The problem of maximizing entropy function

subject to constraints

where

has solution

where are Lagrange multipliers. Finding the distribution which maximizes function (1) subject to constraints generated by equations in (2) is an optimization problem. In the literature, there have been numerous studies that have calculated these multipliers [1] . In this study, we use the MATLAB program to calculate Lagrange multipliers.

If (3) is substituted into (1), the maximum entropy value is obtained:

If distribution is calculated from the data, the moment vector value can be obtained for each moment vector function. Thus, is considered as a functional of and called the functional. Therefore, we use the notation to denote the maximum value of corresponding to

.

Let be the compact set of moment vector functions reaches its least and greatest values in this compact set, because of its continuity property. For this reason,

Consequently,

Distributions and corresponding to the and, respectively, are called and

distributions [1] .

method for a finite set of characterizing moment functions can be defined in following form.

Let be the set of characterizing moment vector functions and all combinations of elements of taken elements at a time be. We note that, each element of is vector with components.

Solving the and the problems require to find vector functions, , where

minimizing and maximizing accordingly with respect to Shannon entropy measure. It should be noted that reaches its minimum value subject to constraints generated by function and all -dimensional vector functions. In other words, minimum value of is least value of values corresponding to

. If gives the minimum value to then distribution corresponding to is called the

distribution. method represents probability distribution in the form of distribution. In a similar way, reaches its maximum value subject to constraints generated by function and all -dimensional vector functions. In other words, maximum value of is greatest value of values corresponding to

. If gives the maximum value to then distribution corresponding to is called the

distribution. method represents probability distribution in the form of distribution. It should be noted that both distributions can be applied in solving proper problems in survival data analysis.

In the present research, the data of the life table for engine failure data (1980) given in Table 1 is considered [10] .

In our investigation, the experiment is planned for 200 numbers of patients surviving at beginning of interval but the presence of censoring from the planning patients 97 individuals stay out the experiment. This situation is taken into account in Table 2.

It should be noted that, the presence of censoring in the survival times leads to a situation where the sum of observation probabilities stands less than 1 for the

survival data. For this reason, in solving many problems, it is required to supplement the sum of observation probabilities up to 1. Since the sum of observed probabilities in Table 2 is 0.8155, according to the number of censoring, supplementary probability is uniformly distributed to each censoring data and corrected probabilities are obtained.

As we noted that above, and distributions can be applied in solving proper problems in survival data analysis. In our investigation as components of characterizing moment vector function

, are chosen. The set of moment functions is chosen from the characteristic moments which are mostly used in Statistics.

Consequently,. For example, if then

gives the least value to and

gives the greatest value to.

The distributions corresponding to

and values are shown in Tables 3-6. In these tables, and distributions corresponding to and are represented with bold font. By virtue of these tables are also obtained, ,

distributions which are shown in Table 7 and Table 8.

In order to obtain the performance of the mentioned distributions, we use various criteria as Root Mean Square Error (RMSE), Chi-Square, entropy values of distributions. The acquired results are demonstrated in Table 9 and Table 10.

All distributions are acceptable to survival data in the sense of Chi ? Square criteria.

In the sense of RMSE criteria each distribution is better than corresponding distribution. Moreover, is nearer to statistical data than and

distributions; each is better than all of distributions. From these results follows that among of distributions, the distribution is more suitable and among of distributions, the distribution is more convenient for statistical data. These results are also corroborated by graphical representation (see Figures 1-4). Consequently, we shall consider Probability Density Function, Cumulative Distribution Function, Survival Function and Hazard Rate for only and distributions.

Although the distribution with the largest number of moment functions tends to fit better, it should be noted that in some cases, the set of moment functions with fewer elements is more informative then a different set of moment functions with more number of elements.

In order to establish availability of GEOD to survival data in the sense of Shannon measure it is required to consider entropy values of GEOD.

From Table 3 it is seen that the (the) distribution is realized by vector function and

.

From Table 4 it is seen that the (the) distribution is realized by vector function

and

.

From Table 5 it is seen that the (the) distribution is realized by vector function

and

.

From Table 6 it is seen that the (the) distribution is realized by vector function

and.

Comparison of GEOD with each other in the sense of Shannon measure shows that along of these distributions is better.

The results of our investigation according to using known characterizing moment vector functions from are summarised in the form of following Corollary.

Corollary 1. If by denote the

(the) distribution corresponding to moment conditions generated by moment functions, then inequality

is fulfilled, when. In other words, entropy value of the (the) distribution depending on the number of moment conditions decreases.

Moreover for any the inequality

takes place.

Now, we calculate the distance between observed distribution

given in Table 2 and distributions

given in Table 7 and Table 8 respectively.

It is known that the Kullback ? Leibler distance between distributions

and is obtained by formula

.

By starting these formula Kullback-Leibler measures for the distance between observed distribution and distributions

are given in Table 11 and Table 12 respectively.

From Table 11 and Table 12 follows that along of GEOD is better in the sense of Kullback-Leibler measure.

The results of our investigation according to using known characterizing moment vector functions from are summarised in the form of following Corollary.

Corollary 2. If observed distribution and

denote the (the) distribution corresponding to moment conditions generated by moment functions, then inequality

is fulfilled, when. In other words, Kullback-Leibler value of the (the) distribution depending on the number of moment conditions decreases.

Moreover for any the inequality

takes place.

In this section survival data analysis is conducted by

distribution since the above acquired investigations is more presentable for survival data among, distributions.

and estimations of Probability Density Function, Cumulative Distribution Function, Survival Function and Hazard Rate are given in Table 13 & Table 14, respectively.

On basis of the results given in Table 13 & Table 14, graphs of, and are demonstrated in Figures 5(a)-(c) & Figures 6(a)-(c).