Optoelectronics, Instrumentation and Data Processing. 2001. - ą 2. - P. 76-88.

 

 

UDC 519.2

 

APPLICATION OF THE NONPARAMETRIC GOOGNESS-OF-FIT TESTS IN TESTING COMPOSITE HYPOTHESES*

 

B.Yu. Lemeshko and S.N. Postovalov

 

1. Introduction

 

Testing for fit of an obtained experimental distribution and a theoretical one of the most common problems of statistical analysis in processing of experimental results. Applying the goodness-of-fit tests, one distinguishes testing for simple and composite hypotheses. A simple hypotheses tested has the form : , where  is the probability distribution function to which the observed sample is tested for fit, and  is the known value of the parameter (either scalar or vector one). A composite tested hypothesis has the form : . In this case, the estimate of the distribution parameter  is calculated by the same sample by which the fit is tested.

While testing the fit by a sample we calculate the value  of statistic of the test used. To obtain the conclusion on accepting or rejecting the hypothesis , we must know the conditional distribution  of statistic  under validity of hypothesis . And if the probability

is sufficiently great, at least , where  is the conditional density, and  is the prescribed significance level (the probability of a first-kind error – to reject the true hypothesis ), then it is usually considered that there are no grounds for rejecting the hypothesis .

The most commonly used goodness-of-fit tests include nonparametric Kolmogorov tests and also  and  Mises tests. The value

,

where  is the empirical distribution function,  is the theoretical distribution function, and  is the sample size, is used as a distance between the empirical and theoretical laws in Kolmogorov test. For testing hypotheses, one usually uses statistic of the form [1]

,

where

, , ,

 are sample values in increasing order, and  is the distribution function, fit to which is tested. The distribution of statistic  in testing the simple hypothesis in the limit obeys Kolmogorov law  [1].

         In tests of the type of , the distance between the hypothetical and the true distributions is considered in the quadratic metric

,

where   is the mathematical expectation operator.

         In choosing  in Mises  tests, one uses a statistic (Cramer – Mises – Smirnov statistic) of the form

.

In testing a simple hypothesis it obeys the distribution  [1].

         In choosing  in Mises  tests, the statistic (Anderson – Darling statistic) has the form

.

In the limit, this statistic obeys the distribution  [1].

          In the case of simple hypotheses, the limiting statistic distributions of the nonparametric Kolmogorov,  and  Mises tests are known for a long time and do not depend on the kind of distribution law observed and its parameters. These tests are said to be “distribution-free” tests. This advantage predetermines common applications of these tests.

 

2. Losing the “distribution–freeness” in testing composite hypotheses

 

While testing the composite hypotheses, when the same sample is used to estimate the parameters of the observed law , the nonparametric goodness-of-fit tests lose the property of “distribution–freeness”. However, the nonparametric test power in testing the composite hypotheses for the same sample sizes is always much higher than that in testing simple ones. And whereas in testing the simple hypotheses the nonparametric Kolmogorov,  and  Mises tests have a lower power compared with the -type tests provided that the latter use the asymptotically optimal grouping [2-5],  in testing the composite hypotheses the nonparametric tests appear to be more powerful. To make use of their advantages, we must merely know the distribution  for the tested composite hypotheses.

The distinctions in the limiting distributions of the same statistics in testing simple and composite hypotheses are such significant that we cannot neglect them. Hence, many publications [6-8] warned against inaccurate application of the goodness-of-fit tests in testing composite hypotheses.

Paper [9] was the pioneer in investigating the limiting statistic distributions of the nonparametric goodness-of-fit tests in testing the composite hypotheses. The literature presents several approaches to using the nonparametric goodness-of-fit tests in the case of testing the composite hypotheses. When the size of a sample is large, it can be divided into two parts , and use one part for estimating the parameters and the other part for testing the fit [10]. In some particular cases, the limiting statistic distributions were analyzed analytically [11], the percent points of the distributions were constructed by statistical simulation [12-15]. For approximate calculation of the probabilities of “fit” of the form of  (achievable significance level), some authors constructed formulas that give sufficiently good approximations for small values of the corresponding probabilities [16-20]. Authors of [21-24] investigated the statistic distributions of the nonparametric goodness-of-fit tests and constructed models of these distributions by means of computer analysis of statistical regularities.

It has been found that in composite hypotheses tests, the conditional distribution law of the statistic  is affected by a number of factors determining the hypotheses complexity: the form of the observed law  corresponding to the true hypothesis ; the type of the parameter estimated and the number of parameters to be estimated; sometimes, it is a concrete value of the parameter (e.g., in the case of gamma-distribution); the method of parameter estimation.

For example, Figure 1 illustrates  as a function of form of the observed law  corresponding to hypothesis  for Kolmogorov test. Figure 2 shows distributions of the Mises  statistic for test of fit to Weibull distribution using various estimation methods: maximum likelihood estimates (MLE) and minimum distance (MD) estimates obtained by minimizing the value of the statistic used in the test.

Distributions of the goodness-of-fit statistics depend substantially on the parameter estimation method. Strictly speaking, each type of estimates for a concrete hypothesis tested is associated with its own limiting statistic distribution . Applying the nonparametric goodness-of-fit tests, one must consider the estimation method used. For the maximum likelihood method the statistic distributions  depend strongly on the law corresponding to hypothesis . The scatter of the distributions  when using MD estimates minimizing the test statistic is much less dependent on the law , corresponding to hypothesis .

          For the MD estimates minimizing the test statistic, the empirical distributions  corresponding to different hypotheses  have the minimal scatter. Hence, we may speak about certain “distribution-freeness” for the tests considered. If we use only this fact as a background, it would seem that only such estimation methods must be applied for testing composite hypotheses. However, investigation of the power of the tests considered using different estimation methods has shown that for close alternatives the tests have the greatest power in the case of using MLE.

Fig. 1. Distribution functions  of statistic  of Kolmogorov test:

1 – for simple hypothesis testing; 2-5 – for calculating MLE of two parameters of Laplace, normal, Cauchy, and logistic distributions, respectively.

 

 

Fig. 2. Distribution functions of statistic of the Mises  test for test of fit to Weibull distribution: 1 – for simple hypothesis testing; 2 – for calculating MLE of two parameters of the distribution; 3 – for calculating MD estimates of two parameters.

 

For small sample sizes the distributions  depend on . However, we observe a substantial dependence of the statistic distribution on  only for small sample sizes. Investigation has shown, that for  the distributions  are sufficiently close to the limiting , and we may neglect the dependence on .

 

3. Construction of approximations for the limiting statistic distributions

 

The limiting statistic distributions of the nonparametric tests and tables of the percent points constructed by the present moment are bounded by a rather narrow range of composite hypotheses.

The infinite number of random variables that can be met in practice cannot be described by a limited subset of models of distribution laws that are most frequently used to describe real observations. Any researcher can propose (construct) a parametric distribution model of his own for a concrete observed variable, i.e., such a model that describes most adequately this random variable, from his point of view. Upon estimating by the given sample the model parameters, it becomes necessary it test the hypothesis on adequacy of the sample observations and the constructed law by means of the goodness-of-fit tests. The next problem consists in knowledge of the limiting statistic distribution corresponding to the given composite hypothesis.

Constructing of the limiting distribution by analytical methods is an extremely complicated problem. It is most suitable to use the method of computer analysis of statistical regularities. The method showed good results in simulating the test statistic distributions [21-24].

For this purpose, we must according to the law  simulate  samples of the same size  as the sample for which we must test hypothesis : , and then for each of  samples calculate estimates of the same law parameters and the value of statistic  of the corresponding goodness-of-fit test. As a result, we will obtain a sample of values of statistic  with the distribution law  for the hypotheses . With this sample for a considerable number  we can construct a sufficiently smooth empirical function of distribution , that can be used in order to conclude whether we must accept hypothesis . If it is necessary, using  we can construct an approximate analytical model approximating , and then, on the basis of this model, make the decision on the hypothesis tested.

Investigation has shown that a good analytical model for  is often represented by one of the following laws: log-normal, gamma distribution, Su-Johnson distribution, and Sl-Johnson distribution [23-24]. At least, basing on the limiting number of distribution laws, we can always construct a model in the form of a mixture of laws.

Implementation of such procedure of computer analysis of statistic distributions contains neither difficulties of principal nor practical difficulties at present. The level of computing allows one to obtain quickly results of simulation, and an engineer who can write programs is able to implement the algorithm. In this paper, we constructed models approximating the limiting statistic distributions for some composite hypotheses with the use of MLE and MD estimates.

      Table 1 contains a list of distributions relative to which we can test composite fit hypotheses using the constructed approximations of the limiting statistic laws. The statistic distribution models constructed by applying the method of computer analysis of statistical regularities are presented in Tables 2–7 (Table 2, Table 3, Table 4, Table 5, Table 6, Table 7). Tables of the models of the limiting statistic distributions, including tables of percent points, for a wider class of tested composite hypotheses are listed on WEB-site http://www.ami.nstu.ru/~headrd/seminar/nonparametric/start2.htm  [25].

 

Table 1

 

Random variable distribution

Density function

Exponential

Seminormal

Rayleigh

Maxwell

Laplace

Normal

Log-normal

Cauchy

Logistic

Extreme-value (maximum)

Extreme-value (minimum)

Weibull

Gamma-distribution

 

 

         In tables 2–7, that contain the distributions  recommended for testing composite hypotheses,  denotes the log-normal distribution with the density function

,

  denotes the gamma distribution with the density function

 ,

 denotes the Sl-Johnson distribution with the density function

,

 denotes the Su-Johnson distribution with the density function

.

          As an example, we will show how strong is the change in the probability  for the same value of statistic in the case of simple and composite hypotheses. For illustration, we will use the Mises  test.

Example. Let we test a hypotheses on fit to Weibull distribution and the calculated statistic value . Hence, in the case of a simple hypotheses on the basis of the distribution  [1] we find, that . If using the sample we calculate MLE of two distribution parameters, then a good approximation of the limiting distribution (see Table 4) is the log-normal distribution lnN(-2.9541,0.5379), and the corresponding probability . For MD estimates in a similar situation, the most suitable model (see Table 5) is the Su-Johnson distribution Su(-1.5326, 1.4446, 0.0147, 0.0188) according to which .

 

4. Conclusion

 

To test composite hypotheses and choose (or construct) statistic distributions  of goodness-of-fit tests, one should take into account all factors that affect the statistic distribution law: the form of the law observed; the type of the parameter estimated and the number of the parameters; sometimes, a concrete parameter value; the method of parameter estimation.

The constructed approximations of the limiting statistic distributions of the nonparametric goodness-of-fit tests extend the region of correct application of these tests and may be recommended for construction of statistical regularities when it is impossible to solve the problem analytically.

 

References

 

1.      L.N. Bolshev and N.V. Smirnov, Tables of Mathematical Statistics (in Russian), Nauka, Moscow, 1983.

2.      V.I. Denisov, B.Yu. Lemeshko, and E.B. Tsoi, Optimal Grouping, Parameter Estimation, and Design of Regression Experiments (in Russian), NSTU, Novosibirsk, 1993.

3.      B.Yu. Lemeshko, Nadezhnost i Kontrol Kachestva, no. 8, p. 3, 1997.

4.      B. Yu. Lemeshko, Zavod. Lab., vol. 64, no. 1, p. 56, 1998.

5.      V.I. Denisov, B.Yu. Lemeshko, and S.N. Postovalov, Application Statistics. Rules for test of Fit of Experimental Distribution to Theoretical one. Methodical Recommendations, Part 1. The Chi-squire Tests, NSTU, Novosibirsk, 1998.

6.      A.I. Orliv, Zavod. Lab., vol. 51, no. 1, p. 60, 1985.

7.      B. V. Bondarev, Zavod. Lab., vol. 52, no. 10, p. 62, 1986.

8.      E.V. Kulinskaya and N.E. Savvushkina, Zavod. Lab., vol. 56, no. 5, p. 96, 1990.

9.      Kac M., Kiefer J., Wolfowitz J., Ann. Math. Stat., vol. 26, p. 189, 1955.

10.  Durbin J., Lect. Notes Math., vol. 566, p. 33, 1976.

11.  G.V. Martynov, Omega-Square Tests (in Rissian), Nauka, Moscow, 1978.

12.  E.S. Pearson, and H.O. Hartley, Biometrica tables for Statistics, University Press, Cambridge, 1972, vol. 2.

13.  M.A. Stephens, Journ. Roy. Statist. Soc., vol. B32, p. 115, 1970.

14.  M.A. Stephens, Journ. Amer. Statist. Assoc., vol. 69, p. 730, 1974.

15.  M. Chandra, N.D. Singpurwalla, and M.A. Stephens, Journ. Amer. Statist. Assoc., vol. 76, p. 375, 1981.

16.  Yu. N. Tyurin, Izv. AN SSSR. Ser. Mat. vol. 48, no. 6, p. 1314, 1984.

17.  Yu. N. Tyurin, and N.E. Savvushkina, Izv. AN SSSR. Ser. Tekhn. Kibernetika, no. 3, p. 109, 1984.

18.  Yu. N. Tyurin, Author’s Abstracts of Dissertation for Phys.-Math. Sci. Dr., MSU, Moscow, 1985.

19.  N.E. Savvushkina, Sb. Trudov VNII Sistemnykh Issledovanii, no. 8, 1990.

20.  Yu. N. Tyurin, and A.A. Makarov, Computer Data Analysis (in Russian), INFRA-M, Finansi i Statistika, Moscow, 1995.

21.  B.Yu. Lemeshko, and S.N. Postovalov, Nadezhnost i Kontrol Kachestva, no. 11, p. 3, 1997.

22.  B.Yu. Lemeshko, and S.N. Postovalov, Proc. of the IV Internat. Conf. on Actual Problems in Electronics Instrument Building, Novosibirsk, vol. 3, p. 12, 1998.

23.  B.Yu. Lemeshko, and S.N. Postovalov, Zavod. Lab., vol. 64, no. 3, p. 61, 1998.

24.  B.Yu. Lemeshko, and S.N. Postovalov, Application Statistics. Rules for testing for Fit of Experimental Distribution to Theoretical One. Methodical Recommendations, Part II. Nonparametric Tests. (in Russian), NSTU, Novosibirsk, 1999.

25.  http://www.ami.nstu.ru/~headrd/

 

The Novosibirsk State Technical University

 

E-mail: headrd@fpm.ami.nstu.ru


 

 

Table 2

Approximation of Limiting Kolmogorov Statistic Distributions

 when Maximum Likelihood Estimates are Used

Random variable distribution

Estimation of scale parameter

Estimation of shift parameter

Estimation of two parameters

Exponential

lnN(-0.3422,0.2545)

 

 

Seminormal

g(4.1332,0.1076,0.3205)

 

 

Rayleigh

lnN(-0.3388,0.2621)

 

 

Maxwell

lnN(-0.3461,0.2579)

 

 

Laplace

g(3.7580,0.1365,0.3163)

g(4.6474,0.0870,0.3091)

lnN(-0.3690,0.2499)

g(4.4525,0.0761,0.3252)

lnN(-0.4358,0.2276)

Normal

g(3.7460,0.1385,0.3142)

lnN(-0.4172,0.2272)

g(4.9014,0.0691,0.2951)

lnN(-0.4825,0.2296)

Log-normal

g(3.0622,0.1577,0.3547)

Su(-2.0328, 2.3642, 0.2622, 0.4072)

Su(-1.8093, 1.9041, 0.1861, 0.4174)

Cauchy

Su(-3.3278, 2.2529, 0.2185, 0.2858)

g(4.8247,0.0874,0.2935)

lnN(-0.5302,0.2427)

Logistic

g(3.2167,0.1476,0.3538)

Su(-2.8534, 3.0657, 0.2872, 0.3199)

lnN(-0.5611,0.2082)

Extreme-value (maximum)

g(3.3841,0.1439,0.3509)

g(4.1008,0.0997,0.3269)

g(4.9738,0.0660,0.3049)

Extreme-value (minimum)

g(3.3841,0.1439,0.3509)

g(4.1008,0.0997,0.3269)

g(4.9738,0.0660,0.3049)

Weibull

g(3.3841,0.1439,0.3509)

**

g(4.1008,0.0997,0.3269)

*

g(4.9738,0.0660,0.3049)

Note. ** - we estimated the Weibull distribution form parameter, * - the Weibull distribution scale parameter.

 


Table 3

Approximation of Limiting Distributions of Kolmogorov Minimum Statistic

 when MD Estimates Minimizing Statistic SK are Used

Random variable distribution

Estimation of scale parameter

Estimation of shift parameter

Estimation of two parameters

Exponential

g(4.4983,0.0621,0.2891)

 

 

Seminormal

g(4.2884,0.0705,0.3072)

 

 

Rayleigh

g(4.8579,0.0639,0.2900)

 

 

Maxwell

g(5.3106,0.0581,0.2865)

 

 

Laplace

g(3.0431,0.1355,0.3182)

g(5.0103,0.0602,0.2968)

lnN(-0.5358,0.2122)

Su(-2.1079, 2.4629, 0.1661, 0.3340)

lnN(-0.6970,0.1952)

Normal

g(3.2458,0.1343,0.3072)

lnN(-0.5469,0.2152)

lnN(-0.7236,0.1837)

Log-normal

g(3.2458,0.1343,0.3072)

lnN(-0.5469,0.2152)

lnN(-0.7236,0.1837)

Cauchy

g(3.4398,0.1255,0.3022)

lnN(-0.5182,0.2268)

Su(-1.6929, 2.5234, 0.1892, 0.3607)

lnN(-0.6946,0.1938)

Logistic

Su(-2.6522,1.8288, 0.1738, 0.3384)

g(3.6342,0.1284,0.2772)

Su(-3.8497, 3.2770, 0.2136, 0.2607)

lnN(-0.5511,0.2045)

lnN(-0.7389,0.1771)

Su(-2.5093, 3.1277, 0.1932, 0.3041)

Extreme-value (maximum)

g(3.5424,0.1203,0.2975)

Su(-1.9028, 2.3972, 0.2227, 0.389)

Su(-1.3144, 2.2480, 0.1616,0.3858)

lnN(-0.7174, 0.1841)

Extreme-value (minimum)

g(3.5424, 0.1203,0.2975)

Su(-1.9028, 2.3972, 0.2227, 0.389)

Su(-1.3144, 2.2480, 0.1616,0.3858)

lnN(-0.7174, 0.1841)

Weibull

g(3.5424, 0.1203, 0.2975)

**

Su(-1.9028, 2.3972, 0.2227, 0.389)

*

Su(-1.3144, 2.2480, 0.1616,0.3858)

lnN(-0.7174, 0.1841)

Note. ** - we estimated the Weibull distribution form parameter, * - the Weibull distribution scale parameter.


Table 4

Approximation of Limiting Distributions of Mises  Statistic

 when Maximum Likelihood Estimates are Used

Random variable distribution

Estimation of scale parameter

Estimation of shift parameter

Estimation of two parameters

Exponential

Su(-1.8734,1.2118, 0.0223, 0.0240)

 

 

Seminormal

Sl(0.9735,1.1966, 0.1531, 0.0116)

 

 

Rayleigh

Su(-1.5302,1.0371, 0.0202, 0.0299)

 

 

Maxwell

Su(-2.0089,1.2557, 0.0213, 0.0213)

 

 

Laplace

Sl(0.9719,0.9805,0.2347, 0.0139)

Su(-2.0821,1.2979, 0.0196, 0.0200)

Su(-1.6085,1.2139, 0.0171, 0.0247)

Normal

Su(-2.2550,0.9569, 0.0152, 0.0212)

lnN(-2.7536,0.5610)

lnN(-2.9794,0.5330)

Log-normal

Sl(1.0669,1.0010, 0.2537, 0.0144)

lnN(-2.7271,0.6092)

Su(-1.6292, 1.1541, 0.0144, 0.0234)

Cauchy

Sl(1.0086,1.0539, 0.2282, 0.0064)

Sl(1.1230,1.2964, 0.1383, 0.0105)

Sl(1.2420,1.2833, 0.1135, 0.0064)

Logistic

Sl(0.9982,1.0287, 0.2303, 0.0126)

Sl(1.3982,1.3804, 0.1205, 0.0102)

lnN(-3.1416,0.4989)

Extreme-value (maximum)

Sl(1.0056, 1.0452, 0.2296, 0.0137)

lnN(-2.5818,0.6410)

lnN(-2.9541,0.5379)

Extreme-value (minimum)

Sl(1.0056, 1.0452, 0.2296, 0.0137)

lnN(-2.5818,0.6410)

lnN(-2.9541,0.5379)

Weibull

Sl(1.0056, 1.0452, 0.2296, 0.0137)

**

lnN(-2.5818,0.6410)

*

lnN(-2.9541,0.5379)

Note. ** - we estimated the Weibull distribution form parameter, * - the Weibull distribution scale parameter.


Table 5

Approximation of Limiting Distributions of Mises  Statistic

when MD Estimates Minimizing Statistic  are Used

Random variable distribution

Estimation of scale parameter

Estimation of shift parameter

Estimation of two parameters

Exponential

Su(-1.9324,1.1610, 0.0134, 0.0203)

 

 

Seminormal

Su(-1.5024,1.0991, 0.0173, 0.0256)

 

 

Rayleigh

Su(-1.4705,1.1006, 0.0164, 0.0259)

 

 

Maxwell

Su(-1.7706,1.2978, 0.0188, 0.0220)

 

 

Laplace

Sl(1.0117, 0.9485, 0.2162, 0.0137)

lnN(-2.8601,0.5471)

lnN(-3.2853,0.4666)

Normal

Sl(1.0477, 0.9883, 0.2356, 0.0112)

lnN(-2.8649,0.5668)

lnN(-3.2715,0.4645)

Log-normal

Sl(1.0477, 0.9883, 0.2356, 0.0112)

lnN(-2.8649,0.5668)

lnN(-3.2715,0.4645)

Cauchy

Sl(1.2759, 1.0437, 0.2825, 0.0089)

lnN(-2.8577,0.5739)

lnN(-3.2603,0.4874)

Logistic

Sl(1.0898,1.0225, 0.2399, 0.0096)

lnN(-2.8831,0.5367)

lnN(-3.2915,0.4592)

Extreme-value (maximum)

Sl(1.0771, 1.0388, 0.2065, 0.0109)

Su(-1.5348,1.1226, 0.0166, 0.0252)

Su(-1.5326, 1.4446, 0.0147, 0.0188)

lnN(-3.2627,0.4680)

Extreme-value (minimum)

Sl(1.0771, 1.0388, 0.2065, 0.0109)

Su(-1.5348,1.1226, 0.0166, 0.0252)

Su(-1.5326, 1.4446, 0.0147, 0.0188)

lnN(-3.2627,0.4680)

Weibull

Sl(1.0771, 1.0388, 0.2065, 0.0109)

**

Su(-1.5348,1.1226, 0.0166, 0.0252)

*

Su(-1.5326, 1.4446, 0.0147, 0.0188)

lnN(-3.2627,0.4680)

Note. ** - we estimated the Weibull distribution form parameter, * - the Weibull distribution scale parameter.


 

 

Table 6

Approximation of Limiting Distributions of Mises  Statistic 

when Maximum Likelihood Estimates are Used

Random variable distribution

Estimation of scale parameter

Estimation of shift parameter

Estimation of two parameters

Exponential

Su(-2.8653,1.4220, 0.1050,0.1128)

 

 

Seminormal

Su(-2.5603,1.3116, 0.1147,0.1330)

 

 

Rayleigh

Su(-2.5610,1.4003, 0.1174,0.1337)

 

 

Maxwell

Su(-2.6064,1.4426, 0.1190,0.1285)

 

 

Laplace

Sl(0.3148, 1.0999, 0.6901, 0.1093)

Su(-2.5528,1.4006, 0.1216,0.1358)

Su(-2.8942,1.4897, 0.0846,0.1131)

Normal

Su(-2.3507,1.0531, 0.1012,0.1595)

Su(-3.1202,1.5233, 0.0874,0.1087)

Su(-2.7057,1.7154, 0.1043,0.0925)

Log-normal

Su(-2.4168, 1.1296, 0.1151, 0.1560)

lnN(-0.8052, 0.5123)

Su(-2.3966, 1.5967, 0.1012, 0.1179)

Cauchy

Su(-2.4935, 1.0789, 0.0923, 0.1458)

Su(-2.8420,1.3528, 0.1010,0.1221)

Su(-2.3195,1.1812, 0.0769,0.1217)

Logistic

Sl(0.3065, 1.1628, 0.7002, 0.0930)

Su(-3.5408,1.6041, 0.0773,0.0829)

lnN(-1.1452,0.4426)

Extreme-value (maximum)

Su(-2.5427, 1.1057, 0.0960, 0.1569)

Su(-2.5550, 1.3714, 0.1152, 0.1289)

Su(-2.4622, 1.6473, 0.1075, 0.1149)

Extreme-value (minimum)

Su(-2.5427, 1.1057, 0.0960, 0.1569)

Su(-2.5550, 1.3714, 0.1152, 0.1289)

Su(-2.4622, 1.6473, 0.1075, 0.1149)

Weibull

Su(-2.5427, 1.1057, 0.0960, 0.1569)

**

Su(-2.5550, 1.3714, 0.1152, 0.1289)

*

Su(-2.4622, 1.6473, 0.1075, 0.1149)

Note. ** - we estimated the Weibull distribution form parameter, * - the Weibull distribution scale parameter.


 

Table 7

Approximation of Limiting Distributions of Mises  Statistic

when MD Estimates Minimizing Statistic  are Used

Random variable distribution

Estimation of scale parameter

Estimation of shift parameter

Estimation of two parameters

Exponential

Su(-2.6741,1.4068, 0.0958,0.1230)

 

 

Seminormal

Su(-2.6752,1.3763, 0.0952,0.1280)

 

 

Rayleigh

Su(-2.2734,1.3473, 0.1101,0.1496)

 

 

Maxwell

Su(-2.2759,1.3988, 0.1171,0.1514)

 

 

Laplace

Su(-2.3884,1.0811, 0.0948, 0.1548)

Su(-2.7267,1.4972, 0.1044,0.1239)

Su(-2.4334,1.6104, 0.0902,0.1123)

Normal

Su(-2.4180, 1.0702, 0.0957, 0.1464)

Su(-2.7639, 1.5393, 0.1102, 0.1115)

Su(-2.5746, 1.7505, 0.0979, 0.1043)

lnN(-1.1651,0.4271)

Log-normal

Su(-2.4180, 1.0702, 0.0957, 0.1464)

Su(-2.7639, 1.5393, 0.1102, 0.1115)

Su(-2.5746, 1.7505, 0.0979, 0.1043)

llnN(-1.1651,0.4271)

Cauchy

Su(-2.5043,1.1355, 0.1035,0.1384)

Su(-2.7029,1.5179, 0.1188,0.1100)

Su(-2.1046,1.4364, 0.0929,0.1301)

lnN(-1.1043,0.4692)

Logistic

Sl(0.3223,1.1159,0.6836, 0.0953)

Su(-2.3007,1.0135, 0.0906,0.1593)

Su(-2.6212,1.4318, 0.0932,0.1370)

Su(-3.0152,1.7751, 0.0800,0.0898)

Extreme-value (maximum)

Su(-2.4454, 1.1083, 0.0968, 0.1459)

Su(-2.6557, 1.4282, 0.1024, 0.1254)

Su(-2.1580, 1.5446, 0.0941, 0.1279)

Extreme-value (minimum)

Su(-2.4454, 1.1083, 0.0968, 0.1459)

Su(-2.6557, 1.4282, 0.1024, 0.1254)

Su(-2.1580, 1.5446, 0.0941, 0.1279)

Weibull

Su(-2.4454, 1.1083, 0.0968, 0.1459) **

Su(-2.6557, 1.4282, 0.1024, 0.1254) *

Su(-2.1580, 1.5446, 0.0941, 0.1279)

Note. ** - we estimated the Weibull distribution form parameter, * - the Weibull distribution scale parameter.

 

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