
  	$U^2$	0.189	0.203	0.189	0.209	0.204	0.221	0.2	0.199	0.239
  	D	0.384	0.399	0.393	0.403	0.395	0.414	0.41	0.417	0.431
  	V	0.38	0.393	0.385	0.383	0.39	0.402	0.38	0.387	0.433
  	L	0.466	0.475	0.481	0.472	0.476	0.468	0.442	0.469	0.496
  LN(3,1)	$A^2$	0.503	0.533	0.509	0.526	0.474	0.519	0.52	0.542	0.543
  	$W^2$	0.472	0.488	0.468	0.493	0.489	0.496	0.478	0.502	0.513
  	$U^2$	0.185	0.199	0.188	0.207	0.201	0.214	0.197	0.202	0.232
  	D	0.384	0.404	0.38	0.409	0.405	0.414	0.398	0.415	0.436
  	V	0.381	0.399	0.372	0.393	0.399	0.401	0.372	0.387	0.417
  	L	0.455	0.49	0.476	0.473	0.488	0.462	0.451	0.469	0.461
  Chi(5)	$A^2$	0.526	0.516	0.52	0.526	0.454	0.531	0.523	0.543	0.553
  	$W^2$	0.492	0.481	0.468	0.491	0.472	0.502	0.48	0.502	0.522
  	$U^2$	0.203	0.197	0.191	0.207	0.188	0.223	0.197	0.209	0.227
  	D	0.4	0.388	0.387	0.408	0.386	0.415	0.403	0.42	0.44
  	V	0.396	0.385	0.377	0.39	0.382	0.407	0.376	0.397	0.422
  	L	0.477	0.469	0.478	0.478	0.468	0.474	0.452	0.461	0.461
  Log(3,1)	$A^2$	0.521	0.514	0.522	0.533	0.47	0.529	0.519	0.55	0.554
  	$W^2$	0.492	0.482	0.468	0.503	0.492	0.506	0.486	0.505	0.519
  	$U^2$	0.201	0.197	0.189	0.207	0.213	0.212	0.193	0.206	0.234
  	D	0.395	0.394	0.396	0.422	0.404	0.415	0.407	0.423	0.437
  	V	0.388	0.391	0.389	0.403	0.399	0.406	0.383	0.395	0.413
  	L	0.475	0.471	0.483	0.478	0.481	0.469	0.441	0.47	0.476

  and

  Table 8. Power of tests for Weibull distribution based on minimum-order statistic at $\gamma =0.05$.

  Alternatives	Tests	Sample Size (r)
  		30			20			10
  		m			m			m
  		4	3	2	4	3	2	4	3	2
  U(1,3)	$A^2$	0.511	0.518	0.514	0.516	0.523	0.517	0.52	0.536	0.546
  	$W^2$	0.475	0.481	0.492	0.489	0.489	0.485	0.49	0.499	0.512
  	$U^2$	0.197	0.197	0.206	0.207	0.196	0.201	0.207	0.212	0.209
  	D	0.4	0.402	0.405	0.417	0.417	0.411	0.425	0.432	0.445
  	V	0.388	0.399	0.398	0.407	0.415	0.398	0.403	0.402	0.413
  	L	0.475	0.473	0.468	0.456	0.462	0.426	0.434	0.461	0.464
  N(3,1)	$A^2$	0.507	0.526	0.523	0.51	0.518	0.527	0.521	0.541	0.533
  	$W^2$	0.474	0.488	0.492	0.484	0.49	0.495	0.495	0.515	0.499
  	$U^2$	0.196	0.199	0.209	0.211	0.207	0.207	0.209	0.218	0.211
  	D	0.396	0.407	0.42	0.406	0.423	0.404	0.428	0.443	0.431
  	V	0.387	0.405	0.411	0.396	0.418	0.394	0.393	0.42	0.401
  	L	0.476	0.481	0.49	0.452	0.467	0.47	0.426	0.474	0.439
  LN(3,1)	$A^2$	0.518	0.529	0.529	0.516	0.525	0.524	0.531	0.524	0.536
  	$W^2$	0.482	0.483	0.5	0.492	0.494	0.494	0.49	0.496	0.505
  	$U^2$	0.2	0.204	0.211	0.202	0.202	0.202	0.216	0.216	0.213
  	D	0.407	0.403	0.428	0.414	0.428	0.412	0.425	0.429	0.436
  	V	0.394	0.401	0.42	0.405	0.421	0.4	0.417	0.405	0.403
  	L	0.485	0.482	0.494	0.457	0.472	0.468	0.433	0.448	0.457
  Chi(5)	$A^2$	0.525	0.518	0.536	0.508	0.519	0.533	0.516	0.541	0.527
  	$W^2$	0.495	0.477	0.492	0.48	0.49	0.498	0.49	0.508	0.501
  	$U^2$	0.206	0.188	0.215	0.205	0.208	0.211	0.203	0.205	0.212
  	D	0.422	0.396	0.431	0.407	0.425	0.421	0.425	0.436	0.442
  	V	0.409	0.392	0.423	0.398	0.418	0.406	0.395	0.406	0.403
  	L	0.491	0.468	0.504	0.462	0.474	0.482	0.437	0.463	0.45
  Log(3,1)	$A^2$	0.528	0.538	0.517	0.509	0.521	0.521	0.514	0.539	0.539
  	$W^2$	0.494	0.506	0.496	0.489	0.489	0.494	0.49	0.505	0.509
  	$U^2$	0.205	0.206	0.2	0.213	0.204	0.206	0.204	0.214	0.207
  	D	0.422	0.42	0.416	0.417	0.424	0.413	0.425	0.431	0.439
  	V	0.413	0.419	0.407	0.406	0.418	0.399	0.394	0.407	0.404
  	L	0.495	0.493	0.472	0.456	0.477	0.469	0.427	0.447	0.444

  show the power results for the maximum, median, and minimum tests at the significance level $\gamma =0.05$, respectively.
• The power efficiency is used to test the behavior of RSS test statistics relative to SRS test statistics as the following:

  $$eff(RSS,SRS)=\frac{PowerofRSS\left(T^{*}\right)}{PowerofSRS\left(T\right)}.$$

  (27)
  If the $eff(RSS,SRS)>1$, then the test statistics that depend on RSS are more powerful than SRS. The power efficiency results for the maximum, median, and minimum tests at the significance level are presented in

  Table 9. Efficiency of tests for Weibull distribution based on maximum-order statistic at $\gamma =0.05$.

  Alternatives	Tests	Sample Size (r)
  		30			20			10
  		m			m			m
  		4	3	2	4	3	2	4	3	2
  U(1,3)	$A^2$	1.027	1.008	0.979	1.017	1.046	1.034	0.982	0.966	0.976
  	$W^2$	0.918	0.959	0.975	1.022	1.018	0.99	0.969	0.958	0.971
  	$U^2$	1.436	0.966	0.98	1.048	1.043	0.986	0.995	1.047	0.991
  	D	1.027	0.942	0.961	1.603	1.634	1.599	0.955	0.944	0.955
  	V	1.673	1.526	1.566	0.99	0.983	0.985	1.385	1.365	1.382
  	L	1.062	1.03	0.985	1.019	1.036	1.049	1.049	0.996	1.02
  N(3,1)	$A^2$	1.006	1.006	1.015	1.019	1.042	1.006	1.006	1.007	1.009
  	$W^2$	0.974	0.972	1.006	1.01	1.029	0.977	0.988	1.018	1.012
  	$U^2$	1.358	0.967	1.028	0.943	1.014	0.957	1.048	1.039	1.005
  	D	1.015	0.975	1.045	1.529	1.602	1.559	1.009	0.993	0.986
  	V	1.619	1.583	1.664	0.951	0.981	0.966	1.446	1.442	1.425
  	L	1.015	0.992	1.002	1.022	1.05	1.02	1.077	1.052	1.039
  LN(3,1)	$A^2$	1.077	1.063	1.059	0.996	0.994	0.974	1.02	1.009	0.996
  	$W^2$	0.964	1.043	1.092	0.973	0.973	0.947	0.998	1.006	0.994
  	$U^2$	1.63	1.033	1.179	0.919	0.973	0.933	1.065	1.014	0.991
  	D	1.077	1.051	1.113	1.436	1.471	1.486	1.007	1	0.998
  	V	1.746	1.712	1.801	0.905	0.919	0.945	1.393	1.383	1.367
  	L	1.12	1.104	1.073	1	0.983	0.988	1.095	1.061	1.045
  Chi(5)	$A^2$	1.045	1.025	1.019	0.989	1.002	1.002	1.007	1.047	1.009
  	$W^2$	1.017	0.979	1.025	0.994	0.978	0.984	1	1.058	1.022
  	$U^2$	1.528	0.97	1.066	0.982	1	0.954	1.048	1.13	1.014
  	D	1.037	0.958	1.035	1.553	1.549	1.609	1.005	1.04	0.995
  	V	1.717	1.592	1.713	0.944	0.93	0.986	1.478	1.55	1.468
  	L	1.034	1.023	1.002	1.026	1.023	1.049	1.047	1.085	1.033
  Log(3,1)	$A^2$	1.058	1.023	0.998	0.987	1.032	0.993	1.021	1.039	1.015
  	$W^2$	1.049	0.981	0.994	0.951	1.018	0.94	1.02	1.059	1.043
  	$U^2$	1.515	1.015	1	0.892	1.009	0.901	1.096	1.096	1.067
  	D	1.067	0.953	1.012	1.444	1.56	1.458	0.991	1.009	1.016
  	V	1.69	1.512	1.607	0.909	0.984	0.934	1.487	1.498	1.523
  	L	1.096	1.028	1	0.981	1.027	1.002	1.074	1.047	1.049

  ,

  Table 10. Efficiency of tests for Weibull distribution based on median-order statistic at $\gamma =0.05$.

  Alternatives	Tests	Sample Size (r)
  		30			20			10
  		m			m			m
  		4	3	2	4	3	2	4	3	2
  U(1,3)	$A^2$	0.965	1.027	1.013	0.99	0.879	1.027	0.938	0.976	0.987
  	$W^2$	0.957	1.01	0.957	0.97	0.957	1.02	0.911	0.959	0.994
  	$U^2$	0.868	0.975	0.99	0.981	0.948	0.995	0.87	0.981	1.093
  	D	0.92	0.983	0.964	1.564	1.553	1.634	0.887	0.919	0.948
  	V	1.49	1.598	1.562	0.934	0.971	0.995	1.247	1.294	1.365
  	L	0.972	1.03	1.047	1	1.004	1.019	0.989	1.02	1.033
  N(3,1)	$A^2$	0.983	1.006	0.996	1.015	0.882	1.017	0.967	1.004	1.009
  	$W^2$	0.949	0.988	0.978	1	0.994	1.033	0.966	1.002	1.036
  	$U^2$	0.892	0.958	0.892	0.995	0.971	1.052	0.966	0.961	1.155
  	D	0.95	0.988	0.973	1.544	1.513	1.586	0.953	0.97	1.002
  	V	1.538	1.591	1.559	0.93	0.947	0.976	1.333	1.358	1.519
  	L	0.977	0.996	1.008	1.024	1.033	1.015	1.005	1.066	0.673
  LN(3,1)	$A^2$	0.996	1.055	1.008	0.981	0.884	0.968	0.959	1	1.002
  	$W^2$	1.013	1.047	1.004	0.967	0.959	0.973	0.937	0.984	1.006
  	$U^2$	1.005	1.082	1.022	0.928	0.901	0.96	0.908	0.931	1.069
  	D	0.985	1.036	0.974	1.461	1.446	1.479	0.919	0.958	1.007
  	V	1.614	1.691	1.576	0.908	0.921	0.926	1.24	1.29	1.39
  	L	1.009	1.086	1.055	0.981	1.012	0.959	1.023	1.063	1.045
  Chi(5)	$A^2$	1.025	1.006	1.014	0.974	0.841	0.983	0.974	1.011	1.03
  	$W^2$	1.017	0.994	0.967	0.972	0.935	0.994	0.964	1.008	1.048
  	$U^2$	1.03	1	0.97	0.95	0.862	1.023	0.947	1.005	1.091
  	D	0.993	0.963	0.96	1.534	1.451	1.56	0.942	0.981	1.028
  	V	1.65	1.604	1.571	0.913	0.895	0.953	1.353	1.428	1.518
  	L	1.008	0.992	1.011	1.019	0.998	1.011	1.009	1.029	1.029
  Log(3,1)	$A^2$	1.014	1	1.016	0.998	0.88	0.991	0.974	1.032	1.039
  	$W^2$	1.012	0.992	0.963	0.981	0.959	0.986	0.99	1.029	1.057
  	$U^2$	1.005	0.985	0.945	0.928	0.955	0.951	0.928	0.99	1.125
  	D	0.973	0.97	0.975	1.523	1.458	1.498	0.947	0.984	1.016
  	V	1.54	1.552	1.544	0.944	0.934	0.951	1.383	1.426	1.491
  	L	1.015	1.006	1.032	0.998	1.004	0.979	0.989	1.054	1.067

  and

  Table 11. Efficiency of tests for Weibull distribution based on minimum-order statistic at $\gamma =0.05$.

  Alternatives	Tests	Sample Size (r)
  		30			20			10
  		m			m			m
  		4	3	2	4	3	2	4	3	2
  U(1,3)	$A^2$	0.983	0.996	0.988	0.989	1.002	0.99	0.942	0.971	0.989
  	$W^2$	0.971	0.984	1.006	0.992	0.992	0.984	0.946	0.963	0.988
  	$U^2$	0.966	0.966	1.01	0.986	0.933	0.957	0.963	0.986	0.972
  	D	0.969	0.973	0.981	1.623	1.623	1.599	0.957	0.973	1.002
  	V	1.546	1.59	1.586	0.99	1.01	0.968	1.361	1.358	1.395
  	L	1.011	1.006	0.996	0.976	0.989	0.912	0.962	1.022	1.029
  N(3,1)	$A^2$	0.975	1.012	1.006	0.983	0.998	1.015	0.968	1.006	0.991
  	$W^2$	0.963	0.992	1	0.992	1.004	1.014	0.992	1.032	1
  	$U^2$	0.925	0.939	0.986	1.005	0.986	0.986	1.01	1.053	1.019
  	D	0.98	1.007	1.04	1.556	1.621	1.548	0.995	1.03	1.002
  	V	1.567	1.64	1.664	0.961	1.015	0.956	1.379	1.474	1.407
  	L	0.998	1.008	1.027	0.98	1.013	1.02	0.968	1.077	0.998
  LN(3,1)	$A^2$	1.026	1.048	1.048	0.963	0.979	0.978	0.98	0.967	0.989
  	$W^2$	1.034	1.036	1.073	0.965	0.969	0.969	0.961	0.973	0.99
  	$U^2$	1.087	1.109	1.147	0.906	0.906	0.906	0.995	0.995	0.982
  	D	1.044	1.033	1.097	1.479	1.529	1.471	0.982	0.991	1.007
  	V	1.669	1.699	1.78	0.935	0.972	0.924	1.39	1.35	1.343
  	L	1.075	1.069	1.095	0.948	0.979	0.971	0.982	1.016	1.036
  Chi(5)	$A^2$	1.023	1.01	1.045	0.941	0.961	0.987	0.961	1.007	0.981
  	$W^2$	1.023	0.986	1.017	0.95	0.97	0.986	0.984	1.02	1.006
  	$U^2$	1.046	0.954	1.091	0.94	0.954	0.968	0.976	0.986	1.019
  	D	1.047	0.983	1.069	1.53	1.598	1.583	0.993	1.019	1.033
  	V	1.704	1.633	1.763	0.932	0.979	0.951	1.421	1.46	1.45
  	L	1.038	0.989	1.066	0.985	1.011	1.028	0.975	1.033	1.004
  Log(3,1)	$A^2$	1.027	1.047	1.006	0.953	0.976	0.976	0.964	1.011	1.011
  	$W^2$	1.016	1.041	1.021	0.953	0.953	0.963	0.998	1.029	1.037
  	$U^2$	1.025	1.03	1	0.955	0.915	0.924	0.981	1.029	0.995
  	D	1.039	1.034	1.025	1.505	1.531	1.491	0.988	1.002	1.021
  	V	1.639	1.663	1.615	0.951	0.979	0.934	1.422	1.469	1.458
  	L	1.058	1.053	1.009	0.952	0.996	0.979	0.957	1.002	0.996

  , respectively.

The following are clear from Table 5:

• The power of SRS for all test statistics varies for different values of r, and it decreases when $\gamma$ decreases.
• The AD test statistic has the highest power for all alternative hypotheses.
• The W test statistic has the smallest power for all alternative hypotheses.

The following are clear from Table 6, Table 7 and Table 8:

• The power of maximum-, median-, and minimum-order statistics for all test statistics varies for different values of r when the set size increases.
• The highest power at all alternative hypotheses is the AD test statistic, and the smallest power at all alternatives is the W test statistic.

The following are clear from Table 9, Table 10 and Table 11:

• For different values of set sizes, the efficiencies of all test statistics vary.
• At sample sizes of 10 and 30, the K test statistic has the maximum efficiency for all alternative hypotheses while, at sample size 20, the Kolmogorov–Smirnov test statistic has the maximum efficiency for all alternative hypotheses.

6. Concluding Remarks

This article proposes many modified goodness-of-fit tests for the Weibull distribution based on the empirical distribution function. The suggested RSS tests are compared with their SRS equivalents. The relevant test statistics’ critical values are computed for each scheme. The power of the suggested goodness-of-fit tests is compared using a variety of alternatives. A Monte Carlo simulation study demonstrated that the Liao and Shimokawa test statistic’s critical value is the highest, while the Watson test statistic’s critical value is the lowest for SRS and RSS. The Kolmogorov–Smirnov test statistic has lower critical values than the Kuiper test statistic. The powers of a set of modified EDF goodness-of-fit tests can be significantly increased if the sample is obtained using the RSS. When the set size grows, the RSS’s power varies for various values of r for all test statistics. The Anderson–Darling test statistic has the maximum power for all alternative hypotheses, whereas the Watson test statistic has the lowest power for all alternative hypotheses. For varying set size values, the efficacy of all test statistics varies. For all alternative hypotheses, the Kuiper test statistic is the most effective for sample sizes of 10 and 30. However, at a sample size of 20, the Kolmogorov–Smirnov test statistic has the highest effectiveness for all alternative hypotheses.