Bayesian Estimation for Nadarajah-Haghighi Distribution Based on Upper Record Values

This paper discusses maximum likelihood and Bayes estimation of the two unknown parameters of Nadarajah and Haghighi distribution based on record values. Different Bayes estimates are derived under squared error, balanced squared error and general entropy loss functions by using Jeffreys' prior information and extension of Jeffreys' prior information. It is observed that the associated posterior distribution appears in an intractable form. So, we have used Tierney and Kadane approximation method to compute these estimates. Finally, numerical computations are presented based on generated record values using R software.

). (x f An observation j X will be called an upper record value if its value exceeds that of all previous observations.Thus, In a sequence of events, the event value that exceeds all previous values is of particular importance in the scientific and applied fields and so their values are recorded.In sporting events, for example, focus attention is usually on recording results that exceed their predecessor, as the hydrologists usually tend to monitor the higher values of the floods.Also, the meteorologists usually concern with upper and lower record temperatures.For more details on the concept of record values and their application, see, Ahsanullah (2004) and Arnold et al. (1998).The statistical treatment of the record values was introduced for the first time by Chandler (1952).Many studies on record values and their associated statistical inference have been done for some distributions by several authors such as Selim (2012) who studied Bayesian estimation of Chen distribution based on record values.Nadarajah and Haghighi (2011) introduced a new extension of the exponential distribution as an alternative to the gamma, Weibull and the exponentiated exponential distributions.The corresponding cdf and pdf are as follows: Jeffreys' prior exhibits many nice features that make it an attractive reference prior.Jeffreys' prior has the property of being approximately non informative in the sense of Box and Tiao (1973), who motivated Jeffreys' prior by introducing the notion of data translated likelihood.Bernardo (1979) showed that, under certain conditions, Jeffreys' prior is an optimal reference prior in the sense that it maximizes the missing information.Kass (1989Kass ( , 1990) ) emphasized that a main feature of Jeffreys' prior is that it is a uniform measure in an information metric, which can be regarded as the natural metric for statistical inference.In addition, for problems involving scale and location parameters, Jeffreys ' (1946, 1961), Box and Tiao (1973), Bernardo (1979) and Kass (1990) pointed out that Jeffreys' multivariate prior should be modified as suggested by Jeffreys' (1946Jeffreys' ( , 1961)).Kumar et al. (2017) discussed the Bayes estimators under symmetric and asymmetric loss functions using Markov Chain Monte Carlo (MCMC) technique to compare the performance of the proposed methods based on record values.Gencer and Saraçoğlu (2016) compares the approximate Bayes estimators under different loss functions for parameters of odd Weibull distribution and the Bayes estimators of the parameters of Erlang distribution under different prior distributions have been obtained by Haq and Dey (2011).For detail survey one may refer to Jung and Chung (2018) from NH distribution with cdf (1.1) and pdf (1.2).The likelihood associated with record data is given by (Ahsanullah, 2004) 2) The log-likelihood function may then be written as Taking derivatives with respect to  and  of (2.3) and equating them to zero, we obtain the likelihood equations for  and  to be The Equations (2.4) and (2.5) cannot be solved analytically for  and .
 Therefore, we may use R software (using optim function, see, R Core Team (2018)) to solve these equations and find the MLE's of the unknown parameters  and .

Bayesian estimation
For Bayesian estimation of the parameters  and  , prior distributions are needed.If once prior knowledge about the parameter is available, it is suitable to make use of an informative prior but in a situation where one does not have any prior knowledge about the parameter and cannot obtain vital information from experts in this regard, then a noninformative prior will be a suitable alternative to use, (Guure et al., 2013).A commonly used reference prior in Bayesian analysis is Jeffreys' prior.It is obtained by applying Jeffreys' rule, which is to take the prior density to be proportional to the square root of the determinant of the Fisher information matrix (Lavanya and Alexander, 2016).In this study, Jeffreys' priors are use and these are as follows ( ) The joint prior of parameters are ( ) We propose an extension of Jeffreys' prior information such that, ( ) where c is the hyper parameter that is assumed to be non-negative and known.When 1 c = , we have the standard Jeffreys' prior information and undefined when 0. c = Since our knowledge on the parameters is limited as a result of which a Jeffreys' prior information approach is employed on both parameters.It is important that one ensures the prior does not significantly influence the final result.If our limited or lack of knowledge influences the results, one may end-up giving wrong interpretation which could affect whatever it is we seek to address.

Bayes estimation under squared error loss function
In this section, we obtain Bayes estimates of  and  using SEL and BSEL functions.Let any function of  and  be ( )  = The Bayes estimate of  under the loss ( ) SEL function is a symmetric function and was introduced by Legendre (1805).Let any function of  and  be ( ) The SEL function is as follows: where  is the estimate of the parameter . Under the above loss function, the Bayes estimator ˆSEL  of θ is given by ( ) where E  stands for posterior expectation.In this case, Bayes estimator of ( ) , u  under SEL function which is a symmetric loss function is obtained as follows: where ( ) is log-likelihood function and ( )

Bayes estimation under general entropy loss function
GEL function is an asymmetric function and was suggested by Calabria and Pulcini (1996).Dey and Liao (1992) have studied about Bayes estimation under GEL function.Let any function of  and  be ( ) The GEL function with parameter k is given by where  is the estimate of the parameter . Under the above loss function, the Bayes estimator ˆGE  of  is given by ( ) where E  stands for posterior expectation.The proper choice for k is a challenging task for an analyst because it reflects the asymmetry of the loss function.In this case, Bayes estimator of ( ) u  under GEL function which is an asymmetric loss function is obtained as follows: It is difficult to solve the equations (2.12) and (2.14) in closed form.Because of this reason, the Bayes estimators of the parameters  and  can be obtained using TK approximation method.

3.
Tierney-Kadane approximation Tierney and Kadane (1986) is one of the methods to find the approximate value of the mathematical explanations as the ratio of two integrals given in equations (2.12) and (2.14).
We consider posterior expectation of ( )

(
) is logarithm joint prior distribution and is defined as follows: For Jeffreys' prior information, ( and for extension of Jeffreys' prior information, We can approximate the function ( ) Ix into an explicit expression by applying the TK approximation method.In sequel, we consider the functions defined by Now, we assume that ( ) We then approximate ( ) Here, ||  and ||    denote the determinants of negative inverse hessian of ( )  are defined as follows: Likewise the corresponding second-order derivatives are obtained by

Tierney-Kadane Bayes estimator of ( )
, u  under squared error loss function Bayes estimator of ( ) , u  under SEL function is defined as follows: Bayes estimators for parameters  and ,  using equation (3.12) under SEL functions are obtained as follows: i) 1 1 n Similarly, we can compute the corresponding Bayes estimates of unknown parameters  and  under the BSEL function.

  =+
Bayes estimators for parameters  and  using equation (3.15) under GEL functions are obtained as follows: i) In order to compute || GEL    , we first obtain the following expressions    , the Bayesian estimates of the parameters are derived with respect to three loss functions, SEL, BSEL and GEL functions.Also, we have considered the values of Jeffreys' extension prior 3.5, 4.5 c = and 5.5.The choice of the extension of Jeffreys' prior information is subjective since it is used to consider the proportion in which one will prefer the prior to influence the posterior density function.
The Bayes estimator of  and  under SEL and BSEL function using extension Jeffreys' prior information are shown in the Table 2.The MLE's calculated are also given in the same table.The Bayes estimator of  and  under SEL and BSEL loss function using Jeffreys' prior information are shown in the Table 4.Moreover, The Bayes estimator of  and  under GEL function using extension Jeffreys' prior information and Jeffreys' prior information are shown in the Table 3 and Table 5.Also, the PE is shown along with the estimators.Conclusion Theoretical results of the study are explained numerically in the above Tables.Here, MLE's and Bayes estimators based on generated record values are obtained.In Table 2 and Table 3, we observe that if the value of c increases then we observed that we have the good estimators under different loss functions.From Tables 2, 3, 4 and 5, we see that, the PE of the Bayes estimates under SEL function using extension Jeffreys' prior information and Jeffreys' prior information are more than PE of the Bayes estimates under BSEL function using extension Jeffreys' prior information and Jeffreys' prior information.So we observe that Bayes estimates under BSEL function using extension Jeffreys' prior information and Jeffreys' prior information perform good compared to the SEL function using extension Jeffreys' prior information and Jeffreys' prior information.Moreover, the performances of all the estimators of BSEL function are improved when the value of  increases.Mostly, MLE's have more PE as compare to Bayes estimators under SEL function using extension Jeffreys' prior information and MLE's have less PE as compare to Bayes estimators under SEL function using Jeffreys' prior information.Mostly, MLE's have more PE as compare to Bayes estimators under GEL function using extension Jeffreys' prior information for both negative and positive values of k and MLE's have less PE as compare to Bayes estimators under GEL function using Jeffreys' prior information for both negative and positive values of .
k Finally, we observe that Bayes estimators under extension Jeffreys' prior information perform good as compared to the Bayes estimators under Jeffreys' prior information.

F
of independent and identically distributed (iid) random variables from an absolutely continuous distribution function (cdf) ) (x and probability density function (pdf)  is called the upper record times.
shape parameters, respectively.Henceforth, Nadarajah and Haghighi distribution will be denoted by (NH) distribution.Singh et al. (2015) discussed classical and Bayesian estimations for NH model under progressive type-II censored data.The maximum likelihood estimators (MLE's) and Bayes estimators of the unknown parameters of NH distribution under progressive type-II censored data with binomial removals have also been obtained by Singh et al. (2014).

where 0 
is a known estimator of  and 0 1.   This loss function reduces to the squared error, when 0.

,u
 with respect to the distribution ( )

u
 is any function of  and  , ( ) ,| lx  is defined in equation (2.3).

)
In order to compute ||GEL    , we first obtain the following expressions Pak.j.stat.oper.res.Vol.XV No.In order to illustrate the usefulness of the inference procedures discussed in the previous sections, we generate five sets of record values of sizes 50, 100, 150, 200 and 250 from ( )

j.stat.oper.res. Vol.XV No.1 2019 pp217-230 219 2.1 Maximum likelihood estimator
, Pandey and Kumari (2018a, b), Kayal et al. (2016) and Faizan and Sana (2018) amongst others.The approximate Bayes estimators of the unknown parameters of NH distribution using Lindley's approximation method based on record values have been obtained by Selim (2018).In this paper, MLE's and approximate Bayes estimators of the parameters of the NH distribution based on upper record values are obtained under squared error loss (SEL), Pak.

Table 1 .
Generated upper record values for different sample sizes.

Table 2 .
MLE's and Bayes estimates using extension Jeffreys' prior information for SEL and BSEL function.

Table 3 .
Bayes estimates using extension Jeffreys' prior information for GEL function.

Table 4 .
Bayes estimates using Jeffreys' prior information for SEL and BSEL function. Pak.j.