Statistical Inference for Inverted Kumaraswamy Distribution Based on Dual Generalized Order Statistics

In this paper, the shape parameters, reliability and hazard rate functions of the inverted Kumaraswamy distribution are estimated using maximum likelihood and Bayesian methods based on dual generalized order statistics. The Bayes estimators are derived under the squared error loss function as a symmetric loss function and the linear-exponential loss function as an asymmetric loss function based on dual generalized order statistics. Confidence and credible intervals for the parameters, reliability and hazard rate functions are obtained. All results are specialized to lower record values, also a numerical study is presented to illustrate the theoretical procedures.


Introduction
The concept of generalized order statistics (gos) was introduced by Kamps (1995) as a unified models for ordered random variables which produce several models as a special case. These models play an important role in statistics in general and in reliability theory and life testing in particular. Since its inception gos has attracted number of statisticians as distribution specific results obtained for gos can be used to obtain the results for other models of ordered random variables as special case. The random variables that are decreasingly ordered cannot be integrated into this framework. Burkschat et al. (2003) studied the dual generalized order statistics (dgos) that enables a common approach to descending ordered random variables as reversed ordered order statistics, lower record models and lower Pfeifer records. Some applications in reliability theory, such as, times of failure of technical components or systems, the failure of some components of the system can be more or less strongly influence life-length distribution of the remaining components in the system. Also as a model for successively largest insurance claims, highest water-levels or highest temperatures. For more details, see Burkschat et al. (2003), Khaledi and Kochar (2005), Jaheen and Al Harbi (2011), Khan and Khan (2012), MirMostafaee et al. (2016) and Mahdizadeh (2016).
Let (1, , , ), (2, , , ), … , ( , , , ) are n dgos from an absolutely cumulative distribution function (cdf) with corresponding probability density function (pdf). Then, the joint pdf has the form (1, , , ),…, ( , , , ) ( (1) , … , ( ) ) = (∏ −1 =1 ) [∏ ( ( ( ) )) ( ( ) ) −1 =1 ] ( ( ( ) )) −1 Assuming T is a random variable distributed as IKum distribution with shape parameters; > 0 and > 0, denoted by T~IKum ( , ). Then the pdf, cdf, reliability function (rf) and hazard rate function (hrf) are given, respectively, by and Fatima et al. (2018) proposed the exponentiated IKum distribution; they derived some statistical properties of this distribution and used the ML method to estimate the parameters. Mohie El-Din and Abu-Moussa (2018) estimated the unknown parameters of the IKum distribution based on general progressive Type II censored data using ML and Bayesian methods. Also, ZeinEldin et al. (2019) introduced the Type I half-logistic IKum distribution, some statistical properties of this distribution are derived. The method of ML estimation, methods of least squares and weighted least squares estimation and method of Cramer-von Mises minimum distance estimation are used to estimate the parameters of this distribution. Usman and ul Haq (2020) introduced the Marshall-Olkin extended IKum distribution, sub models were showed of this generalization. They derived explicit expressions for major mathematical properties of this distribution and they estimated the parameters using the ML method. This paper is organized as follows: In Section 2, ML estimators of the parameters, rf and hrf based on dgos are obtained. Bayes estimators of the parameters, rf and hrf based on dgos under squared error (SE) and linear exponential (LINEX) loss functions are derived in Section 3. Also, credible intervals for the parameters, rf and hrf are obtained. A numerical study is presented in Section 4.

Maximum Likelihood Estimation Based on Dual Generalized Order Statistics
In this section, the ML method is used to estimate the parameters, rf and hrf of the IKum distribution based on dgos. The asymptotic variance-covariance matrix of the ML estimators for the parameters and and the asymptotic 100 (1-)% confidence intervals for and are obtained.
The natural logarithm of the likelihood function is given by Considering that the two parameters and are unknown and differentiating the log likelihood function partially in (8) with respect to and , one obtains and Equating the derivatives (9) and (10) to zero, one can obtain the ML estimator of Then the ML estimator of the parameter can be obtained numerically by substituting (11) in (10).

Maximum likelihood estimation for the reliability and hazard rate functions
The invariance property of the ML estimation can be used to obtain the ML estimators ̂( 0 ) and ĥ ( 0 ), for a given time 0 , just replacing the parameters and by their corresponding ML estimators, as given below and

Asymptotic variance -covariance matrix of the maximum likelihood estimators
The asymptotic variance -covariance matrix, , of the ML estimators for and is the inverse of the observed Fisher information matrix, , using the second derivatives of the logarithm of the likelihood function as follows: The asymptotic normality of the ML estimation can be used to compute the two sided approximate 100 (1-)% confidence intervals for and as follows: Also, the asymptotic 100 (1-)% confidence intervals for rf and hrf are given by where (1− 2 ) is standard normal percentile and (1 − 2 ) is the confidence coefficient.

Bayesian Estimation Based on Dual Generalized Order Statistics
The Bayesian approach is considered to estimate the parameters, rf and hrf of the IKum distribution based on dgos. The Bayes estimators are obtained under the SE and LINEX loss functions to estimate (point and credible intervals) of the parameters, rf and hrf of the IKum distribution based on dgos.

Bayesian estimation under squared error loss function
In this subsection, the Bayes estimators of the shape parameters, rf and the hrf based on dgos are obtained under SE loss function.
Assuming that the parameters and of the IKum distribution are random variables with a joint bivariate prior density function that was considered by AL-Hussaini and Jaheen (1992) as where and the prior of is The joint prior pdf of and ; will be obtained by substituting (20) and (21) in (19) as given below The joint posterior of and can be derived by using (6) and (22) as follows: ( , | ) ∝ ( , | ) ( , ) .
hence, the joint posterior distribution of and is given by where and are given by (7) , also Under SE loss function the Bayes estimators for the parameters , , rf and hrf are given, respectively, by their marginal posterior expectations using (25) as shown below and To obtain the Bayes estimates of the parameters, rf and hrf, (27)-(30) should be solved numerically. Since, the posterior distribution is given by (25), then a 100 (1-) % credible intervals for and is ( ( ), ( )), respectively, where and (34)

Bayesian estimation under linear exponential loss function
In this subsection, the Bayes estimators of the shape parameters, rf and hrf based on dgos are obtained under LINEX loss function. Under the LINEX loss function, the Bayes estimators for the shape parameters , , rf and hrf are given, respectively, by and To obtain the Bayes estimates of the parameters, rf and hrf, (35)-(38) should be solved numerically.

Numerical Results
This section aims to illustrate the theoretical results of the ML and Bayesian estimation under SE and LINEX loss functions. Numerical results are presented for the IKum distribution based on lower record values through a simulation study and some applications.

Simulated example
The  Table 6 displays the Bayes averages and 95% confidence intervals of the rf and hrf at 0 = 0.5, 1, from the IKum distribution based on lower record values for different samples of Rv= 5, 9, and NR = 10000.

Applications
In this subsection, three applications to real data sets are provided to illustrate the importance of the IKum distribution based on lower records. Table 7 displays ML averages of the parameters, rf, hrf and ERs from IKum distribution for the real data based on lower records. The averages of the Bayes estimates for the parameters, their ERs and the credible intervals based on informative prior are given in Table 8. To check the validity of the fitted model, Kolmogorov-Smirnov goodness of fit test is performed for each data set and the p values in each case indicates that the model fits the data very well.
I. The first application is a real data set obtained from Hinkley (1977 The third application is the vinyl chloride data obtained from clean upgrading, monitoring wells in mg/L; this data set was used by Bhaumik et al. (2009). The data is: 5.1, 1.      Table 5: Bayes averages of the parameters and their estimated risks and credible intervals based on lower records( = . , = . , = )   Table 7: ML averages of the parameters, rf, hrf and estimated risks from IKum distribution for the real data based on lower records