# A Novel Goodness-of-Fit Test for Cauchy Distribution

Authors: A. Pekgör
Journal: Journal of Mathematics
 (2023)

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## Abstract

Recently, several goodness-of-fit tests for Cauchy distribution have been introduced based on Kullback–Leibler divergence and likelihood ratio. It is claimed that these tests are more powerful than the well-known goodness-of-fit tests such as Kolmogorov–Smirnov, Anderson–Darling, and Cramér–von Mises under some cases. In this study, a novel goodness-of-fit test is proposed for the Cauchy distribution and the asymptotic null distribution of the test statistic is derived. The critical values of the proposed test are also determined through a Monte Carlo simulation for different sample sizes. The power analysis shows that the proposed test is more powerful than the current tests under certain cases.


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## Body

## 1. Introduction

The Cauchy distribution, which has many applications, is one of the most popular heavy-tailed models in statistics. For instance, the Cauchy distribution can been used to model the points of impact of a fixed straight line of particles emitting from a point source [1], the energy of an unstable state in quantum mechanics [2], the lifetime of natural broadening [3], the observations for spinning objects [4], the value-at-risk in financial risk management [5], the polar and non-polar liquids in porous glasses [6], the energy of an unstable state [7], the contact resistivity [8], the hypocenters on focal spheres of earthquakes [9], the velocity differences induced by different vortex elements [10], and the energy width of the state [11]. Since the Cauchy distribution has a wide range of applications, goodness-of-fit (GOF) tests for the Cauchy distribution have been studied by many researchers (see, for example, [12–16]). This paper aims to propose a novel goodness-of-fit test for the Cauchy distribution and compare the power performance with some already existing tests.This paper is organized as follows. In Section2, the proposed GOF test statistic for Cauchy distribution is introduced and the critical values are simulated for different sample sizes and significance levels. The asymptotic null distribution of the proposed test statistic is derived. An R function is provided for computing the simulated and approximate p values. The power performance of the current and proposed test procedure is studied and some remarks are given in Section 3. In Section 4, two illustrative examples are given. In Section 5, some concluding remarks are provided.


## 2. Proposed Test Procedures

LetX be a Cauchy random variable with probability density function (pdf)(1)fx;μ,σ=1πσ1+x−μ/σ2,x∈ℝ,and cumulative distribution function (cdf)(2)Fx;μ,σ=12+1πtan−1x−μσ,x∈ℝ,where μ∈ℝ is the location parameter and σ>0 is the scale parameter. The Cauchy distribution with parameter μ=0 and σ=1 is called the standard Cauchy distribution. Let us consider the index(3)Rθ,p=ξp3−ξp2ξp4−ξp1,where p=p1,p2,p3,p4, 0<p1<p2<p3<p4<1, θ=μ,σ is the parameter vector, and ξp=F−1p;μ,σ is the p-th quantile for any distribution. It is clear that the index in equation (3) lies in the interval 0,1 and the index is constant for all of μ and σ for the Cauchy distribution with cdf in equation (2). These properties are also valid for the symmetric distributions with location-scale parameters μ and σ. In other words, the index in equation (3) is not affected by the location shift and scale shift for the symmetric distributions. In this paper, we focus on developing a GOF test for the Cauchy distribution. The index in equation (3) for Cauchy distribution is denoted by(4)RCμ,σ,p=ξp3C−ξp2Cξp4C−ξp1C,where ξpC, is the p-th quantile of Cauchy distribution with parameters μ and σ. The choice of the vector p=p1,p2,p3,p4 is an issue in developing a GOF test based on the index in equation (3). For instance, choosing p1 and p4 as in equation (6) by maximizing the distance between the indexes of the Cauchy and normal distributions when the p2=0.25 and p3=0.75 can be thought of maximizing the ability of the test to detect the difference between the Cauchy and normal distributions. For Cauchy distribution, we consider(5)RCμ=0,σ=1,p∗=0.02217468,0.25,0.75,0.97782532=0.069776725.In equation (5), p2=0.25 and p3=0.75 are prefixed, and p1 and p4 are obtained as 0.02217468 and 0.97782532, respectively, by maximizing the distance(6)ξ0.75C−ξ0.25Cξp4C−ξp1C−Φ−10.75;0,1−Φ−10.25;0,1Φ−1p4;0,1−Φ−1p1;0,1,with respect to p1 and p4, where Φ−1⋅;μ,σ is the inverse cdf of normal distribution with mean μ and standard deviation σ. Since the normal distribution is a well-known distribution with a moderate tail, the absolute difference of the indexes for Cauchy and normal distribution is maximized to maximize the power compared to the normal alternative. It is noted that the optimality of p can also be discussed for discriminating Cauchy and the other distributions. That is, this idea can, of course, be extended to other alternative distributions. The values of the index Rθ,p∗ in equation (3) for different distributions are presented in Table 1.Table 1
Some values of indexRθ,p∗ for several distributions.
DistributionsRθ,p∗Lomax (1, 0.5)0.00700Pareto (0.5, 1)0.00700Weibull (0.2, 1)0.01752Pareto (0.75, 1)0.03061Gamma (0.1, 1)0.03353Lomax (1, 1)0.06050Pareto (1, 1)0.06050Cauchy (2, 3)0.06978Cauchy (0, 1)0.06978Pareto (1.25, 1)0.08848Pareto (1.5, 1)0.11227Weibull (0.5, 1)0.12677Lomax (1, 2)0.14819Loglogistic (2, 3)0.17792t (2)0.17792Log normal (0, 1)0.19817Laplace (0, 1)0.22247Laplace (2, 3)0.22247t (3)0.22893Gamma (0.5, 1)0.23355Loglogistic (4, 3)0.25412Exp (1)0.29015Logistic (2, 3)0.29015Logistic (0, 1)0.29015t (10)0.30443Gumbel (0, 1)0.30624Gamma (2, 1)0.31560Normal (0, 1)0.33544Normal (2, 3)0.33544Weibull (1.5, 1)0.34225Weibull (2, 1)0.35577Kumaraswamy (2, 5)0.38539Beta (2, 2)0.42212Beta (2, 1)0.43578Beta (1, 2)0.43578Arcsine (−1, 1)0.44510Uniform (0, 1)0.52320Uniform (−10, 10)0.52320Kumaraswamy (1, 1)0.52320Kumaraswamy (0.5, 0.5)0.68950Beta (0.5, 0.5)0.70883Beta (0.1, 0.1)0.99831In the simulation study, we considerp2=0.25 and p3=0.75 with p1,p4=0.20,0.80, 0.10,0.90, 0.05,0.95, 0.025,0.975, 0.02217468,0.97782532, and 0.01,0.99. In the simulation study, it is observed that the value of p=0.02217468,0.25,0.75,0.97782532 can be chosen in practice.It is clear from Table1 that the index in equation (3) less than 0.069776725 indicates that the distribution has a heavier tail than the Cauchy distribution, and most of the values of the index in equation (3) are greater than 0.069776725. Form this point of view, a test statistic for GOF test of the Cauchy distribution based on sample quantiles can be proposed. Let X=X1,X2,…,Xn be a random sample of size n. Let H¯0 and H¯1 be hypotheses:H¯0:X comes from the Cauchy distribution.H¯1:X does not come from the Cauchy distribution.In order to testH¯0 against H¯1, the GOF test statistic is defined as(7)APP=ξp3X−ξp2Xξp4X−ξp1X,where ξpX is a p-th p∈0,1 sample quantile based on sample X. The proposed statistic is also the estimator of RCμ,σ,p in equation (4). The test statistic APP in equation (7) lies between 0,1, and small and large values of APP indicate that the data come from a non-Cauchy distribution. The test function, based on the test statistic APP, can be defined as(8)ϕx=0,c1<APp<c2,1,otherwise,where c1 and c2 can be determined by using simulation for a fixed sample size and significance level. Although we define a statistic here, it should not be overlooked that infinitely many test statistics are proposed that cannot be counted with the change in p.We use the R function “hdi” to get the simulated critical valuesc1 and c2. Table 2 shows the simulated critical values with 100000 simulations for significance level 0.05 and sample sizes n=10,15,20,25,30,40,50,70,100, and 150. The six cases p1=0.20,0.25,0.75,0.80, p2=0.10,0.25,0.75,0.90, p3=0.05,0.25,0.75,0.95, p4=0.025,0.25,0.75,0.975, p5=p∗=0.02217468,0.25,0.75,0.97782532, and p6=0.01,0.25,0.75,0.99 are considered for APP test in this table. Since the null distribution of the test statistic APP is asymmetric, equal-tail critical values do not give a better power performance. Therefore, the critical values c1 and c2 are determined by the highest density interval (HDI) bounds. The HDI is the interval which contains the required mass such that all points within the interval have a higher probability density than points outside the interval. The R function “hdi” [17] in library “HDInterval” is used to get the critical values. The simulated critical values are given in Table 2 for α=0.05. The remaining tables for α=0.01 and α=0.10 are presented in the supplementary file (see Tables 12 and 13).Table 2
Some critical values for the tests at significance levelα = 0.05.
Statisticsn = 10n = 15n = 20n = 25n = 30n = 40n = 50n = 70n = 100n = 150KS0.27079160.23135510.19602410.18199030.16322720.14172600.12776480.10812830.09099650.0744883A20.92596341.00858270.98438201.06813601.02203671.03556931.04024401.04130231.06408481.0513005W20.12995120.14751820.13613090.14877710.14006760.14102530.14119780.14120480.14347350.1423438D12.08584491.67333861.46581101.34769901.24345161.07811970.94826170.75440320.57913090.4209202D21.27156731.20515581.15838161.13080741.10341021.06914281.04356801.00530050.96706290.9260567D31.32833061.13762170.97612070.85930120.76336750.62795170.53391760.40842650.30279730.2142463D40.84011140.80186330.74069430.70037430.65435840.58870130.53698090.45731840.37937600.3014972D51.75543741.45763611.26441641.15347961.04383660.87148140.74267380.56506210.41187290.2836562D61.36303531.24165091.11006231.01774000.92419200.79336680.69607750.55562020.42850930.3131210D71.11234571.00192220.86543910.82388270.73948870.66412900.61431140.56121890.52233200.4884361Dn,λ0.15142540.14935080.15096260.16731230.15968960.15786390.16051700.15998280.16249870.1617073ZA3.76116393.67939673.61160253.57683863.53851783.49346453.46282413.42256583.39000733.3613262ZC12.626151014.840283115.880426817.434501917.876107018.963806119.988034720.965571922.538608223.7266435ZK1.92642082.28843512.51293282.80476622.89847913.08379943.24516143.43033653.65307163.8507040MA20.61217050.61416990.61824480.61946280.62357000.62257320.62564310.62299110.63092460.6280443APp1(0.4349739, 0.9349241)(0.5352918, 0.9313266)(0.5137147, 0.9337732)(0.4858210, 0.9476388)(0.5190890, 0.9200224)(0.5438602, 0.9098861)(0.5567621, 0.8964914)(0.5762001, 0.8767661)(0.5964245, 0.8553014)(0.6195406, 0.8344835)APp2(0.0322042, 0.6905924)(0.0831989, 0.6543933)(0.1101955, 0.6189048)(0.1138894, 0.5910641)(0.1458912, 0.5791273)(0.1603971, 0.5397994)(0.1740842, 0.5147346)(0.1946389, 0.4836451)(0.2159806, 0.4607291)(0.2341656, 0.4321215)APp3(0.0000071, 0.5241523)(0.0000146, 0.4289383)(0.0130453, 0.4193312)(0.0260534, 0.4029930)(0.0287134, 0.3626796)(0.0458372, 0.3405257)(0.0564848, 0.3164474)(0.0666044, 0.2858383)(0.0805256, 0.2669331)(0.0935328, 0.2434633)APp4(0.0000022, 0.4826172)(0.0000068, 0.3667168)(0.0000036, 0.2996265)(0.0000049, 0.2606636)(0.0000049, 0.2362534)(0.0066456, 0.2321035)(0.0115683, 0.2118949)(0.0177521, 0.1792437)(0.0258474, 0.1622622)(0.0333565, 0.1435291)APp5(0.0000022, 0.4786860)(0.0000064, 0.3615346)(0.0000032, 0.2931898)(0.0000042, 0.2524953)(0.0000038, 0.2254776)(0.0000017, 0.2009124)(0.0091070, 0.2045620)(0.0126942, 0.1669218)(0.0199464, 0.1524896)(0.0276135, 0.1335041)APp6(0.0000019, 0.4624685)(0.0000051, 0.3415483)(0.0000023, 0.2693888)(0.0000026, 0.2246124)(0.0000089, 0.1932879)(0.0000004, 0.1546240)(0.0000024, 0.1308791)(0.0000029, 0.1041793)(0.0031911, 0.0997057)(0.0053126, 0.0793512)In the following, we obtain the asymptotic null distribution of the test statisticAPP. Let X be the random sample from the Cauchy distribution with a pdf in equation (1). One can write(9)nξp1Xξp2Xξp3Xξp4X−ξp1ξp2ξp3ξp4⟶dN40,Σ,where f and F are pdf and cdf of a Cauchy distribution with parameters μ and σ, ξpi=F−1pi;μ,σ, i=1,2,3,4, and(10)∑=p11−p1f2ξp1p11−p2fξp1fξp2p11−p3fξp1fξp3p11−p4fξp1fξp4p21−p2f2ξp2p21−p3fξp2fξp3p21−p4fξp2fξp4p31−p3f2ξp3p31−p4fξp3fξp4p41−p4f2ξp4⋅Let us consider the statisticAPP which is a tool for testing GOF to Cauchy distribution, where 0<p1<p2<p3<p4<1. Using the result in equation (9) and Cramér’s delta rule, the asymptotic null distribution of APP can be obtained as(11)nAPP−ξp3−ξp2ξp4−ξp1⟶dN0,σp2,where(12)σp2=κ5κ6,with(13)κ1=−p4−ξp3+ξp22−1+p4fξp32+2p3fξp4−ξp3+ξp2−ξp4+ξp1−1+p4fξp3−p3fξp42−ξp4+ξp12−1+p3,κ2=−−ξp3+ξp2−1+p4fξp3+−ξp4+ξp1fξp4−1+p3,κ3=κ1fξp22+2κ2fξp3fξp4p2−ξp4+ξp1fξp2−p2fξp32fξp42−ξp4+ξp12−1+p2,κ4=−ξp3+ξp2−1+p4fξp3−−ξp4+ξp1fξp4−1+p3fξp2+−ξp4+ξp1fξp4fξp3−1+p2,κ5=κ3fξp12+2κ4p1fξp2fξp3fξp4−ξp3+ξp2fξp1−p1fξp22fξp32fξp42−ξp3+ξp22−1+p1,κ6=−ξp4+ξp14fξp12fξp22fξp32fξp42.It is noticed thatσp2 is invariant for the values of Cauchy parameters μ and σ. Since the null distribution of the test statistic is not symmetric, the p value of the test is not well defined. Rohatgi and Saleh [18] indicated that “if the distribution is not symmetric, the p value is not well defined in the two-sided case, although many authors recommend doubling the one-sided p value.” Although the exact null distribution of APP is not symmetric, the asymptotic distribution in equation (11) is symmetric. Then, we can obtain an approximate p value as(14)2×min1−ΦAPP;RC0,1,p,σp2n,ΦAPP;RC0,1,p,σp2n⋅The R function shown in the supplementary file (see Figure11) can be used to compute the p values in equation (14) by using simulation and asymptotic distribution in equation (11).The approximation behavior of the asymptotic distribution is demonstrated in Figure1. It is observed that large sample size is needed to desirable approximation. However, simulation study indicates that approximate p values are close to simulated p values for moderate sample cases.Figure 1
The approximation behavior of the asymptotic distribution ofAPp5. (a) n = 250. (b) n = 500. (c) n = 1000. (d) n = 5000.
(a)(b)(c)(d)The empirical type-I error rates are given for the Cauchy distribution with different parameters in the supplementary file (see Table11). It is observed that these simulated type-I error rates are not affected by the Cauchy parameters, which indicates that the statistic APP can be used for GOF test of Cauchy distribution.


## 3. Power Analysis

LetX=X1,X2,…,Xn be a random sample from a population with cdf G⋅ and pdf g⋅ and the corresponding order statistic be X1<X2<⋯<Xn. The null and alternative hypotheses of the GOF test can be expressed as(15)H0:Gx=Fx;μ,σforGx≠Fx;μ,σfor anyμ∈ℝ,σ>0,someμ∈ℝ,σ>0,H1:Gx≠Fx;μ,σfor anyμ∈ℝ,σ>0,where F is a cdf in equation (2).In this study, we consider some current test procedures for testingH0 against H1. These test statistics are described in the following. The well-known test statistics Kolmogorov–Smirnov (KS), Anderson–Darling A2, and Cramér–von Mises W2 are given, respectively, by(16)KS=maxi=1,2,…,nmaxin−F0Xi;μ^,σ^,F0Xi;μ^,σ^−i−1n,A2=−n−2n∑i=1ni−0.5logF0Xi;μ^,σ^+n−i+0.5log1−F0Xi;μ^,σ^,W2=112n+∑i=1nF0Xi;μ^,σ^−i−0.5n2,where μ^ and σ^ are reasonable estimates of the location and scale parameters. In simulation study and numerical example, we use(17)μ^=ξ0.5X,σ^=12ξ0.75X−ξ0.25X,to estimate the parameters μ and σ which are in closed form. Recently, some GOF tests for Cauchy distribution based on Kullback–Leibler divergence are introduced in [14, 15]. Below, these test statistics are presented, which are obtained by using different kinds of Shannon entropy estimators. These statistics are defined as(18)Di=exp−Hi−1n∑j=1nlogfXj;μ^,σ^,i=1,2,…,7,where Hi’s are(19)H1=1n∑i=1nlogn2mXi+m−Xi−m,H2=−1n∑i=1nlog1nh∑j=1nKXi−Xjh,H3=−1n−m∑i=1n−mlogn+1mXi+m−Xi+∑i=mn1i−logn+1m,H4=1n∑i=1nlogndimYi+m−Yi−m,H5=−1n∑i=1nlog∑j=i−mi+mXj−X¯ij−in∑j=i−mi+mXj−X¯i2,H6=∑i=1nF^yXi+m−F^yXi−m∑j=1nF^yXj+m−F^yXj−m×logXi+m−Xi−mF^yXi+m−F^yXi−m,H7=−1n∑i=1nlog1/nh∑j=1nKXi+m−Xj/h−1/nh∑j=1nKXi−m−Xj/h2⋅Furthermore,m is the window size which is given in [15], K⋅ is a symmetric kernel function, and h (in the simulation study, it is fixed at 1.06 s n−1/5 where s is the sample standard deviation) is the bandwidth:(20)di=1+i+1m−im2,1≤i≤m,2,m+1≤i≤n−m,1+n−im+1,n−m+1≤i≤n,and Yi’s are(21)Yi−m=a+i−1mX1−a,1≤i≤m,Yi=Xi,m+1≤i≤n−m,Yi+m=b−n−imb−Xn,n−m+1≤i≤n,a=x¯−ks,b=x¯+ks,x¯=1n∑i=1nxi,s2=1n−1∑i=1nxi−x¯2,X¯i=12m+1∑j=i−mi+mXj,i=1,2,…,n,F^yXi=n−1nn+1i+1n−1+Xi−Xi−1Xi+1−Xi−1,F^yX1=1−F^yXn=1n+1⋅Gürtler and Henze [12] proposed a test based on the empirical characteristic function which is given by(22)Dn,λ=2n∑j=1n∑k=1nλλ2+Yj−Yk2−4∑j=1n1+λ1+λ2+Yj2+2n2+λ,where Yj=Xj−μ^/σ^,j=1,2,…,n.Zhang [13] introduced three GOF test statistics based on likelihood ratio, and these are given by(23)ZK=maxi=1,…,ni−0.5logi−0.5nF0Xi;μ^,σ^+n−i+0.5logn−i+0.5n1−F0Xi;μ^,σ^,ZA=−∑i=1nlogF0Xi;μ^,σ^n−i+0.5+log1−F0Xi;μ^,σ^i−0.5,ZC=∑i=1nlog1/F0Xi;μ^,σ^−1n−0.5/i−0.75−12⋅Villaseñor and González-Estrada [16] proposed a test based on the Anderson–Darling test which is given by(24)MA2=−n−1n∑i=1n2i−1logpi+2n+1−2ilog1−pi,where pi=1−exp−vi/v¯,vi=−logF0xi;θ^MLE, and v¯=∑i=1nvi/n.In this section, we use a simulation study to evaluate the power performance of the proposed test and compare it with those existing test procedures. The power of different test procedures is estimated with 100000 trials with significance level of 5% for different alternatives. The six casesp1=0.20,0.25,0.75,0.80, p2=0.10,0.25,0.75,0.90, p3=0.05,0.25,0.75,0.95, p4=0.025,0.25,0.75,0.975, p5=0.02217468,0.25,0.75,0.97782532, and p6=0.01,0.25,0.75,0.99 are considered in the simulation study. The following distributions and the corresponding R packages are considered in the simulation study. We refer to [19–22] for the R libraries “stats,” “extraDistr,” “distr,” and “flexsurv,” respectively.(i)
The distributions used in R library “stats”: normal,t, logistic, beta, gamma, uniform, exponential, and Weibull.(ii)
The distributions used in R library “extraDistr”: Pareto, Laplace, Gumbel, and Lomax.(iii)
The distributions used in R library “distr”: arcsine.(iv)
The distributions used in R library “flexsurv”: loglogistic.In the power study, in addition to the well-known distributions, data are generated from other distributions for the alternative hypothesis. SupposeX1,X2,X3, and X4 are independent random variables with distributions t (2), normal (0, 1), uniform (−1, 1), and Cauchy (0, 1), respectively. Let us consider the following transformations having possible heavily tailed distributions (HTD): X1/X2, X2/X1, X3/X2, 1/X1, X2/X3, 1/X2, 1/X3, and X1+X4. The distributions of these transformations are denoted by HTDii=1,2,…,8, respectively.The simulation results are presented in Figures2 and 3. The other power analysis results are available in the supplementary file (see Tables 1–10 and Figures 1–10). From these tables and figures, we observe that if the underlying distributions are symmetric, the proposed test procedure has a good performance among the tests considered here for almost all the sample sizes. The test based on APP gives desired power values for almost all the sample sizes when the data come from distributions with lighter tails (say Rθ,p>0.3). For large sample cases, the APP test has a similar power performance compared to the other test procedures. The APP test does not work well when the data come from a distribution with heavier tails than the Cauchy distribution for small and moderate sample cases. However, APP1 and APP2 have a better power than APP3, APP4, APP5, and APP6.Figure 2
The empirical powers for different alternative distributions (1/2).Figure 3
The empirical powers for different alternative distributions (2/2).For the detailed comparison, the following findings may be illustrative. For small sample cases,APP tests are competitive with Di tests. Also, APP tests are better than the tests based on empirical cumulative distribution function ecdf and Z tests. For moderate and large sample cases, APP tests are better than Di tests. It should be noted that the Z tests are better compared to other tests. APP tests have better performance for the large sample and large Rθ,p index cases. When the Rθ,p index value increases, it is observed that the APP test outstrips the other tests. The power of the APP tests has bad power when the distribution has a heavier tail than the Cauchy distribution (that is, Rθ,p<0.06978. However, APP test has the desired power when the sample size increases. The index Rθ,p for the alternative distribution becomes smaller, and p2 and p4 should be chosen large and small, respectively.Figure4 shows the performance plots of the APP test statistics versus the Rθ,p index values of the alternative distributions. Figure 4 gives detailed information to compare the APP tests and to observe the behavior of the APP test for the selection of p. The power of APP test is low when Rθ,p is around RCθ,p=0.06978. It can be seen that the power of the APP test statistic increases as Rθ,p becomes larger than RCθ,p for all p. When Rθ,p is less than RCθ,p and the sample size is small, the powers of the APP1 and APP2 test statistics are higher than those of the other APP test statistics. When Rθ,p is greater than RCθ,p, it can be seen that the powers of the APP5 and APP6 test statistics outperform the other APP tests.Figure 4
Power graphs based onRθ,p index for different sample sizes. (a) n = 10. (b) n = 50. (c) n = 100. (d) n = 150.
(a)(b)(c)(d)


## 4. Illustrative Examples

In this section, for illustrative purposes, 30 returns of closing prices of the German Stock Index (DAX) are considered. The data, presented in Table3, are also used in [14, 15]. The GOF test statistics and the corresponding p values are presented in Table 4. The proposed statistic and related p value have been calculated using the AP.test function available in the supplementary file (see Figure 11). Considering Figure 5 and Table 4, the null hypothesis that the data follow the Cauchy distribution is not rejected at significance level of 0.05.Table 3
The DAX data.
−0.09179−0.00560−0.00514−0.00092−0.000220.001180.001900.002450.008750.01468−0.00662−0.00525−0.00479−0.000510.000500.001620.002000.002680.011080.01496−0.00576−0.00518−0.00424−0.000310.000740.001780.002100.006800.011750.05207Table 4
Test statistic and correspondingp values from real data.
Test statisticsStatisticsp valuesCritical valuesKS0.126320.678200.16323A20.497680.747811.02204W20.076320.717040.14007D10.66129>0.051.24345D20.84347>0.051.10341D30.25500>0.050.76337D40.30236>0.050.65436D50.38581>0.051.04384D60.35753>0.050.92419D70.46086>0.050.73949Dn,λ0.05107>0.050.15969ZA3.34629>0.053.53852ZC5.76070>0.0517.87611ZK1.34297>0.052.89848MA20.46401>0.050.62357APp10.58944>0.05(0.51909 0.92002)APp20.41181>0.05(0.14589 0.57913)APp30.34518>0.05(0.02871 0.36268)APp40.13172>0.05(0.00000 0.23625)APp50.11149>0.05(0.00000 0.22548)APp60.06708>0.05(0.00001 0.19329)Figure 5
Cauchy Q-Q plot of real data.In order to observe the ability of the GOF tests, we generate data fromtdistribution with 3 degrees of freedom which is close to the Cauchy distribution. The simulated data are presented in Table 5. Q-Q plot in Figure 6 is also provided to show that the data do not fit the Cauchy distribution. The results are given in Table 6. It is clear from Table 6 that D1, D2, D4, D5, D6, D7, Dn,λ, ZA, MA2, APP3, APP4, APP5, and APP6 reject the null hypothesis at significance level of 0.05.Table 5
Simulated dataset.
−7.39751−1.00157−0.58343−0.36200−0.239410.020790.294620.775501.039882.03169−2.19172−0.98322−0.56932−0.34136−0.213010.057710.448900.809101.108122.37101−1.37478−0.86532−0.54262−0.33343−0.182080.129100.684280.877481.149732.60013−1.34010−0.84523−0.48439−0.32489−0.013470.266250.692590.887751.519232.97099−1.21350−0.70272−0.46140−0.314380.018320.287710.702580.943001.754193.00023Figure 6
Cauchy Q-Q plot of simulated data.Table 6
Test statistic and correspondingp values from simulated data.
Test statisticsStatisticsp valuesCritical valuesKS0.107050.578370.12776A20.806480.475731.04024W20.098100.596160.14120D10.99278∗<0.050.94826D21.14771∗<0.051.04357D30.53262>0.050.53392D40.57399∗<0.050.53698D50.77456∗<0.050.74267D60.72989∗<0.050.69608D70.67559∗<0.050.61431Dn,λ0.16731∗<0.050.16052ZA3.48292∗<0.053.46282ZC19.86963>0.0519.98803ZK2.42161>0.053.24516MA20.87240∗<0.050.62564APp10.88455>0.05(0.55676 0.89649)APp20.49504>0.05(0.17408 0.51473)APp30.36006∗<0.05(0.05648 0.31646)APp40.28362∗<0.05(0.01157 0.21189)APp50.27440∗<0.05(0.00911 0.20456)APp60.17727∗<0.05(0.00000 0.13088)


## 5. Conclusion

In this article, a novel GOF test procedure based on quantiles is proposed for Cauchy distribution. The simulation study reveals that the proposed GOF test procedure has a good power performance and it can be used for testing, whether the data come from the Cauchy distribution or not. An R function is provided to compute the test statistic and the approximatep value. As a future study, the statistic APP can be extended for GOF testing for any continuous distributions without a shape parameter. The introduced test can also be extended under different types of censoring schemes such as progressive or progressive first-failure censoring.



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*Source: 9200213-2023-03-14.xml*