Discriminatiing Between Degum Distribution and Burr-III Distribution

Received: 12/Sept/2018, Accepted: 03/Oct/2018, Online: 31/Oct/2018 Abstract Test statistics based on Likelihood function , population Quantiles are suggested to discriminate between Dagum distribution and Burr-III distribution. Because of the non tractability of their exact sampling distributions, the percentiles of the proposed test statistics are tabulated with the help of simulated sampling distributions of the test statistics . The power of the test statistics are also tabulated and a comparative study w.r.t the powers for a given sample and the level of significance are worked out.


I. INTRODUCTION
Having experienced with the lack of analytical expression for the classical maximum likelihood estimation (MLE) of parameters in Dagum distribution, we propose to study whether any other standard model be an alternative to Dagum distribution with a reasonably admissible risk. Accordingly, we have chosen Burr type III model to test whether it can be an alternative to Dagum distribution. This aspect is viewed as the problem of discriminating between Dagum and Burr III models where Dagum is null population. We have considered the principle of Likelihood ratio criterion in a practically usable way. Hence we may call our procedure as likelihood ratio type procedure. The distinction is-in the likelihood procedure to the classical MLE is used for both null and alternative populations. In the likelihood ratio type procedure we use any admissible estimators of concerned models as given by [1,2,3].

The
Probability density function (pdf) of Dagum distribution is given by The Dagum distribution is a skewed, uni-modal distribution on the positive real line.
The probability density function (pdf) of Burr-III distribution is given by The likelihood function of Burr-III distribution is We are therefore motivated to study whether Dagum distribution is a reasonable alternative to Bur-III distribution at least for the sake of adopting the analytical, powerful inferential characteristics of the data following Burr-III distribution. With this backdrop we suggested two different test statistics to discriminate Dagum distribution(with a=0.5,p=0.5 and b=2/3) and Burr-III distribution(with c=0.5 and k=0.5). The pdfs and cdfs plotted for the mentioned values of both the distributions are given below The basic distribution characteristics of Dagum distribution(DD) and its Properties are Presented in Section I. The rest of the paper is organized as follows. The proposed test statistics based on ratio of likelihood functions along with their percentiles and power values are explained in section II and Population quantiles explained in section III along with their percentiles and power values. Summary and Conclusions are given in section IV.

RATIO
Let us consider Dagum distribution as the null population say ( 0 P ), the Bur-III distribution is considered as the alternative population say     These tables indicate that even with the help of a small sample of size as small as 2 the power remains to be at more than 99%. It is therefore concluded that the T1 statistic proposed in this section cannot discriminate between the null and alternative population with a high power values.  As we can see from table -3, the percentile points of T 2 increase as the smple size increases as well as the significance level increases. The power of the test statistic T 1 is also tabulate for 3 different levels of significance (1%,2.5%,5%) at the sample sizes n=2,3 … 10.by simulating sampling from P 1 and using the values of T 2 .

III. TEST STATISTICS BASED ON QUANTILES
The count of T 2 values that fall beyond the table values of   table -3.These are given in Table-4.

IV. COMPARATIVE STUDY&CONCLUSION
The powers given in table 2 & 4 for the test statistics T 1 and T 2 respectively indicate that the two statistics do not discriminate between Dagum & Burr-III distributions significantly. i.e there is much similarity between these two distributions for larger values of n. If the sample size increases the Dagum distribution and Burr-3 distribution cannot be distinguished with respect to proposed test statistics. We may say that for a data from Dagum distribution and the application of Dagum distribution is quite complicated, the methodology of Burr-III distribution can be used in large samples.