Optimal Asymptotic Tests for Nakagami Distribution

Rayleigh distribution, Nakagami distribution Abstract: Nakagami distribution is often used to model positive valued data with right skewness. The distribution includes some familiar distributions as special cases such as Rayleigh and Half-normal distributions. In real life applications, one of the simpler model may be sufficient to describe data. The aim of this paper is to adapt tests of goodness of fit of the Rayleigh distribution against Nakagami


Introduction
Characterization of wireless channels plays an important role in designing a reliable wireless communication system.Fading is occured during transmittion of signals from transmitter to receiver.In the literature there are many statistical distribution to determine behavior of fading of signals such as Rician, Rayleigh, lognormal, Weibull distributions.Nakagami distribution is one of the most common distribution among these distributions and the distribution was proposed by Nakagami [1] to model fading of radio signals.
Applications of Nakagami distribution have been carried out in many scientific fields such as engineering, hydrology and medicine.For example, Sarkar et al. [2,3] use the distribution to derivate unit hydrographs for estimating runoff in hydrology.Shankar et al. [4] and Tsui et al. [5] apply the Nakagami distribution to model ultrasound data in medical imaging studies.Also, the statistical characteristics of "Moving Pictures Expert Group" (MPEG) is modelled by Nakagami distribution [6].Carcole and Sato [7] and Nakahara and Carcole [8] show the usefullness of the Nakagami distribution to deal with high-frequency seismogram envelopes.Ozonur et al. (9) analyze performance of the goodness of fit tests for Nakagami distribution.
The probability density function of Nakagami distribution is ( ) where  0.5 m is the shape parameter and 0 is the scale parameter.
The mean and variance of the distribution are given by following equations respectively:

( )  C
statistic reduces to the score statistic, when maximum likelihood estimators of nuisance parameters, which are n consistent, are used [10].

Many authors have shown that the ( )  C
or score test is asymptotically equivalent to the likelihood ratio test [11,12].( )  C or score tests have some advantages such as maintaining a preassigned level of significance approximately [13], requiring estimates of the parameters only under the null hypothesis.Also, the tests often are calculated easily [14].
These tests are all asymptotically optimal.They provide tests with good properties in large samples [15].Although there are various studies including these goodness of fit tests [16,17], these tests have not been taken into consideration for Nakagami distribution.Goodness of fit problem of the distribution is considered in the study due to pervasive usage in many scientific areas as mentioned above.In this context, the main focus of this study is to test goodness of fit of Rayleigh distribution against Nakagami distribution.

Test Statistics
In this section asymptotically optimal goodness of fit tests such as likelihood ratio, Neyman's ( ) The likelihood ratio test ( ) H is given as follows: ( ) ( ) the maximum likelihood estimates of parameters under alternative hypothesis are obtained as Here the notation ( ) . The statistic LR asymptotically follows a chi-square distribution with one degree of freedom under the null hypothesis.

( ) C α and Score Tests
Rao [10] introduced score test as an alternative to likelihood ratio test.Neyman [18] proposed ( ) ; ' which has an asymptotic Chi-square distribution with 1 degree of freedom.

Simulation Study
In this section, we compare performance of the test statistics -sided Gaussian distribution when = 12 m .Nakagami probability densities are plotted for different/various m and  parameter combinations in Figure 1.

Figure 1 .
Figure 1.Nakagami densities for various parameter combinations.In real applications, a one-parameter distribution may be sufficient to analyze data avoiding unnecessary complications.Aim of the study is to adapt tests of goodness of fit of the Rayleigh distribution against Nakagami distribution.Specifically we adapt likelihood ratio test, Neyman's ( )  C test and Rao's score test in the study.Monte Carlo simulation study is conducted to compare these tests in terms of empirical size and power and concluding remarks are given.( )  C test is developed by regressing the residuals of the score function for the parameters of interest on the score function for the nuisance parameters.The nuisance parameters are then replaced by some n consistent estimates ( n is number of observations).

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s score test are obtained to compare Rayleigh distribution against Nakagami distribution.random sample from a Nakagami distribution with probability density function given in Equation (1) with the parameter vector The log-likelihood function under the model (1) is given by of  under the full model and under the null hypothesis respectively.By solving the following equations

1
m is derivative of digamma function ( )  m .Define ( ) is the asymptotic covariance matrix of m[11].The Rao's score statistic is as follows: the parameter  , under null hypothesis, is replaced by its moment estimate  = 2 ˆ4 mm x , which are n consistent, the distribution of C T is also asymptotically Chi-square distribution with 1 degree of freedom.On the other hand, if  is replaced by its LR in terms of Type I errors and powers of tests by using statistical software R 3.4.1.Firstly, critical values of the goodness of fit tests are obtained by simulating 10000 samples of size n from Nakagami distribution with = 1.m Using critical values, type I errors are calculated by generating 5000 random samples from Nakagami distribution with 1 m = for various combinations of levels, sample sizes and scale parameters.In the simulation study, we consider sample sizes = 20,30,50 n and scale parameters =0.

Table 1 .
Type I errors of goodness of fit tests for different scale parameters, nominal levels and sample sizes As seen inTable 2-4, as the sample size increases powers of all test statistics increase.Also, powers of test statistics increase as the value of

Table 2 .
Powers of goodness of fit tests for

Table 3 .
Powers of goodness of fit tests for

Table 4 .
Powers of goodness of fit tests for and a likelihood ratio statistic LR are adapted to test goodness of fit of the Rayleigh distribution against two parameter Nakagami distribution.Performance of the test statistics are compared in terms of type I error and power of test by Monte Carlo simulation study.Simulation study shows that LR test statistic is the most powerful test.Powers of test statistics are not affected by scale parameter  and power results increase as shape parameter increases.Although test statistic is the least powerful test among the test statistics.Finally, our recommendation is to use LR statistic for all sample sizes and/or m parameters.
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