Power Comparisons of EDF Goodness-of-Fit Tests Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

EDN: BTGNPN УДК 519.2

Power Comparisons of EDF Goodness-of-Fit Tests

Djahida Tilbi*

Departement of mathematics Laboratory of Probability and Statistics LaPS

Skikda, Algeria

Received 15.11.2022, received in revised form 26.12.2022, accepted 20.02.2023

Abstract. In this article, the power of common goodness-of-fit (GoF) statistics is based on the empirical distribution function (EDF) where the critical values must be determined by simulation. The statistical power of Kolmogorov-Smirnov Dn, Cramer-von Mises W2, Watson U2, Liao and Shimokawa Ln, and Anderson-Darling A2 statistics were investigated by the sample size, the significance level, and the alternative distributions, for the generalized Rayleigh model (GR). The exponential, the Weibull, the inverse Weibull, the exponentiated Weibull, and the exponentiated exponential distributions were considered among the most frequent alternative distributions.

Keywords: generalized Rayleigh distribution, Kolmogorov-Smirnov test, the Cramer-von Mises test (C-VM), Anderson-Darling test (A-D), Watson test (W), Liao and Shimokawa test (LS).

Citation: D. Tilbi, Power Comparisons of EDF Goodness-of-Fit Tests, J. Sib. Fed. Univ. Math. Phys., 2023, 16(3), 308-317. EDN: BTGNPN.

Introduction

Statistical analysis means investigating trends, patterns, and relationships using quantitative data. It is an important research tool used by scientists, governments, businesses, and other organizations. Many statistical analysis tools rely on assumptions of underlying distributions. The goodness-of fit problem is to validate such assumptions before applying those tools to data, therefore it arises in applications of many statistical approaches. Many goodness of-fit tests (GoF) have been developed, and most of them are based on the empirical distribution functions (EDF), the old one, being the Kolmogorov-Smirnov (K-S) statistic Dn (Kolmogorov 1933). Later, Cramer-Von Mises W2 statistics have been shown to be more powerful than a K-S test statistic (Dn) against a large class of alternative hypotheses. The Anderson-Darling statistic A2 (Anderson and Darling 1954) can be considered as a limiting distribution of W2 and it gives more weight to the tails than the statistic (Dn) does (see Darling 1957). Watson (1961a, 1962b) proposed a new test statistic U2 as a generalization of Cramer-Von Mises test statistic W2. Another new test statistic Ln, is developed by Liao et Shimokawa (1999) and applied for testing the GoF.

Let (Xi,..., Xn) be a random sample from the distribution F(x) = P(X ^ x). The main problem is that of testing hypotheses about F of the form:

{ H0 : F(x) = F0(x) \ Hi : F(x) = Fo(x) '

where F0(x) is a known distribution function. The EDF is defined as

^ . . number of observations < x 1 v^ , .

Fn(x) = -— = "]C !(Xi < x), (1)

n n z—'

i=1

*d.tilbi@univ-skikda.dz © Siberian Federal University. All rights reserved

where I is an indicator function. Almost surely, the EDF Fn(x) converges uniformly to the distribution function F(x) (more detail, see the Glivenko-Cantelli theorem).

Many authors have addressed the problem of testing the null hypothesis in (1) when X follows a specified model, The EDF statistics are not distributed, but in the case of unknown parameters, their distribution will depend not only on the sample size but also on the hypothetical distribution. Using numerical methods, they developed modified test statistics, replacing the unknown parameters with their estimates. We find, for example, both Hassan from generalized exponential distribution (2005) and Al-Zahrani from Top-Leone distribution (2012) are obtained critical values for GoF tests based on a random sample and on the EDF tests. According to the critical tables which have been obtained by certain authors such as for example (for the two-and three parameter Weibull distributions (Evans, Johnson, and Green 1989), for the generalized Frechet distribution (Abd-Elfattah, Fergany, and Omima 2010) for the double Exponential distribution (Lemeshko and Lemeshko 2011a), it is particular that the statistic A2 of AD test is the most powerful EDF test.

The generalized Rayleigh (GR) distribution plays an important role in the analysis of reliability and survival data (see, Kundu and Raqab 2007, Rao and Gadde Srinivasa 2014). This distribution was introduced by Surles and Padgett (2001). Originally, Mudholkar and Srivastava (1993), Mudholkar and al. (1995) proposed several distributions called the Burr distributions, whose generalized Rayleigh (GR) distribution is a special case of those of Burr Type X. Depending on the values of the parameters, Kundu and Raqab (2005) used different estimation methods for simple data so that Al-Khedhairi et al. (2007) calculated the estimators on grouped data and censored data. Fathipour et al. (2013) and Rao (2014) interested in estimating the weakness of the components described by GR distributions. Note that modified chi-square goodness-of-fit tests for this distribution have been developed for complete data and for censored data (D. Tilbi and Seddik-Amour 2017).

In this article, we explore the GoF for the generalized Rayleigh model with unknown parameters. After replacing the unknown parameters by their maximum likelihood estimates, we use R software and Monte Carlo methods, to provide tables of GoF critical values of the modified statistics Dn, Ln, , U% and AA% based on the FDE for this model. Finally, the power of these statistics is studied using alternative distributions (Weibull and exponential).

1. Generalized Rayleigh model

The Rayleigh distribution is widely used to model events that occur in different fields such as medicine, social and natural sciences. For instance, it is used in the study of various types of radiation, such as sound and light measurements. It is also used as a model for wind speed and is often applied to wind-driven electrical generation. Recently, Surles and Padgett (2001) considered the two parameter Burr Type X distribution by introducing a shape parameter and correctly named it as the generalized Rayleigh (GR) distribution. This distribution was studied by Mohammad Z. Raqab and Mohamed T. Madi (2011). If the random variable X has a two parameter GR distribution, then it has the cumulative distribution function (cdf)

F(x; a, X) = (1 - e-(Xx)2 )a, x> 0, a > 0, X > 0, (2)

and probability density function (pdf)

f (x; a,,X) = 2aX2xe-(Xx)2 (1 - e-(Xx)2 )a—, x> 0, a > 0, X > 0, (3)

where a and X are shape and inverse scale parameters, respectively. We denote the GR distribution with shape parameter a and inverse scale parameter X as GR(a, X). Its hazard and

reliability functions are

. 2aX2xe-(Xx)2 (1 - e-(Xx)2 )a-1 h(x; a, X) = --—--,, ,2N---. (4)

v ' ' ' 1 — (1 — e-(Xx)2 )a v >

S(x; a, X) = 1 — (1 — e-(Xx)2 )a. (5)

1.1. Maximum likelihood estimates

Suppose that X1,X2, AAA, Xn is a random sample from GR(a, X). Then the log-likelihood function of the observed sample is

n n n

L(x; a, X) = n ln2 + n ln a + 2n ln X + J^ln x, — X2 ^ x2 + (a — 1) ^ln(1 — e-(Xx)2). (6)

i=1 i=1 i=1

The MLEs of a and X say a and A, respectively, can be obtained as the solutions of the following equations

dL

da = - + Eln(! - e-(Xx)2 ) = ° (7)

da a

i=1

dL 2n ^ 2 w x2e-(Xx)

9X = A - ^ 5>2 + 2A(a - 1 - e-(Xx)2 = ° (8)

i=1 i=1

We obtain

E?=i ln(1 - e-(Xx)2)'

(9)

i=1

and A can be obtained as the solution of the nonlinear equation g(X) = 0, where

g(X) = dL(x a,X) = ^ — 2X f x2 — 2X( ^ fn f ^Xl.

yy ' dX X = 1 \vEn=1 ln(1 — e-(Xx)2) y 1 — e-(Xx)2

Therefore, X can be obtained as solution of the nonlinear equation of the form H(X) = X, where

r2p-(Xx)2

H (A) = 2n

2A ]T x2 ^„=i in(i- e-(Xx)2) +1) g 1 -

(10)

Since, A is a fixed point solution of the non-linear equation (10), therefore, it can be obtained using an iterative scheme as H(Aj) = Aj+1, where Aj is the jth iterate of A. The iteration procedure should be stopped when |Aj - Aj+1| is sufficiently small. Once we obtain A, then A can be obtained from (9).

n

2. GoF statistics based on the EDF

A goodness of fit test based on the empirical function (EDF), when the parameters are estimated, is called a modified goodness of fit test. The most popular nonparametric goodness-of-fit tests, namely; the Kolmogorov-Smirnov Dn, Cramer-von-Mises W2, Anderson-Darling A2, Watson U2, and Liao-Shimokawa Ln test statistics. The critical values of the modified statistics did not exist in the statistical literature prior to the last decades. Through simulations, some authors have provided critical table values for classical models and some of their generalizations (for more details, see Lemeshko and Lemeshko 2011b). In this paper, using the Monte Carlo method and the R software, we offer tables of critical values of Dn, W2, A2, U2, and Ln statistics for the generalized Rayleigh model when the parameters are unknown.

2.1. K-S test statistics Dn

The most popular GoF test is the Kolmogorov-Smirnov K-S test. The test statistic Dn is defined as

Dn = max[D+; D"j,

where

and

D+ = max

1<i<n

- - F (xi) n

D-

max

1<i<n

F(xi) -

i - 1

with xi is the order statistic. For Rayleigh model GR(a, A) the Dn statistic becomes

D+ = max

1<i<n

- - (1 - e

n

-(Xxi)2 )&

(11)

and

D- = max

1<i<n

(1

- e-(XXi)2 )â -

i - 1

(12)

where a and A are the maximum likelihood parameter estimators of the unknown parameters.

2.2. C-VM test statistics W

2

The Cramer-von Mises test is an alternative to the Kolmogorov-Smirnov test (1933). C-VM test statistic W2 may be considered as the sum of the quadratic differences between the empirical distribution function (EDF) and the theoretical cumulative distribution function (CDF). It is defined as

W2 = ih +1 <*> - <i3>

So, for the GR(a, A) distribution, we obtain

W2 = -L + Y ((1 - e-(^xi)2 )a - an 12n ) 2n i=1 4

(14)

2.3. A-D test statistics A2

The A-D test statistic A2 was developed by Anderson and Darling (1954) as a limiting distribution of the test of C-VM as in n^ to. The A2 is given by

A2 = -n - 1

i=1

(2i - 1)(ln(F (xi))+ln(1 - F (xi))) .

(15)

We obtain the test statistic for GR(a, A) as follows

An = -n - n^

(2i - 1)(ln((1 - e-(Xx<)2 )a) +ln(1 - (1 - e-(Xx<)2)a))

(16)

n

2.4. W test statistics U2

Watson test statistic U2 was developed for distributions which are cyclic and in 1961 it is based on the empirical distribution function. U2 is a generalization of the C-VM test statistic. It is defined by

U2 = »'2 + ¿( ^ - ^ 2 (17)

i=1

£( ^ -1)-

i=i v 7

The explicit form of this statistic for the GR(a, A) model is

« ^+¿( (i - - - 2)'- <18)

i=iv 7 ¿=iv 7

2.5. LS test statistics Ln

The Liao-Shimokawa statistic measures the average of all weighted distances over the entire range of the data. For more details, we refer to Liao and Shimokawa (1999). The test statistic is given by

n max ( — - F(xi),F(xi) - --- )

Ln = -= V i n J. (19)

v^ ^F(xi)[l - F(xi)] V ;

For the distribution of GR(a, A), Ln becomes

n max f— - (l - )a, (l - )a - ——

Ln = V i ^ - (20)

Vn i=i y/(l - e-(XXi)2 )&[l - (l - e-(XXi)2 )&]

3. Critical values

The purpose of this paper is to provide critical adjustment values of the modified statistics Dn, An, Wn, Un and Ln for the generalized Rayleigh distribution when the parameters are unknown and replaced by their maximum likelihood estimates of the non grouped data. For this, we use Monte Carlo simulation method and R software to generate l0,000 samples of different sizes n.

Under the null hypothesis H0 that a sample X = X1,X2,..., Xn belongs to generalized Rayleigh model, we calculated the values of the various fit testing statistics mentioned above. To this end, the following steps are used to calculate the critical values for each statistic of the fit tests at different levels of significance a = 0.0l, 0.05, 0.l0, 0.l5 and 0.25 and sample sizes n = 5, l0, l5, 20, 30, 50 and l00:

Step 1. Generate n random variables U(0, l) independent U1,U2,... ,Un.

Step 2. For given values of the parameters a and A, we set xi = F-1(Ui).

Then (x1, x2,..., xn) is the required sample size n of the GR distribution.

Step 3. Use the generated sample to estimate the unknown parameters using the maximum likelihood estimators given by (9) and (10).

Step 4. The unknown parameter estimators were used to determine the hypothetical cumulative distribution function of the GR distribution.

Step 5. The statistical tests Dn, Ln, Wn, Un and An mentioned above are calculated for each generation random sample of different sizes.

Step 6. This procedure was repeated 10,000 times independently. Therefore, we got 10,000 values for each proposed test statistic. These values have been classified at different levels of significance 0.01, 0.05, 0.10, 0.15 and 0.25 are shown in the Tab. 1.

Table 1. Critical values for K-S, C-VM, A-D, W and LS tests

Sample test Significance level a

size n statistics 0.01 0.05 0.10 0.15 0.25

5 Dn 0.0000 0.0006 0.0020 0.0053 0.0300

W2 *v n 0.0006 0.0061 0.0185 0.0312 0.0593

An 0.0187 0.0700 0.1400 0.1990 0.3590

U2 ^ n 0.0005 0.0044 0.00102 0.0199 0.0412

Ln 0.0138 0.0449 0.0655 0.1022 0.1114

10 Dn 0.0000 0.0004 0.0017 0.0050 0.0111

Wn2 n 0.0004 0.0054 0.0182 0.0309 0.0587

A2 0.0156 0.0706 0.1359 0.1986 0.3840

Un2 n 0.0004 0.0031 0.0099 0.0185 0.0391

Ln 0.0125 0.0395 0.0592 0.0965 0.1072

15 Dn 0.0000 0.0004 0.0016 0.0049 0.0101

Wn2 0.0003 0.0048 0.0163 0.0305 0.0575

An 0.0152 0.0762 0.1293 0.1836 0.3570

Un2 n 0.0004 0.0029 0.0079 0.0178 0.0352

Ln 0.0120 0.0345 0.0522 0.0960 0.1066

20 Dn 0.0000 0.0004 0.0015 0.0047 0.0100

Wn2 n 0.0003 0.0043 0.0140 0.0304 0.0569

An 0.0147 0.0657 0.1297 0.1788 0.3470

Un2 n 0.0004 0.0026 0.0072 0.0174 0.0332

Ln 0.0115 0.0338 0.0452 0.0865 0.0987

30 Dn 0.0000 0.0003 0.0011 0.0045 0.0100

Wn2 n 0.0003 0.0042 0.0133 0.0289 0.0565

An 0.0145 0.0700 0.1150 0.1755 0.1986

Un2 n 0.0003 0.0020 0.0063 0.0170 0.0325

Ln 0.0111 0.0332 0.0434 0.0799 0.0977

50 Dn 0.0000 0.0003 0.0009 0.0034 0.0079

Wn2 0.0002 0.0039 0.0126 0.0286 0.0559

An 0.0129 0.0561 0.1132 0.1707 0.2590

Un2 n 0.0002 0.0014 0.0039 0.0143 0.0291

Ln 0.0101 0.0245 0.0398 0.0592 0.0923

100 Dn 0.0000 0.0001 0.0006 0.0030 0.0067

Wn2 n 0.0001 0.0032 0.0120 0.0284 0.0530

An 0.0125 0.0524 0.1087 0.1585 0 .2500

Un2 n 0.0001 0.0012 0.0036 0.0137 0.0278

Ln 0.0097 0.0231 0.0341 0.0564 0.0878

From the table, we noticed that:

•For each statistical test, the power increases monotonically as the sample size increases and the level of significance increases.

•The Anderson-Darling Ann statistical test is the most powerful of the proposed fit tests. •The statistical test of Komogorov-Smirnov Dn is the least powerful among the fit tests proposed.

4. Simulation study

In this section, we performed a power comparison between Dn, Ln, W2, Un and An statistics for the GR model with unknown parameters. For this, we simulated l0, 000 random samples of different sizes n = l0, 20, 50 and l00, for each test at the significance level a = 0.05 and from each of the alternative distributions:

1. The Exponential distribution Exp(A), with probability density function

fx (x,A) = A exp(-Ax), and its cumulative distribution function is

FExp(x, A) = l - exp(-Ax). (21)

2. The Weibull distribution Wei(j, a), with probability density function

f (x; Y, a) = 7-1 exp(-(x )Y),

a \a/ a

and its cumulative distribution function is

Fwei(x; Y,a) = l - exp ( - (^Y). (22)

3. The Inverse Weibull distribution InWei(a,Y), with probability density function

f (x; y, a) = YaY x-(Y+1) exp ( - (§) ) ^, and its cumulative distribution function is

FinWei(x; y, a) = exp ( - a^) ). (23)

4. The Exponentiated Weibull distribution ExpWei(a,Y, A), with probability density function

f (x; y, a, A) = aYAYxY-1(l - exp(-AxY))a, and its cumulative distribution function is

FExpWei(x; Y, a, A) = (l - exp(-AxY))a. (24)

5. The Exponentiated Exponential distribution EE(a, A), with probability density function

fEE(x; a, A) = aA(l - exp(-Ax))a-1 exp(-Ax),

and its cumulative distribution function is

Fee(x; a, A) = (l - exp(-Ax))a. (25)

The power results of tests statistics Dn, Ln, W^, Un and An, for each alternative distribution at significance level a = 0.05 are presented in Tab. 2. From the table, we notice that:

•According to the test power values for the different statistics, are indicating that the generalized Rayleigh model is distinct from competing distributions of all sizes of the sample. •The power of the test statistic increases as the sample size increases.

The modified test statistics Dn, Ln, W2, Un and An provided in this work and their critical values can detect the difference betwen the GR model and different alternatives with high Power.

Table 2. Power of statistics tests for GR distribution where Exp, Wei, InWei, ExpWei and EE are the alternative distributions

Alternatives test Sample size n

statistics 10 20 50 100

Dn 1.0000 1.0000 1.0000 1.0000

W2 0.1016 0.3653 0.9119 0.9997

Exponential Al 0.4158 0.7834 0.9976 1.0000

Exp(1) U 2 ^ n 0.1004 0.3202 0.9164 0.9786

Ll 0.1059 0.3728 0.9993 1.0000

Dn 1.0000 1.0000 1.0000 1.0000

W2 *v n 0.0995 0.3542 0.9080 0.9998

Weibull A 0.0644 0.0603 0.0539 0.0495

Wei(1, 2) U ^ n 0.0244 0.0282 0.0393 0.0450

Ll 0.0159 0.0228 0.0324 0.0445

Dn 0.9249 0.9992 1.0000 1.0000

W *v n 0.1059 0.3101 0.9463 0.9459

Inverse Weibull A 0.8286 0.9324 0.9981 1.0000

InWei(1, 2) U ^ n 0.1083 0.3089 0.9059 0.9228

Ll 0.1055 0.3076 0.9034 0.9210

Dn 0.9999 0.9996 1.0000 0.9999

W 0.1035 0.3588 0.9997 0.9995

Exponentiated Weibull A 0.9999 0.9998 0.9992 0.8853

ExpWei(1, 2, 3) U n 0.1030 0.3438 0.9127 0.9960

Ln 0.1011 0.3298 0.9037 0.9860

Dn 1.0000 1.0000 1.0000 1.0000

W n 0.1055 0.3676 0.9087 0.9994

Exponentiated Exponential A 0.0592 0.0726 0.0639 0.0597

EE(1, 2) U y n 0.0548 0.0526 0.0611 0.0684

Ln 0.0539 0.0523 0.0601 0.0672

Conclusion

We have provided critical values for the statistics Dn, Ln, , and An for the generalized Rayleigh model when the parameters are unknown. The 1 and 2 tables given in this manuscript can be used to check whether the sample data fits this pattern which helps practitioners to choose the appropriate pattern for their analysis.

We would like to thank the editorial board and referees for their suggestions useful which improved this manuscript greatly.

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Сравнение мощностей тестов согласия EDF

Джахида Тилби

Кафедра математики Лаборатория вероятностей и статистики LaPS

Скикда, Алжир

Аннотация. В этой статье сила общей статистики согласия (СоР) основана на эмпирической функции распределения (ЕБР), где критические значения должны быть определены путем моделирования. Статистическая мощность Колмогорова-Смирнова Вп, Крамер-фон Мизеса Ш2, Ватсона и2, Ляо и Симокавы Ь„ , и статистика Андерсона-Дарлинга А исследовалась по размеру выборки, уровню значимости и альтернативным распределениям для обобщенной модели Рэлея (СИ). Экспоненциальное, Вейбулла, обратное Вейбулла, экспоненциальное Вейбулла и экспоненциальное распределения были рассмотрены среди наиболее частых альтернативных распределений.

Ключевые слова: обобщенное распределение Рэлея, критерий Колмогорова-Смирнова, критерий Крамера-фон Мизеса (С-УМ), критерий Андерсона-Дарлинга (ЛБ), критерий Ватсона (W), критерий Ляо и Симокавы (ЬЯ).