# Characterization and Goodness-of-Fit Test of Pareto and Some Related Distributions Based on Near-Order Statistics

Authors: Masoumeh Akbari
Journal: Journal of Probability and Statistics
 (2020)

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## Abstract

In this paper, a new definition of the number of observations near thekth order statistics is developed. Then some characterization results for Pareto and some related distributions are established in terms of mass probability function, first moment of these new counting random variables, and using completeness properties of the sequence of functions xn,0<x<1,n≥1. Finally, new goodness-of-fit tests based on these new characterizations for Pareto distribution are presented. And the power values of the proposed tests are compared with the power values of well-known tests such as Kolmogorov–Smirnov and Cramer-von Mises tests by Monte Carlo simulations.


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## Body

## 1. Introduction

Let order statisticsX1:n≤X2:n≤⋯≤Xn:n be the nondecreasing order of independent and identically distributed (iid) random variables X1,X2,…,Xn with cumulative distribution function (cdf) FX⋅. The order statistics have important roles in different areas of statistics and probability. In reality, some kinds of order statistics are more applicable. For example, in actuarial science, the distribution of the minimum of the two lifespans of the couple is important for insurance policy to make decisions. In industries, specifically in reliability and survival analysis, order statistics are used to solve problems. Meteorology, hydrology, and so on are other fields of applications of order statistics. Interested readers can study the detail of theory and application of order statistics, for instance, in Arnold et al. [1]. Let FX⋅ be discrete cdf. For the first time, Eisenberg et al. [2] defined the quantity Kn=∑i=1nIXi=Xn:n as the number of winners in a golf competition. They studied sufficient conditions under which Kn converges to 1. After that Pakes and Steutel [3] considered similar notions for continuous cdf as follows:(1)Kna=∑j=1nIXn:n−a,Xn:nXj,where a>0 is a constant. In fact, Kna counts the number of observations in the left-hand neighbourhood of the sample maximum with fixed distance “a.” Later, Kna was developed to the number of observations near the kth-order statistics as follows:(2)K−n,k,a=#j∈1,2,…,n;Xj∈Xk:n−a,Xk:n,where the support of K−n,k,a is 0,1,…,k−1. Similarly, the number of observations in the right-hand neighbourhood of the kth order statistics was defined as(3)K+n,k,a=#j∈1,2,…,n;Xj∈Xk:n,Xk:n+a.The support ofK+n,k,a is 0,1,…,n−k. The probability mass function (pmf) of K−n,k,a is as follows:(4)PK−n,k,a=j=k−1j∫SXηLx,ak−j−11−ηLx,ajdFk:nx,where ηLx,a=FXx−a/FXx and Fk:n⋅ is the cdf of Xk:n. Also, the pmf of K+n,k,a is given by(5)PK+n,k,a=j=n−kj∫SXηUx,an−k−j1−ηUx,ajdFk:nx,where ηUx,a=F¯Xx+a/F¯Xx. For more details, one can refer to Dembińska et al. [4].More results of the description of their distributions, asymptotic properties, and their generalization have been investigatede.g., Pakes and Li [5], Li [6], Pakes ([7,8]), Balakrishnan and Stepanov ([9,10]), Dembińska et al. [4], and Dembińska ([11–13]). So far, few researchers have addressed the issue of statistical inference based on near-order statistics, e.g., Müller [14], Hashorva and Hüsler [15], Akbari et al. [16], and Akbari and Akbari [17]. In the present paper, a new version of near-order statistics is first defined. Then some characterization results as a statistical tool in goodness-of-fit (GOF) test for some continuous distributions are obtained. The results of this paper are organized as follows. Section 2 contains preliminary results. The characterization results of the paper are included in Section 3. Finally, in Section 4 are introduced two tests for goodness-of-fit tests for Pareto distribution. The critical values of the proposed test statistics are computed by Monte Carlo simulations. Also, their power is compared with those computed by well-known tests such as Kolmogorov–Smirnov and Cramer–von Mises tests by simulations. All simulations are carried out by using R 3.6.3 and with 10000 replications.


## 2. Preliminary Results

LetX1,…,Xn be iid random variables from continuous cdf FX⋅ with support SX, and FX be one of the distribution functions Pareto, uniform, or power function. Constructing characterizations for such FX by the pmfs or moments of some functions of random variables is a common way. But it is not possible by counting random variables K± as the number of observations on the location-type neighbourhood of the certain order statistics defined in equations (4) and (5), because their pmfs do not have a closed-form expression. Therefore, new types of number of observations falling within the left-hand and right-hand of neighbourhood of the specific order statistics, as an extension to scale-type neighbourhood, are introduced, respectively, as follows:(6)K−′n,k,a=#j∈1,2,…,n;Xj∈aXk:n,Xk:n,where 0<a<1 and(7)K+′n,k,b=#j∈1,2,…,n;Xj∈Xk:n,bXk:n,where b>1.Proposition 1.
Using the same arguments given in Dembińska et al. [4], it is concluded that the pmfs of new counting random variables K−′n,k,a and K+′n,k,b are the same as the pmfs of K−n,k,a and K+n,k,b, respectively, with ηLx,a=FXax/FXx and ηUx,a=F¯Xbx/F¯Xx, that is,(8)PK−′n,k,a=j=k−1j∫SXFXaxFXxk−j−11−FXaxFXxjdFk:nx,(9)PK+′n,k,b=j=n−kj∫SXF¯XbxF¯Xxn−k−j1−F¯XbxF¯XxjdFk:nx.On the other hand, by simple algebra calculations, the first moment ofK−′n,k,a and K+′n,k,b can be derived, respectively, as follows:(10)EK−′n,k,a=k−11−n!k−1!n−k!∫SXFXaxFXk−2x1−FXxn−kdFXx=k−1−nk−1n−1k−1∫01FXaFX−1uuk−21−un−kdu,(11)EK+′n,k,b=n−k−kn−knk∫SXF¯XbxFXk−1xF¯Xn−k−1xdFXx=n−k−kn−knk∫01F¯XbFX−1uuk−11−un−k−1du.Here, some results as examples of special cases of Pareto and power function distributions are reported that will be useful for obtaining further results in the next section.Example 1.
LetX1,X2,…,Xn be iid random variables from Pareto (α, β). So their survival distribution functions is given by(12)F¯x=βxα,x≥β.
Then from equation (9), the pmf of K+′n,k,b related to this sequence is as follows:(13)PK+′n,k,b=j=n−kj∫β∞1bαn−k−j1−1bαjdFk:nx=n−kj1bαn−k−j1−1bαj.
Equation (13) shows that K+′n,k,b has binomial pmf with parameters n−k and 1−1/bα. Thus,(14)EK+′n,k,b=n−k1−1bα,which does not depend on scale paremeter β.Example 2.
It is well-known that ifX∼ Pareto (α, β), then random variable Y=1/X has power function distribution with following cdf:(15)FYy=βyα,0≤y≤1β.
The notation power (α, β) is used for power function distribution with parameters α and β. It is also called generalized uniform distribution because it is standard uniform cdf at β=1 and α=1. From (8), the pmf of K−′n,k,a for random variables X1,…,Xn that are distributed as power (α, β) is concluded as(16)PK−′n,k,a=j=k−1j∫01/βaαk−j−11−aαjdFk:nx=k−1jaαk−j−11−aαj.
According to (16), K−′n,k,a has binomial pmf with parameters k−1 and success probability 1−aα. So(17)EK−′n,k,a=k−11−aα.
As we know, the uniform distribution function on interval0,1 is a special case of power (1,1) with following cdf:(18)Fx=x,0<x<1.
Therefore from (16), the pmf of K−′n,k,a when Xi∼U0,1,i=1,…,n, equals(19)PK−′n,k,a=j=k−1j∫01ak−j−11−ajdFk:nx=k−1jak−j−11−aj,and its expectation is(20)EK−′n,k,a=k−11−a.


## 3. Characterizaion Results

In this section, some characterization results based on distributional properties of near-order statisticsK−′n,k,a and K+′n,k,b for some continuous distributions are established in terms of property of sequence of complete functions. Thus, in the sequel, some notions and theorems related to this theory are reminded.Definition 1.
A sequenceϕnn≥1 of elements of a Hilbert space ℋ is called complete if the only element which is orthogonal to every ϕn is the null element, that is(21)f,ϕn=0,for everyn≥1,implies f null.
The notation⋅,⋅ denotes the inner product of ℋ. In the present paper, the Hilbert space L20,1 with the following inner product being considered:(22)f,g=∫01fxgxdx,where f and g are real-valued square integrable functions on 0,1. One of the sequences of complete functions in L20,1 is xn,n≥1 which is used in this paper.
The following theorem is known as Müntz theorem that states the necessary and sufficient condition for completeness of the subsequencexnj,j≥1,n1<n2<⋯.Theorem 1 (Higgins [18], page 95).
Sequencexn1,xn2,…,1≤n1<n2<⋯ in L20,1 is complete if and only if(23)∑j=1∞nj−1=∞.For more details about Hilbert space and complete sequences, refer to Higgins [18]. Pareto is one of the distributions that have many applications in economics and actuturial sciences. So far, a lot of properties and characterization of it based on order statistics or their functions have been obtained, for example, Lee and Chang [19], Afify [20], Ahsanullah and Shakil [21], Ahsanullah et al. [22], and Nofal and El Gebaly [23]. In the following theorem, some characterizations for Pareto law in terms of K+′n,k,b are established.Theorem 2.
LetX1,X2,…,Xn be iid random variables with continuous cdf FX⋅ whose support is β,∞. FX⋅ is Pareto (α,β) cdf, if and only if one of the following conditions holds:(a)
There existsj0∈0,1,… such that for all n≥j0+1,b>1 and for k=n−j0, we have(24)PK+′n,k,b=j0=1−1bαj0.(b)
For alln≥k,b>1, and a fixed k≥1, we have(25)EK+′n,k,b=n−k1−1bα.Proof.
IfX−sequence has Pareto (α,β) cdf, by the use of equations (13) and (14), one can easily obtain parts (a) and (b). Let part (a) hold. Using pmf of K+′n,k,b and the assumptions of (a), the equality in (a) can be rewritten as(26)∫β∞1−F¯XbxF¯Xxj0dFk:nxdx=1−1bαj0.
The right-hand side of equation (26) can be expressed as(27)1−1bαj0∫01n!k−1!n−k!1−uk−1un−kdu.
On the other hand, replacingdFk:nx with n!/k−1!n−k!FXk−1xF¯Xn−kxdFXx, the equality in (26) can be stated as(28)∫β∞1−F¯XbxF¯Xxj0FXk−1xF¯Xn−kxdFXx=1−1bαj0∫011−uk−1un−kdu.
Taking the change of variableu=1−FXx in the left-hand side of equation (28), it is deduced:(29)∫011−F¯XbFX−11−uuj01−uk−1un−kdu=1−1bαj0∫011−uk−1un−kdu.
By the assumption “k=n−j0” and after some algebra simplifications in the aforementioned equality, it is concluded that(30)∫01u−F¯XbFX−11−uj0−u1−1bαj01−un−j0−1du=0.
Sinceu−F¯XbF−11−u≤2 and u1−1/bα≤2 , by Minkowski’s inequality, we have(31)u−F¯XbFX−11−uj0−u1−1bαj0∈L20,1.
If (30) holds for all n≥j0+1, by the completeness property of the sequence 1−un−j0−1,n≥j0+1, the following identity can be derived(32)u−F¯XbFX−11−uj0−u1−1bαj0=0.
Hence,(33)F¯XbFX−11−uu=1bα.
By takingt=FX−11−u in (33), it can be rewritten as(34)F¯XbtF¯Xt=1bα.
If (34) holds for all b>1 and t≥β, by the use of the method of solution given in Aczél [24], it is concluded that function F¯Xt=ct−α is the genaral solution of (34). Because t≥β and F¯X⋅ is a survival distribution function, the constant c will be βα. So, the proof is completed.
Suppose that part (b) holds. Then from equation (11), it is deduced that(35)n!k−1!n−k!∫01F¯XbFX−1uuk−11−un−k−1du=1bα.
Since1/bα=1/bα∫01n!/k−1!n−k!uk−11−un−kdu, the last equality, after some simplifications, can be expressed as(36)∫01uk−1F¯XbFX−1u1−u−1bα1−un−kdu=0.
The rest of the proof is similar to the proof of part (a).
So far, some results of characterization of power function distribution have been obtained. For example, it was characterized by Ahsanullah et al. [25] through lower records. Also Khan and Khan [26] and Lim and Lee [27] characterized it based on dependency property of lower records. Tavangar [28] presented a characterization of it using dual generalized order statistics. Now, in the next theorem new characterization results of power function distribution are proved.Theorem 3.
Suppose thatX1,X2,…,Xn are iid continuous random variables from cdf FX⋅ with support 0,1/β. Then Xi’s have power function distribution with cdf (15) if and only if one of the following statements holds.(a)
For alln≥k and 0<a<1, there exists j0∈0,1,… such that for k=j0+1 and some α>0, we have(37)PK−′n,k,a=j0=1−aαj0.(b)
For a fixedk≥1 and for all n≥k,0<a<1 and some α>0, we have(38)EK−′n,k,a=k−11−aα.Proof.
By supposingXi’s have power function distribution with cdf (15), from equations (7) and (16), parts (a) and (b) can be easily concluded. Let condition (a) be satisfied, then(39)0=PK−′n,k,a=j0−1−aαj0=∫01/β1−FXaxFXxj0dFk:nx−1−aαj0=n!k−1!n−k!∫01/β1−FXaxFXxj0FXk−1x1−FXxn−kdFXx−1−aαj0=n!k−1!n−k!∫011−FXaFX−1uuj0uk−11−un−kdu−1−aαj0n!k−1!n−k!∫01uk−11−un−kdu=∫01uk−11−FXaFX−1uuj0−1−aαj01−un−kdu.
Since1−FXaFX−1u/u≤2 and 1−aα≤2, the quantity uk−11−FXaFX−1u/uj0−1−aαj0 belongs in L20,1 by Minkowski’s inequality. The completeness property of the sequence 1−un−k,n≥k and equation (39) result in(40)1−FXaFX−1uuj0−1−aαj0=0.
This is equivalent to(41)FXaFX−1uu=aα.
Taking the change of variablet=FX−1u in (41) gives(42)FXatFXt=aα.
The functionFXx=cxα is the general solution of above functional equation. This completes the proof of (a). In a similar way, if (b) holds, one can easily prove that the parent population is power function distribution.
The results of Theorem3 can also be observed directly from Theorem 2 by noticing that X∼ power (α, β) if and only if Y=1/X∼ Pareto (α, β). Therefore, for 0<a<1(43)KX−′n,k,a=#j∈1,2,…,n,Xj∈aXk:n,Xk:n=#j∈1,2,…,n,1Yj∈a1Yn−k+1:n,1Yn−k+1:n=#j∈1,2,…,n,Yj∈Yn−k+1:n,1aYn−k+1:n=KY+′n,n−k+1,1a,where K−′n,k,a with superscript X presents the number of observations near the kth-order statistics related to X-sequence. According to relationship between distributions of power function and standard uniform that is mentioned before, from Theorem 3, the following results without proof are stated.Corollary 1.
LetX1,X2,…,Xn be iid continuous random variables from cdf FX⋅ that is supported on 0,1/β. Then, FX⋅ is standard uniform distribution if and only if one of the following statements holds:(a)
For alln≥k and 0<a<1, there exists j0∈0,1,… such that for k=j0+1, we have(44)PK−′n,k,a=j0=1−aj0.(b)
For a fixedk≥1 and for all n≥k and 0<a<1, we have(45)EK−′n,k,a=k−11−a.Remark 1.
According to Theorem1, it is not necessary that the assumptions of Theorem 2 hold for all “n≥j0+1” or “n≥k.” This fact is also true for Theorem 3 and Corollary 1. So, the results of Theorems 2 and 3 and Corollary 1 hold if their assumptions provide for any increasing subsequence ni≥j0+1,i≥1 or ni≥k,i≥1 such that the equality (23) holds.Remark 2.
Some characterization results of two-parameter exponential distribution have been obtained based on counting random variableK+n,k,a by Akbari and Akbari [17]. Their results of characterizations in Section 2 can also be derived directly by Theorems 2 and 3. For considering this claim, suppose X be a random variable having two-parameter exponential distribution with parameters (μ, α), denoted by Exp (μ,α), and the following cdf:(46)F¯Xx=e−αx−μ,x>μ.
SinceX∼Exp (μ,α) if and only if Z=eX∼Pareto (α,eμ), the following relationship holds between KX+ and KZ+′. For a>0,(47)KX+n,k,a=#j∈1,…,n,Xj∈Xk:n,Xk:n+a=#j∈1,…,n,Zj∈Zk:n,eaZk:n=KZ+′n,k,ea.
Also,X∼Exp (μ,α) if and only if T=e−X∼power (α,eμ). So for a>0,(48)KX+n,k,a=#j∈1,…,n,Xj∈Xk:n,Xk:n+a=#j∈1,…,n,Tj∈e−aTn−k+1:n,Tn−k+1:n=KT−′n,n−k+1,e−a.


## 4. Goodness-of-Fit Test Results

So far, many results of GOF tests for different distributions using characterization results have been obtained. For example, Rizzo [29], Obradovi et al. [30], and Volkova [31] obtained GOF tests for Pareto distribution in terms of its different characteristic properties. According to Nikitin [32], tests based on characterization results are usually more efficient than other tests, because the unique feature of the same distribution has been used in constructing test statistics.LetX1,…,Xn be iid random variables from continuous distribution function FXx. For testing null hypothesis H0:FXx=F0x=1−x/βα,x≥β against H1:FXx≠F0x for some x, are presented two test statistics based on characterization results of Theorem 2. According to part (a) of this theorem, the null hypothesis H0 will be rejected if there exists “n” such that for all j≤n−1 and k=n−j, equation (24) is not satisfied, i.e., the value of quantity(49)PK+′n,k,b−1−1bαj,to be large. From (9) and assumption k=n−j, the above expression is equivalent to(50)∫01FXn−j−1xF¯Xx−F¯Xbxj−F¯Xx1−1bαjdFXx.ReplacingFXx by Fnx, the emprical distribution function, a point estimator of (50), can be considered as(51)DPj,n=1n∑i=1nFnn−j−1xi:nFnbxi:n−Fnxi:nj−1−Fnxi:n1−1bαj=1n∑i=1nin+1n−j−1FnbXi:n−in+1j−1−in+11−1bαj.Therefore, the test statistic that its large value will be rejected,H0, is given by(52)DP=∑m=1n∑j=0n−1DPj,m.With the same discussion, the another test statistic based on the part (b) of Theorem2 can be as(53)DE=∑m=1n∑k=1mDEk,m,where DEk,n=1/n∑i=1ni/n+1k−11−i/n+1n−k1−FnbXi:n−1−i/n+11/bα.It is obvious thatDP and DE are free of scale parameter of Pareto distribution and their large values reject H0.In the rest of this section, the power values of two test statisticsDP and DE will be compared with well-known tests, namely, Kolmogorov–Smirnov and Cramer–von Mises tests which their statistics are, respectively,(54)D=maxD+,D−,where D+=max1≤i≤ni/n−F0Xi:n and D−=max1≤i≤nF0Xi:n−i−1/n, and(55)W2=112n+∑i=1n2i−12n−F0Xi:n2.Since Pareto distribution is long tail, following alternative distributions that are long tail on the right-hand side are considered for comparison of power values of statisticsDP,DE,D, and W2.(i)
The Weibull distribution with densitya/bx/ba−1exp−x/ba, denoted by Wa,b(ii)
The gamma distribution with density1/λrΓrxr−1exp−x/λ, denoted by Gr,λ(iii)
The lognormal distribution with density1/2πσxexp−lnx−μ2/2σ2, denoted by LNμ,σ(iv)
The uniform distribution with density1/10,0≤x≤10, denoted by U(v)
The log-gamma distribution with densityexpx−a/b−expx−a/b/b, denoted by LGa,bSince it is not easy to find the null distribution ofDP,DE,D, and W2, Monte Carlo simulations with 10000 replications are used for calculating their power values and critical values at 5 percent significance level. Tables 1 and 2 show the results for null distribution Pareto (1,2) and Pareto (2,2), respectively. Because the statistics DP and DE have the parameter b, one can choose an optimal b to maximize corresponging power values. So, these values are calculated and shown in the tables. Unfortunately there is no accurate method to find these values and depend on the support of null and alternative distributions.Table 1
Monte Carlo power estimates of the tests at 5 percent significant level with null distribution function Power (1,2).
Sample sizeTestLG (2,8)LN (0,1)G (3,1)W (0.5,2)U5DE0.9999 (0.0494)0.892 (0.0475)0.7619 (0.05)0.8969 (0.048)0.8698 (0.0498)b = 2.1b = 1.66b = 1.69b = 1.68b = 1.67DP0.9999 (0.05)0.892 (0.0495)0.7619 (0.05)0.8969 (0.05)0.8698 (0.05)b = 2.22b = 1.42b = 1.39b = 1.65b = 1.39D0.757 (0.05)0.752 (0.05)0.3217 (0.05)0.618 (0.05)0.121 (0.05)W20.428 (0.05)0.588 (0.05)0.3217 (0.05)0.367 (0.05)0.08 (0.05)7DE1 (0.05)0.9626 (0.499)0.8711 (0.05)0.963 (0.05)0.9528 (0.05)b = 2.3b = 1.67b = 1.61b = 1.71b = 1.62DP1 (0.05)0.9626 (0.05)0.8711 (0.05)0.963 (0.05)0.9528 (0.05)b = 2.42b = 1.37b = 1.88b = 1.381b = 1.89D0.826 (0.05)0.810 (0.05)0.392 (0.05)0.81 (0.05)0.11 (0.05)W20.833 (0.05)0.766 (0.05)0.496 (0.05)0.764 (0.05)0.094 (0.05)10DE1 (0.05)1 (0.05)1 (0.05)1 (0.05)0.8813 (0.05)b = 2.38b = 1.15b = 1.15b = 1.11b = 2DP1 (0.05)1 (0.05)1 (0.05)1 (0.05)0.8813 (0.05)b = 2.57b = 1.38b = 1.88b = 1.38b = 2.01D0.945 (0.05)0.9066 (0.05)0.559 (0.05)0.9118 (0.05)0.129 (0.05)W20.975 (0.05)0899 (0.05)0.706 (0.050.9071 (0.05)0.121 (0.05)Table 2
Monte Carlo power estimates of the tests at 5 percent significant level with null distribution function Power (2,2).
Sample sizeTestsLG (2,8)LN (0,1)G (3,1)W (0.5,2)U5DE0.9999 (0.049)0.9578 (0.047)0.933 (0.047)0.997 (0.0476)0.975 (0.0487)b = 2.10b = 2.21b = 2.13b = 2.32b = 2.13DP0.9999 (0.05)0.9578 (0.05)0.933 (0.048)0.9969 (0.05)0.9751 (0.05)b = 2.22b = 2.24b = 2.04b = 2.49b = 2.18D0.759 (0.05)0.916 (0.05)0.2352 (0.05)0.767 (0.05)0.414 (0.05)W20.4624 (0.05)0.8089 (0.05)0.1562 (0.05)0.5238 (0.05)0.365 (0.05)7DE1 (0.05)0.9682 (0.05)0.9589 (0.0499)0.9985 (0.05)0.9922 (0.05)b = 2.24b = 2.26b = 2.08b = 2.23b = 2.08DP1 (0.05)0.9682 (0.0499)0.9589 (0.0486)0.9985 (0.05)0.9922 (0.04b = 2.43b = 2.39b = 2.28b = 2.47b = 2.39D0.8 (0.05)0.949 (0.05)0.2243 (0.05)0.809 (0.05)0.491 (0.05)W20.74 (0.05)0.932 (0.05)0.2049 (0.05)0.763 (0.05)0.471 (0.05)10DE1 (0.05)0.9879 (0.05)0.978 (0.05)0.9997 (0.05)0.9977 (0.05)b = 2.38b = 2.11b = 2.14b = 2.11b = 2.15DP1 (0.05)0.9879 (0.05)0.978 (0.05)0.9997 (0.05)0.9977 (0.05)b = 2.57b = 2.57b = 2.56b = 2.94b = 2.55D0.9199 (0.05)0.9883 (0.05)0.283 (0.05)0.912 (0.05)0.647 (0.05)W20.9286 (0.05)0.9883 (0.05)0.285 (0.05)0.921 (0.05)0.675 (0.05)The values in parentheses in the tables refer to estimated significance level. According to the results of the two tables, it is concluded that proposed tests are always more powerful than the other tests. Even in small sample size, the proposed tests perform very well and better than the others.In the following example, it is used real data set to illustrate how the proposed tests can be applied.Example 3 (As an application to real data).
The following data represent the time for break down of a type of electrical insulating material subject to a constant-voltage stress (Nelson [33]).(56)1.97,0.59,2.58,1.69,2.71,25.50,0.35,0.99,3.67,2.07,0.96,5.35,2.90,13.77.
This data recently were used by Tiku and Akkaya [34]. They established that the null hypothesis where data come from exponential distribution cannot be rejected at 10 percent significance level. It is obvious that data come from a distribution with long tail on the right-hand side. So, Pareto distribution can be another suggested distribution for such data. For testing H0:X∼Pareto versus H1:X≁Pareto, first the parameters of Pareto distribution are estimated with shape parameter α=0.51 and scale parameter β=0.35. Then, based on data, the values of the proposed statistics (with b=1.2), Kolmogorov–Smirnov, and Cramer–von Mises statistics have been obtained as follows:(57)DP=0.563,DE=1.958,D=0.286,W2=0.2989.
Also, with 10000 replications based on estimated Pareto distribution, the critical values at 10 percent significance level have been calculated, respectively,(58)0.75,2.3,0.48,1.22.
Hence, the null hypothesis that data come from Pareto distribution cannot be rejected using this data.



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*Source: 4262574-2020-07-15.xml*