Some Statistical Inferences on the Records Weibull Distribution Using Shannon Entropy and Renyi Entropy

In this paper‎, ‎we discuss different estimators of the records Weibull‎ distribution parameters and also we apply the Kullback-Leibler divergence‎ of survival function method to estimate record Weibull parameters‎. ‎Finally‎, ‎these methods have been compared using Monte Carlo simulation‎.


Introduction
In the common practice , the communication theory is necessary in order to quantify and fully study the information mathematically .Information theory is a branch of applied mathematics , electrical engineering and computer science involving the quantification of information .limited informationtheoretic ideas had been developed by Boltzmann (1896), Nyquist (1924), Hartley (1928Hartley ( , 1930) ) and etc .What is known today as the "information theory" has been founded by Shannon in 1948 through introducing ) (X H as the entropy of random variable X .He for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory .Later on , other measures were introduced such as Renyi entropy , relative entropy and Mutual information .Relative entropy (also Kullback-Leibler divergence) was first defined by Kullback and Leibler in 1951 as the directed divergence between two distributions .Also , record values and associated statistics have widely been used in many real life applications , for example in sports , weather , business , etc .Record values are very important in case when observations are difficult to obtain or when they are being destroyed when observations are subjected to an experimental test .Observations are obtained by observing successive maximum (minimum) values .The term record value was first introduced by Chandler (1952).Some inferential statistics based on record value have been discussed by Ahsanullah (1988Ahsanullah ( , 1990)) In statistics , the Weibull distribution is one of the most important continuous probability distributions .It was , first introduced by Weibull in 1939 when he was studying the issue of structural strength .Teymori et al .(2012) introduce a point estimator for shape parameter of upper record Weibull distribution and use maximum-likelihood estimation (MLE) of scale parameter in their method .Also some inferential statistics based on Weibull distribution have been discussed by Yari et al . (2013).
In this paper we use Kullbak-Leibler method for estimation parameters of upper record Weibull and we discuss on point estimator method .The results show moment method estimation (MME) or Kullback-Leibler divergence of survival function estimation (DLS) of  are better than maximum-likelihood estimation (MLE) of  , for estimation  .Now we introduce some basic definitions that play a central role the present paper , using the notation of Arnold et al .(1998) and Yari et al . (2013).
), the upper record values of every new sequence will correspond to lower record values of the original sequence, so in this paper we shall study only upper record values and shall use the notation for the i th upper record statistic.The pdf of the where  is the gamma function.Definition 1.2 Suppose that X be a random variable with pdf ) (X f and with support where  is the indicator function and is the ordered sample.[ The rest of this paper is organized as follows.In the next section we state some properties of record statistics.Section 3 present the upper record value of Weibull distribution and its Entropy.Our approach is illustrated in section 4. Finally, we discuss the estimation of parameters of the upper records Weibull distribution by Monte Carlo simulation in section 5.

Some properties of record statistics
In current section we examine some properties of upper record statistics .See Arnold et al .(1992) and Teimouri et al . (2012).
a) Using ( 1), the chi-square distribution with 2 i degree of freedom and conversion of See more details Arnold et al .(1992) and Teimouri et al . (2012).In the remainder of this specification , we will state a lemma and two theorems which has been presented in the Baratpour et al . (2007).Lemma 2.1 (Ahmadi, 2000) For where  is the Euler's constant.Theorem 2.2 Let ,...... , 2 1 X X be a sequence of iid random variables from Cdf F(x) with pdf f(x) 3 Let Y is a random variable from cdf F(x) with pdf f(x).Under the assumption of Theorem 1, we have: , m is the mode of the distribution and Y=MX , .

The upper record value of Weibull distribution and its Entropy
Now we examine the Shannon's Entropy on the upper record of the Weibull distribution .A continuous random variable X is said to be a two-parameter Weibull distribution with shape parameter α and scale parameter β denoted by and

Estimation of parameters of upper records Weibull distribution
In this paper , we use the notation UW i th  for the i th upper record Weibull distribution and in current section we estimate shape parameter α and scale parameter β by five methods .Suppose that n X X X ,...., , 2 1 is a random sample from (7) with sample size n .

a) Moment method (MME):
Here we provide the MME method of the parameters of a UW i th  distribution .To this purpose , one can show that its mean and variance of the sample are respectively : We consider three cases:  When α is known, then from (a) we have an estimator for β, say  When β is known , in this case we need to solve (a) with respect to α, denoted by **  When both α and β are unknown, first we obtain the population coefficient of variation (CV) from (a) and (b) Then, equating the sample CV with the population CV ( We observe that the population CV is independent of the β) we have . We need to solve (12) to obtain MME of α, denoted by  ~.Then substituting α in (10) we have an estimator for β, denoted by  ~.

b) Maximum-likelihood method (MLE):
Here the maximum likelihood estimators of UW i th  are considered.The log-likelihood function is given by .
We consider three different cases:  α is known: differentiate (13) with respect to β then equating to zero and solving with respect to β .
Then the MLE of β, denoted by  β is known: again differentiate (13) with respect to α then equating by equating by zero solving with respect to α.Then the MLE of α, denoted by MLE  ˆ, will be  Both α and β are unknown: in this case, setting (14) in (13) and then differentiate on it with respect to α then equating to zero and solving with respect to α.Here the MLE of α, denoted by ˆ,will be .e) Kullback-Leibler divergence of survival function method (DLS): To estimate the parameters by this method, we set(8) in ( 4).Then we have ( 1) ) . Now differentiate (!8) with respect to α then equating to zero and solving with respect to α .Then, But we don't have a good estimator for α by this equation since it does not depend on the sample values.Now we suppose that α is known.Now Equation (18) should be minimized respect to β.For this purpose, we have .0 We observe that a close form solution of (19) for β is not possible.

Simulation study
Since the MME , MLE , DLS and Bayesian estimation of the parameters have not closed form so checking the performance of them , theoretically is a difficult task .Therefore , they must be solved numerically .We have done this work by Matlab software .First , we have generated  are the estimated 4 th -UW shape and scale parameters from th k sample.To illustrate the effect of sample size, we carried out a simulation study for some levels of n=5, 6,…, 30.First, we study the behavior of β given in (10), ( 14) and (19).Figure 1 shows the simulation results of  .The following results can be concluded from this figure : 1) According to figure 1.a, the MME, DLS and Bayesian with prior normal methods act better than the rest in terms of bias.MME and Bayesian for  are overestimated whereas DLS is underestimated.
2) According to figure 1.b, mean square error (MSE) of MME method is best for every n.Now we study the behavior of  given in (12), ( 16), (17) and Bayesian method with four prior distribution.Figure 2 shows the simulation results for .The following results can be concluded from this figure: 1) According to figure 2.a, The MME and point estimator methods act better than the rest and are so similar in terms of bias.Also MLE values for  are closed to true  , then this method can be a good estimator for  .
method Here, we assume the previous information of α and β are independent of each other, so) in this approach for normal, uniform and triangular prior distribution by Monte Carlo simulation.d)Point estimator for the shape parameter (PE)Theorem 4. 1 (Teimouri et al. (2012)) Suppose that a sequence of i th upper record from Weibull family are observed.A simple estimator of shape parameter  is given by median of i th upper record values.[5]

2 )
Figure 2.b shows MSE of MME, MLE and point estimator methods are best for every n and very closed to zero.