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Goodness-of-fit tests based on generalized Lorenz curve for progressively Type II censored data from a location-scale distributions
Communications for Statistical Applications and Methods 2019;26:191-203
Published online March 31, 2019
© 2019 Korean Statistical Society.

Wonhee Leea, Kyeongjun Lee1,b

aDepartment of Statistics, Daegu University, Korea;
bDivision of Mathematics and Big Data Science, Daegu University, Korea
Correspondence to: 1Division of Mathematics and Big Data Science, Daegu University, 201, Daegudae-ro, Gyeongsansi, Gyeongsangbuk-do 38453, Korea. E-mail: leekj@daegu.ac.kr
Received November 1, 2018; Accepted February 7, 2019.
 Abstract

The problem of examining how well an assumed distribution fits the data of a sample is of significant and must be examined prior to any inferential process. The observed failure time data of items are often not wholly available in reliability and life-testing studies. Lowering the expense and period associated with tests is important in statistical tests with censored data. Goodness-of-fit tests for perfect data can no longer be used when the observed failure time data are progressive Type II censored (PC) data. Therefore, we propose goodness-of-fit test statistics and a graphical method based on generalized Lorenz curve for PC data from a location-scale distribution. The power of the proposed tests is then assessed through Monte Carlo simulations. Finally, we analyzed two real data set for illustrative purposes.

Keywords : generalized Lorenz curve, goodness-of-fit test, location-scale distribution, Monte Carlo simulation, power, progressive Type II censoring
1. Introduction

The problem of examining how well an assumed distribution fits the data of a sample is of significant and must be examined prior to any inferential process. The estimation of data distribution is used to Q-Q plot and histogram a graph. Also, classical methods involving the goodness-of-fit test statistic use empirical distribution function (EDF) test statistics for goodness-of-fit tests in terms of distance measured between the theoretical cumulative distribution function (CDF) and the EDF over the interval (0, 1). We then reject the hypothesized distribution if the distance is far in some metrics.

In reliability and life-testing studies, the observed failure time data of items are often not available. Lowering the expense and period associated with the tests is important in statistical tests with censored data. Among the censoring method, progressive Type II censored (PC) have become quite popular in reliability and life-testing studies.

The PC arises in a reliability and lifetime-testing experiment as follows. R1 surviving items are eliminated from the test at random after 1st observed failure time; in addition, R2 surviving items are then eliminated from the test at random after the 2nd observed failure time. This process continues until, promptly following the mth observed failure time, all the remaining Rm = nR1 – · · · –Rm–1m items are eliminated from the test. In test, the PC scheme R = (R1, R2, . . . , Rm) is pre-fixed. Consequently, the m ordered observed failure times, which we denote by X1:m:n, X2:m:n, . . . , Xm:m:n, are referred to as PC.

When the observed failure time data are PC data, the goodness-of-fit tests for perfect data can no longer be used. In this motive, the goodness-of-fit test under PC has received the attention from numerous authors.

Balakrishnan et al. (2002) suggested a goodness-of-fit test for the exponential distribution (ED) based on spacings from PC data, while Balakrishnan et al. (2004) subsequently extended their method to general location-scale distributions. Wang (2008) proposed another goodness-of-fit test for the ED under PC data. Pakyari and Balakrishnan (2012) proposed a modification to the EDF goodness-of-fit statistics under PC data. Pakyari and Balakrishnan (2013) recently employed a modification to the classical EDF test statistics based on order statistics, making it suitable for PC data.

In this paper, we suggest a goodness-of-fit test statistics and graphical method based on a generalized Lorenz curve for PC data from a location-scale distribution. The rest of this paper is organized as follows.

The detail explanation of the generalized Lorenz curve and proposed goodness-of-fit test procedures are presented in Section 2. We also propose a new graphical method which uses a generalized Lorenz curve. In Section 3, the power of the proposed test statistics is assessed through Monte Carlo simulations, and their power is compared with that of the test process proposed earlier by Pakyari and Balakrishnan (2013). We found the suggested goodness-of-fit tests to be more powerful than (or as good as) the test statistics of Pakyari and Balakrishnan (2013) for different choices of sample sizes and PC schemes. Section 4 analyzed two real data sets and Section 5 provides the conclusion.

2. Proposed tests

2.1. Generalized Lorenz curve

Lorenz curve provides the means to evaluate income or wealth disparity between two distributions. Atkinson’s theorem (1970) was the first result to give terms under which such a Lorenz inequality comparison has normative significance. In the case of an increasing and strict concave utility function, that theorem indicates that one prefer a distribution with dominating Lorenz curves do not cross. Gastwirth (1971) presented an alternative definition of the Lorenz curve in terms of the inverse of discrete variables as well as continuous ones. Let 꽦 denotes the CDF of income or wealth distribution. Then the income or wealth is assumed to be non-negative. For a given percentile p, let

F-1(p)=infy{yF(y)p}, 듼 듼 듼0p1,

denotes the inverse CDF corresponding to 꽦 . It shall be assumed all through that 꽦 is continuous CDF with finite support.

The Lorenz curves corresponding to the distributions with 꽦 is defined as

L(p)=1μ0F-1(p)xdF(x),

where μ denotes the mean of the distribution with 꽦 .

Atkinson’s theorem has the restriction that both the distributions have same means. In this reason, Shorrocks (1983) extended Atkinson’s theorem to unequal means. Shorrocks (1983) proposed the generalized Lorenz curve as a method to simultaneously account for differences in income mean and income disparity. The generalized Lorenz curve is a standard Lorenz curve scaled up by the mean. In this curve, the scaling reveals a dominant relationship that is not apparent from a checkup of means and Lorenz curves on their own. The height of generalized Lorenz curve shows the incomes grades, while the curvature of the generalized Lorenz curve shows the income inequality grade (Arora and Jain, 2006).

The generalized Lorenz curve corresponding to 꽦 is the Lorenz curve scaled up by the mean μ and is given by

GL(p)=μL(p)=0F-1(p)xdF(x), 듼 듼 듼0p1.

2.2. Proposed tests based on generalized Lorenz curve

Let X1:m:n < X2:m:n < ··· < Xm:m:n be the PC data with PC scheme R = (R1, R2, . . . , Rm) from a location-scale distribution. Also, let the PC data have a location-scale distribution with a probability density function (PDF)


f(x;μ,σ)=1σg(x-μσ),

where g(·) is the known function, but μ and σ are the unknown location scale parameter, respectively.

We want to test whether the PC data comes from a location-scale distribution with PDF (2.1) and in testing the null hypothesis

H0:FFθ 듼 듼 듼for some θΘ={(μ,σ)|-<μ<,σ>0},

where 꽦 = 꽦 (x; μ, σ) denote the distribution function.

If Ui:m:n = 꽦 (Xi:m:n; μ, σ), then pi:m:n = E(Ui:m:n) denote the expected value of the ith PC order statistics from the standard uniform distribution, which is given by

pi:m:n=1-j=m-i+1m{j+Rm-j+1++Rmj+1+Rm-j+1++Rm}.

Generalized Lorenz curve assumed that X is a non-negative wealth or incomes. However, all distributions do not have non-negative support. In order to solve this problem, all values of the ordered PC data were subtracted by the value of the first ordered PC data, and then each result was added. Since a generalized Lorenz curve cannot show the characteristics of the skewed distribution, the above result is multiplied by (1 – pj:m:n). Then mGLC is obtained as


mGLC(pj:m:n)=1DX(m)[i=1jDX(i)+(1-pj:m:n)i=1mDX(i)],

where and j = 1, 2, . . . ,m. Using the percentile points of Gumbel distribution, normal distribution, log-gamma distribution with parameter 4, 8, and ∞, and t distribution with 2, 3, and 4 degrees of freedom (Figure 1).

Let 꽦 denote the CDF of location-scale distribution, an nGLC+ and nGLC are obtained as


nGLC+(pj:m:n)=mGLC(pj:m:n)mGLCF(pj:m:n),nGLC-(pj:m:n)=mGLCF(pj:m:n)mGLC(pj:m:n),

where

mGLCF(pj:m:n)=1DF(m)[i=1jDF(i)+(1-pj:m:n)i=1nDF(i)]

and .

Here, the mGLC, nGLC+, and nGLC are clearly location-scale invariant.

Now, we propose test statistics by using nGLC+ and nGLC.


Gm:n+=max1jm[|1-nGLC+(pj:m:n)|], 듼 듼 듼Gm:n-=max1jm[|1-nGLC-(pj:m:n)|]

and


Gm:n=Gm:n++Gm:n-.

If the data accurately follows a location-scale distribution, we expect the Gm:n+,Gm:n-, and Gm:n test statistics to be zero. Consequently, large values of Gm:n+,Gm:n-, and Gm:n test statistics lead to the rejection of null hypothesis. Therefore, we reject the null hypothesis if the Gm:n+,Gm:n-, and Gm:n test statistics exceed the corresponding upper tail null critical values. Since Gm:n+,Gm:n-, and Gm:n test statistics has a disadvantage in that its distribution theory is difficult; therefore, the percentile points need to be determined through Monte Carlo simulations because the critical values are not available explicitly.

Next, we propose new plot methods using nGLC+ and nGLC,


G+(pj:m:n)=|1-nGLC+(pj:m:n)|, 듼 듼 듼G-(pj:m:n)=|1-nGLC-(pj:m:n)|.

If the data accurately follows a location-scale distribution, the nGLC+ and nGLC are 1, and and will converge with the x-axis. So, we are going to check if the data will follow the locationscale distribution by using the degree of how much the and are apart from the x-axis.

To check the shapes of and , we consider the normal distribution and Gumbel distribution. Also, we consider the t distribution with 2, 3, and 4 degrees of freedom, and log-gamma distribution with shape parameter 4, 8, and ∞ as alternatives. First, we generate 50 data from Xj:m:n = 꽦−1(Pj:m:n) at Gumbel distribution, normal distribution, log-gamma distribution with shape parameter 4, 8, ∞, and t distribution with 2, 3, and 4 degrees of freedom. Next, we draw the and . The results of and for various distributions appear in Figure 2. For testing the normal distribution, and converge with the x-axis at normal distribution data. In t distribution case, and are apart from the x-axis. For testing the Gumbel distribution, and converge with the x-axis at Gumbel distribution data. In log-gamma distribution case, and are apart from the x-axis.

3. Simulation results

In this section, we assess the power of the proposed tests by comparing the simulated power values with those of the test of Pakyari and Balakrishnan (2013).

First, we introduced Pakyari and Balakrishnan’s test statistics (Pakyari and Balakrishnan, 2013) based on the deviation between order statistics. Let vi:m:n = Ui:m:nαi:m:n denote the deviation between the ith order statistic from its expected value based on the PC data. Pakyari and Balakrishnan’s test statistics are obtained as

Cm:n+=max1imvi:m:n, 듼 듼 듼Cm:n-=max1i(-vi:m:n), 듼 듼 듼Cm:n=max(Cm:n+,Cm:n-),Km:n=Cm:n++Cm:n-, 듼 듼 듼Tm:n(1)=i=1mvi:m:n2m, 듼 듼 듼Tm:n(2)=i=1mvi:m:nm.

Note that Cm:n+,Cm:n-, and Cm:n are closely related to the modified Kolmogorov-Smirnov statistics.

We generated 10,000 samples for different choices of sample sizes and PC schemes. For comparative purposes, various PC schemes were used by Balakrishnan et al. (2004) and Pakyari and Balakrishnan (2013) in their study were considered here (PC Scheme Table in Balakrishnan et al. (2004)).

First, we consider a normal distribution with the parent distribution with the PDF

f(x;μ,σ2)=1σ2πexp{-(x-μ)22σ2}, 듼 듼 듼-<x<,-<μ<,σ>0

as the parent distribution.

For testing the normal distribution, the alternative distribution is considered t distribution with PDF

f(x;ν)=Γ[(ν+1)/2]νπΓ(ν/2)(1+x2ν), 듼 듼 듼-<x<,ν>0.

Next, we consider a Gumbel distribution with the parent distribution with the PDF

f(x;μ,σ2)=1σexp{x-μσ}exp[-exp{x-μσ}], 듼 듼 듼-<x<,-<μ<,σ>0

as the parent distribution.

For testing the normal distribution, the alternative distribution is considered log-gamma distribution with PDF

f(x;κ)=κκ-12Γ(κ)exp[κx-κexp{xκ}], 듼 듼 듼-<x<,κ>0.

For a detailed discussion on these distributions, refer to Johnson et al. (1994, 1995).

When the alternative distribution is the distribution from which data are simulated, the rejection probabilities give the power of the test. The power values are presented in Table 1 and Table 2. Table 1 presents the estimated power of the proposed tests when the null hypothesis stipulates normal distribution and the alternative hypothesis corresponds to t distribution with 2, 3, and 4 degrees of freedom. Table 1 shows that the proposed test statistics Gm:n+ and Gm:n possessed better power than Pakyari and Balakrishnan’s (2013) test statistics in a number of situations (indicated in bold). Gm:n+ was found to be better than Pakyari and Balakrishnan’s (2013) test statistics in 46 out of 81 situations, while Gm:n was found better than Pakyari and Balakrishnan’s (2013) test statistics in 31 out of 81 situations. The proposed statistics gained better power as the progressively Type II censored sample size increased. When the data were generated from t distribution with 2 degree of freedom, the proposed statistics gained better power. The proposed statistic Gm:n+ was almost always more powerfull than the proposed statistics Gm:n- and Gm:n.

Table 2 presents the estimated power of the proposed tests when the null hypothesis stipulates Gumbel distribution and the alternative hypothesis corresponds to log-gamma distribution with shape parameters 4, 8 and ∞. When it comes to testing the Gumbel distribution against a log-gamma distribution, the results become somewhat different. Table 2 shows that the proposed test statistics Gm:n- and Gm:n possessed better power than Pakyari and Balakrishnan’s (2013) test statistics in a number of situations (indicated in bold). The proposed test statistics Gm:n- and Gm:n almost always performed better than the Pakyari and Balakrishnan’s (2013) test statistics. Gm:n- was found to be better than Pakyari and Balakrishnan’s (2013) test statistics in 75 out of 81 situations, while Gm:n was found to be better than Pakyari and Balakrishnan’s (2013) test statistics in 52 out of 81 situations. As the PC sample size increased, the proposed statistics gained better power. The proposed statistics gained better power when the data were generated from log-gamma distribution with shape parameter ∞. The proposed statistic Gm:n- was almost always more powerfull than the proposed statistics Gm:n+ and Gm:n.

4. Illustrative examples

For illustrative purposes, we present here two sets of real data analysis using the Pakyari and Balakrishnan’s (2013) test statistics and proposed test statistics. First, the data concerned times to breakdown for an insulating fluid tested at 32kV (Nelson, 1982). We randomly generated a PC sample of size m = 8 from n = 15. The log data and the PC scheme employed are given in Table 3.

The values of the test statistics and the corresponding p-values are presented in Table 4. From Table 4, the p-values given strongly support the null hypothesis of the Gumbel distribution for the data (all p-values are greater than the significance level 0.05). Incidentally, this is in agreement with the findings of Nelson (1982) and Pakyari and Balakrishnan (2013).

It is also able to check as the and . The and of insulating fluid log data appear in Figure 3. From Figure 3, the and of the insulating fluid log data is closed to zero. Thus, and conclude that the insulating fluid log data follows Gumbel distribution.

Next, the data concerned the breaking strength of 23 wire connections. This data was previously studied by Nelson (1982) and King (1971). We randomly generated a PC sample of size m = 10 from n = 20. The data and the PC scheme employed are given in Table 5.

The values of the test statistics and the corresponding p-values are presented in Table 6. From Table 6, shows that the given p-values strongly support the null hypothesis of the normal distribution for the data (all p-values are greater than significance level 0.05) and are agreement with the findings of Nelson (1982), King (1971) and Pakyari and Balakrishnan (2013).

It is also able to check as the and . The and of wire connection strength data are appeared in Figure 4. From Figure 4, the and of wire connection strength data is closed to zero. Thus, and conclude that the wire connection strength data follows normal distribution.

5. Conclusions

The problem of examining how well an assumed distribution fits the data of a sample is of significant and needs to be examined prior to any inferential process. The observed failure time data of items are often not wholly available in reliability and life-testing studies. Lowering the expense and time associated with the tests is important in statistical tests with censored data. Among the censoring method, PC has become quite popular in reliability and life-testing studies. Goodness-of-fit tests for perfect data can no longer be used when the observed failure time data are PC data. Therefore, we propose a goodness-of-fit test statistics and graphical method based on generalized Lorenz curve for PC data from a location-scale distribution.

The proposed test statistics Gm:n+ and Gm:n possessed better power than Pakyari and Balakrishnans (2013) test statistics when the null hypothesis stipulates normal distribution and the alternative hypothesis correspond to t distribution. The proposed statistics gained better power as the PC sample size increased. The proposed statistic Gm:n+ was almost always more powerful than the proposed statistics Gm:n- and Gm:n. The proposed test statistics Gm:n- and Gm:n possessed better power than Pakyari and Balakrishnan’s (2013) test statistics when the null hypothesis stipulates Gumbel distribution and the alternative hypothesis corresponds to log-gamma distribution. The proposed test statistics Gm:n- and Gm:n almost always performed better than the Pakyari and Balakrishnan’s (2013) test statistics. The proposed statistic Gm:n- was almost always more powerful than the proposed statistics Gm:n+ and Gm:n.

We have assumed that the location-scale distributions are Gumbel and normal distribution; however, any other location-scale distribution can also be considered shuch as exponential, half-logistic and Rayleigh distribution.

Acknowledgements

Authors would like to express deep thanks to the Editor-in-Chief and the referees for their helpful comments and suggestions which led to a considerable improvement in the presentation of this paper. This research was supported by the Daegu University Research Grant, 2017.

Figures
Fig. 1. mGLC of various distributions.
Fig. 2. and of various distributions.
Fig. 3. and of insulating fluid log data.
Fig. 4. and of wire connection strength data.
TABLES

Table 1

Estimated power for Student’s t distribution with 2, 3, and 4 degrees of freedom when the null hypothesis is the normal distribution, with α = 0.05

Sch. No. df Cm:n+ Cm:n- Cm:n Km:n Tm:n(1) Tm:n(2) Gm:n+ Gm:n- Gm:n
[1] 2 0.2521 0.0942 0.2690 0.2009 0.2827 0.2653 0.2697 0.0970 0.1556
3 0.1570 0.0677 0.1608 0.1100 0.1700 0.1571 0.1698 0.0757 0.0956
4 0.1179 0.0573 0.1161 0.0775 0.1220 0.1131 0.1258 0.0649 0.0743

[2] 2 0.2493 0.0149 0.1555 0.1390 0.1704 0.1750 0.2443 0.0152 0.1057
3 0.1591 0.0228 0.0922 0.0823 0.0956 0.0973 0.1580 0.0221 0.0602
4 0.1202 0.0278 0.0691 0.0643 0.0712 0.0730 0.1193 0.0261 0.0458

[3] 2 0.2873 0.0138 0.2278 0.1349 0.2379 0.2278 0.2638 0.0139 0.1095
3 0.1801 0.0220 0.1351 0.0771 0.1395 0.1306 0.1623 0.0206 0.0576
4 0.1312 0.0279 0.1000 0.0585 0.0996 0.0924 0.1222 0.0247 0.0430

[4] 2 0.2732 0.1640 0.3368 0.2818 0.3575 0.3465 0.3344 0.1433 0.2684
3 0.1767 0.0976 0.1977 0.1498 0.2113 0.2056 0.2238 0.1059 0.1672
4 0.1333 0.0737 0.1394 0.1023 0.1480 0.1432 0.1681 0.0893 0.1238

[5] 2 0.2883 0.0182 0.2233 0.1757 0.2333 0.2362 0.3452 0.0170 0.2291
3 0.1786 0.0232 0.1295 0.0966 0.1340 0.1313 0.2238 0.0187 0.1330
4 0.1289 0.0274 0.0898 0.0694 0.0937 0.0942 0.1648 0.0238 0.0963

[6] 2 0.0952 0.2990 0.2351 0.2503 0.2805 0.2990 0.3642 0.0948 0.2771
3 0.0495 0.1748 0.1284 0.1403 0.1575 0.1679 0.2448 0.0772 0.1767
4 0.0409 0.1267 0.0882 0.0942 0.1100 0.1179 0.1853 0.0692 0.1317

[7] 2 0.2917 0.2449 0.3717 0.3407 0.4024 0.3961 0.3692 0.1844 0.3213
3 0.1747 0.1358 0.2120 0.1809 0.2307 0.2250 0.2473 0.1347 0.2148
4 0.1269 0.0942 0.1427 0.1188 0.1575 0.1548 0.1916 0.1090 0.1611

[8] 2 0.3022 0.0746 0.2511 0.2143 0.2658 0.2680 0.3899 0.1297 0.3086
3 0.1848 0.0423 0.1411 0.1124 0.1474 0.1495 0.2550 0.0675 0.1884
4 0.1294 0.0359 0.1022 0.0797 0.1032 0.1048 0.1906 0.0477 0.1344

[9] 2 0.1888 0.3016 0.3100 0.3084 0.3580 0.3660 0.3795 0.1779 0.3287
3 0.0985 0.1688 0.1666 0.1573 0.1915 0.1998 0.2547 0.1264 0.2215
4 0.0694 0.1146 0.1109 0.1014 0.1282 0.1332 0.1963 0.1031 0.1646

[10] 2 0.3987 0.0985 0.4124 0.3438 0.4348 0.4164 0.3602 .0814 0.2112
3 0.2492 0.0631 0.2421 0.1779 0.2571 0.2425 0.2259 .0658 0.1170
4 0.1755 0.0532 0.1643 0.1124 0.1764 0.1656 0.1603 .0600 0.0829

[11] 2 0.3425 0.0121 0.2073 0.2006 0.2372 0.2519 0.2943 .0104 0.1654
3 0.2125 0.0178 0.1142 0.1144 0.1304 0.1369 0.1929 .0170 0.0922
4 0.1565 0.0231 0.0818 0.0837 0.0879 0.0923 0.1414 .0216 0.0640

[12] 2 0.2243 0.0507 0.1736 0.1976 0.2571 0.2811 0.4045 0.0075 0.2680
3 0.1199 0.0227 0.0885 0.0929 0.1307 0.1446 0.2626 0.0129 0.1496
4 0.0849 0.0210 0.0640 0.0625 0.0859 0.0925 0.1919 0.0178 0.1009

[13] 2 0.4753 0.2955 0.5458 0.5325 0.5838 0.5823 0.4523 .2183 0.3954
3 0.2994 0.1471 0.3278 0.3063 0.3591 0.3548 0.3096 .1371 0.2560
4 0.2127 0.0972 0.2189 0.1952 0.2422 0.2428 0.2288 .1027 0.1859

[14] 2 0.4975 0.0369 0.4000 0.3512 0.4334 0.4361 0.5370 0.3123 0.4633
3 0.3026 0.0165 0.2228 0.1819 0.2420 0.2426 0.3597 0.1654 0.2881
4 0.2097 0.0191 0.1386 0.1114 0.1530 0.1569 0.2612 0.1067 0.1998

[15] 2 0.2051 0.4403 0.3513 0.3719 0.4325 0.4543 0.5499 0.2212 0.4685
3 0.0826 0.2514 0.1767 0.1897 0.2279 0.2500 0.3734 0.1068 0.2964
4 0.0542 0.1674 0.1090 0.1163 0.1417 0.1603 0.2744 0.0624 0.2064

[16] 2 0.5259 0.4391 0.6198 0.6227 0.6690 0.6716 0.4839 0.3381 0.4580
3 0.3095 0.2205 0.3604 0.3489 0.4048 0.4091 0.3387 0.2141 0.3077
4 0.2098 0.1351 0.2329 0.2180 0.2645 0.2674 0.2547 0.1603 0.2337

[17] 2 0.5191 0.1671 0.4502 0.4194 0.4809 0.4844 0.6110 0.4719 0.5634
3 0.3089 0.0571 0.2416 0.2132 0.2655 0.2682 0.4226 0.2796 0.3699
4 0.2051 0.0342 0.1507 0.1290 0.1657 0.1697 0.3107 0.1884 0.2641

[18] 2 0.5227 0.1991 0.4795 0.4495 0.5115 0.5078 0.5801 0.4442 0.5356
3 0.3164 0.0787 0.2697 0.2402 0.2965 0.2943 0.4047 0.2643 0.3562
4 0.2170 0.0485 0.1753 0.1510 0.1929 0.1939 0.2975 0.1805 0.2557

[19] 2 0.5788 0.2975 0.6264 0.6190 0.6674 0.6634 0.4962 0.2173 0.4232
3 0.3822 0.1402 0.3881 0.3574 0.4280 0.4226 0.3396 0.1245 0.2635
4 0.2686 0.0861 0.2588 0.2348 0.2902 0.2879 0.2502 0.0911 0.1869

[20] 2 0.5498 0.0075 0.4405 0.3891 0.4742 0.4860 0.5434 0.3123 0.4649
3 0.3468 0.0093 0.2435 0.2069 0.2705 0.2742 0.3686 0.1652 0.2901
4 0.2432 0.0144 0.1511 0.1302 0.1724 0.1785 0.2671 0.1068 0.2005

[21] 2 0.2674 0.4125 0.3295 0.4575 0.4744 0.5283 0.6636 0.4549 0.6015
3 0.1095 0.1965 0.1414 0.2236 0.2474 0.2939 0.4615 0.2488 0.3913
4 0.0643 0.1139 0.0787 0.1281 0.1472 0.1870 0.3385 0.1563 0.2719

[22] 2 0.6752 0.5767 0.7532 0.7671 0.8036 0.8063 0.5286 0.4221 0.5077
3 0.4259 0.2963 0.4714 0.4768 0.5335 0.5404 0.3845 0.2738 0.3564
4 0.2870 0.1727 0.3075 0.3027 0.3555 0.3549 0.2934 0.1974 0.2644

[23] 2 0.6723 0.2364 0.6067 0.5688 0.6470 0.6526 0.7508 0.6508 0.7199
3 0.4184 0.0731 0.3340 0.2997 0.3758 0.3833 0.5378 0.4214 0.4980
4 0.2728 0.0350 0.2065 0.1780 0.2312 0.2346 0.4038 0.2862 0.3630

[24] 2 0.5023 0.6206 0.6212 0.6412 0.6853 0.6974 0.5797 0.3829 0.5264
3 0.2472 0.3578 0.3517 0.3644 0.4071 0.4208 0.4113 0.2237 0.3568
4 0.1404 0.2319 0.2202 0.2230 0.2521 0.2624 0.3157 0.1479 0.2645

[25] 2 0.7036 0.6678 0.7824 0.8121 0.8356 0.8377 0.5464 0.4451 0.5247
3 0.4175 0.3670 0.4918 0.5166 0.5589 0.5604 0.3933 0.2989 0.3740
4 0.2674 0.2145 0.3090 0.3205 0.3642 0.3654 0.3035 0.2217 0.2808

[26] 2 0.6517 0.3224 0.6017 0.5986 0.6470 0.6489 0.7391 0.6576 0.7074
3 0.3947 0.1244 0.3322 0.3217 0.3773 0.3746 0.5406 0.4368 0.4996
4 0.2556 0.0644 0.2010 0.1921 0.2362 0.2369 0.4048 0.3060 0.3667

[27] 2 0.6431 0.3751 0.6227 0.6359 0.6808 0.6802 0.6799 0.6003 0.6500
3 0.3934 0.1588 0.3610 0.3561 0.4080 0.4060 0.4933 0.3952 0.4551
4 0.2573 0.0870 0.2257 0.2173 0.2599 0.2586 0.3740 0.2803 0.3362

Table 2

Estimated power for log-gamma distribution with the shape parameter 4, 8, and ∞ when the null hypothesis is the Gumbel distribution, with α = 0.05

Sch. No. df Cm:n+ Cm:n- Cm:n Km:n Tm:n(1) Tm:n(2) Gm:n+ Gm:n- Gm:n
[1] 4 0.0198 0.0878 0.0459 0.0208 0.0182 0.0200 0.0837 0.1045 0.1045
8 0.0139 0.1041 0.0469 0.0159 0.0135 0.0168 0.1041 0.1271 0.1271
0.0074 0.1519 0.0539 0.0121 0.0093 0.0142 0.1610 0.1952 0.1952

[2] 4 0.0342 0.0664 0.0470 0.0489 0.0463 0.0455 0.0520 0.0674 0.0674
8 0.0309 0.0724 0.0458 0.0502 0.0466 0.0452 0.0543 0.0732 0.0729
0.0248 0.0842 0.0452 0.0552 0.0478 0.0461 0.0616 0.0883 0.0879

[3] 4 0.0295 0.0741 0.0462 0.0373 0.0336 0.0337 0.0594 0.0740 0.0740
8 0.0251 0.0829 0.0472 0.0338 0.0316 0.0316 0.0637 0.0827 0.0827
0.0161 0.1016 0.0491 0.0326 0.0282 0.0294 0.0801 0.1055 0.1055

[4] 4 0.0145 0.0930 0.0424 0.0186 0.0166 0.0190 0.0793 0.1174 0.1135
8 0.0103 0.1098 0.0443 0.0169 0.0149 0.0182 0.0995 0.1480 0.1439
0.0050 0.1606 0.0531 0.0166 0.0180 0.0250 0.1745 0.2502 0.2436

[5] 4 0.0279 0.0751 0.0498 0.0464 0.0445 0.0427 0.0501 0.0798 0.0748
8 0.0244 0.0854 0.0503 0.0496 0.0457 0.0455 0.0553 0.0902 0.0843
0.0172 0.1106 0.0541 0.0546 0.0517 0.0523 0.0696 0.1210 0.1137

[6] 4 0.0478 0.0948 0.0686 0.0521 0.0539 0.0529 0.0563 0.1116 0.1029
8 0.0477 0.1127 0.0780 0.0542 0.0582 0.0591 0.0683 0.1403 0.1288
0.0475 0.1593 0.1043 0.0622 0.0784 0.0850 0.1186 0.2273 0.2099

[7] 4 0.0157 0.0965 0.0467 0.0245 0.0236 0.0286 0.0743 0.1422 0.1267
8 0.0115 0.1168 0.0499 0.0246 0.0249 0.0299 0.0948 0.1820 0.1640
0.0080 0.1733 0.0698 0.0310 0.0403 0.0544 0.1851 0.3114 0.2865

[8] 4 0.0230 0.0924 0.0529 0.0425 0.0434 0.0453 0.0495 0.1029 0.0851
8 0.0196 0.1061 0.0558 0.0443 0.0475 0.0503 0.0579 0.1251 0.1053
0.0125 0.1498 0.0708 0.0570 0.0676 0.0747 0.0931 0.1909 0.1646

[9] 4 0.0210 0.0999 0.0513 0.0360 0.0359 0.0410 0.0639 0.1401 0.1192
8 0.0177 0.1225 0.0575 0.0371 0.0391 0.0467 0.0837 0.1798 0.1554
0.0133 0.1667 0.0709 0.0453 0.0524 0.0690 0.1657 0.3079 0.2723

[10] 4 0.0160 0.1106 0.0496 0.0211 0.0153 0.0169 0.1067 0.1265 0.1258
8 0.0110 0.1386 0.0554 0.0178 0.0111 0.0158 0.1397 0.1664 0.1662
0.0048 0.1982 0.0682 0.0179 0.0116 0.0180 0.2453 0.2831 0.2830

[11] 4 0.0346 0.0658 0.0448 0.0540 0.0510 0.0501 0.0503 0.0670 0.0656
8 0.0307 0.0714 0.0452 0.0560 0.0545 0.0526 0.0520 0.0721 0.0705
0.0246 0.0804 0.0463 0.0612 0.0589 0.0553 0.0571 0.0887 0.0851

[12] 4 0.0391 0.0746 0.0606 0.0492 0.0473 0.0464 0.0512 0.0784 0.0742
8 0.0369 0.0848 0.0649 0.0499 0.0477 0.0487 0.0541 0.0884 0.0841
0.0346 0.1016 0.0724 0.0510 0.0517 0.0534 0.0687 0.1158 0.1107

[13] 4 0.0088 0.1331 0.0463 0.0188 0.0152 0.0213 0.1047 0.1762 0.1551
8 0.0056 0.1720 0.0536 0.0216 0.0173 0.0255 0.1504 0.2444 0.2191
0.0033 0.2523 0.0768 0.0327 0.0317 0.0458 0.3205 0.4516 0.4177

[14] 4 0.0223 0.0879 0.0502 0.0540 0.0511 0.0500 0.0492 0.0922 0.0712
8 0.0179 0.1027 0.0531 0.0622 0.0579 0.0550 0.0545 0.1074 0.0851
0.0141 0.1315 0.0606 0.0756 0.0718 0.0715 0.0770 0.1570 0.1234

[15] 4 0.1000 0.0975 0.1136 0.1011 0.0732 0.0648 0.0537 0.1287 0.1030
8 0.1263 0.1189 0.1395 0.1279 0.0884 0.0792 0.0693 0.1663 0.1329
0.1776 0.1622 0.1992 0.1803 0.1302 0.1160 0.1304 0.2871 0.2392

[16] 4 0.0101 0.1354 0.0595 0.0385 0.0361 0.0445 0.1001 0.2078 0.1690
8 0.0084 0.1740 0.0754 0.0484 0.0494 0.0621 0.1504 0.2902 0.2424
0.0101 0.2568 0.1186 0.0781 0.0943 0.1153 0.3562 0.5384 0.4858

[17] 4 0.0164 0.1154 0.0608 0.0587 0.0579 0.0604 0.0514 0.1119 0.0841
8 0.0123 0.1436 0.0723 0.0697 0.0718 0.0753 0.0664 0.1512 0.1080
0.0094 0.1982 0.0964 0.0991 0.1054 0.1114 0.1233 0.2653 0.1989

[18] 4 0.0135 0.1239 0.0642 0.0486 0.0502 0.0561 0.0608 0.1342 0.0988
8 0.0095 0.1569 0.0786 0.0625 0.0666 0.0732 0.0811 0.1819 0.1354
0.0071 0.2253 0.1086 0.0924 0.1059 0.1169 0.1676 0.3409 0.2681

[19] 4 0.0079 0.1565 0.0493 0.0191 0.0118 0.0168 0.1312 0.1969 0.1796
8 0.0042 0.2072 0.0587 0.0213 0.0131 0.0209 0.1942 0.2756 0.2565
0.0021 0.3095 0.0858 0.0376 0.0263 0.0416 0.4020 0.5106 0.4857

[20] 4 0.0249 0.0826 0.0488 0.0580 0.0532 0.0522 0.0480 0.0834 0.0667
8 0.0202 0.0940 0.0513 0.0641 0.0575 0.0578 0.0514 0.0953 0.0775
0.0149 0.1172 0.0569 0.0784 0.0727 0.0684 0.0674 0.1314 0.1047

[21] 4 0.0389 0.0935 0.0757 0.0546 0.0575 0.0541 0.0490 0.1038 0.0809
8 0.0363 0.1117 0.0862 0.0592 0.0632 0.0627 0.0576 0.1304 0.0992
0.0346 0.1406 0.1112 0.0691 0.0778 0.0761 0.0945 0.2091 0.1637

[22] 4 0.0077 0.1668 0.0622 0.0420 0.0449 0.0540 0.1267 0.2540 0.2044
8 0.0056 0.2278 0.0861 0.0597 0.0693 0.0853 0.2079 0.3638 0.3064
0.0097 0.3490 0.1561 0.1056 0.1440 0.1738 0.4846 0.6796 0.6121

[23] 4 0.0161 0.1270 0.0605 0.0660 0.0716 0.0710 0.0560 0.1203 0.0868
8 0.0116 0.1596 0.0727 0.0814 0.0893 0.0919 0.0717 0.1528 0.1188
0.0074 0.2282 0.0996 0.1189 0.1364 0.1426 0.1420 0.2741 0.2156

[24] 4 0.0367 0.1432 0.1017 0.0721 0.0805 0.0815 0.0918 0.2136 0.1623
8 0.0409 0.1875 0.1310 0.0921 0.1099 0.1148 0.1421 0.3013 0.2386
0.0530 0.2742 0.1977 0.1440 0.1853 0.2012 0.3418 0.5488 0.4720

[25] 4 0.0102 0.1730 0.0840 0.0604 0.0740 0.0867 0.1333 0.2618 0.2088
8 0.0133 0.2321 0.1176 0.0895 0.1166 0.1346 0.2188 0.3858 0.3149
0.0339 0.3679 0.2241 0.1639 0.2366 0.2696 0.5248 0.7076 0.6395

[26] 4 0.0136 0.1550 0.0805 0.0768 0.0902 0.0942 0.0671 0.1513 0.1106
8 0.0098 0.2024 0.1045 0.1030 0.1237 0.1305 0.1013 0.2160 0.1587
0.0096 0.2985 0.1654 0.1628 0.2088 0.2244 0.2374 0.4145 0.3326

[27] 4 0.0099 0.1677 0.0818 0.0708 0.0861 0.0943 0.0786 0.1787 0.1312
8 0.0076 0.2223 0.1085 0.0991 0.1243 0.1359 0.1243 0.2557 0.1960
0.0104 0.3400 0.1821 0.1650 0.2248 0.2438 0.3060 0.5012 0.4172

Table 3

Insulating fluid log data and the progressive Type II censoring scheme

i 1 2 3 4 5 6 7 8
xi:m:n −1.6608 −0.2485 −0.0409 0.2700 1.0224 1.5789 2.8718 1.9947
Ri 0 0 3 0 3 0 0 5

Table 4

Test statistics and the corresponding p-values for the insulating fluid log data

Criterion Cm:n+ Cm:n- Cm:n Km:n Tm:n(1) Tm:n(2) Gm:n+ Gm:n- Gm:n
Test statistic 0.0802 0.0683 0.0802 0.1485 0.0025 0.0422 0.0760 0.0706 0.1466
p-value 0.5632 0.4503 0.6482 0.5642 0.5352 0.5158 0.7937 0.8125 0.8042

Table 5

Wire connection strength data and the progressive Type II censoring scheme

i 1 2 3 4 5 6 7 8 9 10
xi:m:n 550 750 950 1150 1150 1150 1350 1450 1550 1850
Ri 0 2 1 0 3 0 0 2 0 2

Table 6

Test statistics and the corresponding p-values for the wire connection strength data

Criterion Cm:n+ Cm:n- Cm:n Km:n Tm:n(1) Tm:n(2) Gm:n+ Gm:n- Gm:n
Test Statistic 0.0952 0.0887 0.0952 0.1839 0.0021 0.0349 0.0587 0.0623 0.1210
p-value 0.6624 0.3699 0.7077 0.5413 0.8011 0.8764 0.8711 0.8562 0.8635

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