On Estimation and Prediction for the Inverted Kumaraswamy Distribution Based on General Progressive Censored Samples

In this article, the problem of estimating unknown parameters of the inverted kumaraswamy (IKum) distribution is considered based on general progressive Type-II censored Data. The maximum likelihood (MLE) estimators of the parameters are obtained while the Bayesian estimates are obtained using the squared error loss(SEL) as symmetric loss function. Also we used asymmetric loss functions as the linear-exponential loss (LINEX), generalized entropy (GE) and Al-Bayatti loss function (AL-Bayatti). Lindely’s approximation method is used to evaluate the Bayes estimates. We also derived an approximate confidence interval for the parameters of the inverted Kumaraswamy distribution. Two-sample Bayesian prediction intervals are constructed with an illustrative example. Finally, simulation study concerning different sample sizes and different censoring schemes were reported.


Introduction
In most of life-testing experiments, the censored samples used when the experimenter wants to terminate the experiment early before all units are failed due to the time limitation and the huge cost of the experiment.Type-I and Type-II are the two basic types of censoring schemes, where in Type-I the experiment is terminated at pre-specified time point and the number of failures is variable, while the experiment under Type-II is terminated after a fixed number of failures. Removing unites at certain time points during the experiment are not allowed in Type-I and Type-II, so Progressive censoring is applicable, such that the experimenter can remove some units at pre-specified time points (Progressive Type-I) or remove units at each failure (Progressive Type-II). For further reading about progressive censoring, see Balakrishnan and Aggarwala (2000) and Balakrishnan (2007) who presented a study on different features of progressive censoring schemes. Now, suppose that we have n units were placed on a lifetime-experiment, suppose that the first Bayesian prediction is an important topic in statistical inference where we try to use the previous data to predict the future observations inside the same population with a specified probability. When the unobserved failures belong two the same sample, then the prediction called One-sample Bayesian prediction, while it is called Two-sample Bayesian prediction when we want to predict by a new sample using an old sample. The Bayesian prediction was discussed by many authors based on different distributions with different types of censored samples as Mohie El-Din and Shafay (2013), they study Bayesian prediction intervals based on progressively Type-II censored data. Shafay and Balakrishnan (2012) This article is organized as follows. In section 2, the likelihood function and the maximum likelihood estimates of  and  are obtained, also the asymptotic confidence intervals are constructed in the same section. In section 3, Bayes estimates for the parameters  and  are obtained using four different loss functions (SEL, LINEX, GE and Al-Bayatti). In section 4, the approximation of Bayesian estimates are obtained using Lindley's approximation method. In section 5, a real data example is constructed to compare the proposed methods. In section 6, simulation study is performed to discover the properties of different estimators proposed in this paper. Finally, the paper is concluded in section 7.

Maximum Likelihood Estimation (MLE)
Suppose that n randomly selected units have a lifetimes follow ) , distribution are put on the lifetime-test at time zero. Based on the general progressively Type-II censoring, then the sample is given by By applying the partial derivatives for (8) with respect to  and  and putting the derivatives equal to zero, then we get: it is obviously that the closed form solution for the parameters  and  from the likelihood equations given in (9) and (10)

Observed Fisher Information
In this subsection, the observed fisher information based on general progressive censoring are observed to construct interval estimates for the parameters of Inverted Kumaraswamy distribution. Using Equations. (9) and (10), we have:

On Estimation and Prediction for the Inverted Kumaraswamy Distribution Based on General Progressive Censored Samples
Pak.j.stat.oper.res. Vol.XIV No.3 2018 pp717-736 Then the asymptotic Variance-Covariance matrix is the inverse of the Fisher information matrix, which is given by, Then the asymptotic confidence intervals for the parameters  and  is given by,  Z is obtained from the table of the standard normal distribution.

Bayesian Estimation
In this section, we derive the Bayesian estimates for the parameters  and  of the Inverted Kumaraswamy distribution ) , based on general progressive Type-II censoring using four different loss functions.The squared error loss function (SEL) which is defined as symmetric loss function, given by: , where ˆ is an estimate of  . The Bayes estimate for  under the loss function 1 L is the ,the second loss function is the linear exponential (LINEX) loss function, it is an asymmetric loss function and defined as 0, 1, The Bayes estimate for any parameter  under the loss function 2 L is given as, the third asymmetric loss function is the generalized entropy function which is given by, and the Bayes estimate under 3 L is given by, Finally, we have Al-Bayatti loss function 4 L which introduced by Al-Bayatti (2002), and given by, , While the Bayes estimate under 4 L is given by, Assume that the parameters  and  are independent variables having Weibull prior are the hyper parameters of the priors. Using (7) and (16), then the posterior distribution of  and  is obtained as The Bayes estimates for the parameters  and  under the squared error loss function 1 L are given by, For the LINEX loss function 2 L , the Bayes estimates for  and  are given by, The Bayes estimates for  and  depending on the generalized entropy loss function 3 L is given by, , Unfortunately, all estimates have the form of ratio of two integrals, and the closed forms for these integrals are not obtained. Therefore, the approximated values for these estimates are computed using the Lindley approximation method. Lindley (1980)was discussed an approximate Bayesian method. His method used to obtain an approximate for a ratio of two integrals. Suppose ) ,

Lindley approximation method
, then the Lindley method defined as, where  and ˆ are the MLE estimators of  and  , respectively. Also, ij u is the second derivative of the function u with respect to i and j , i.e.  û is MLE of the second derivative of ) , (   u with respect to  . while the other terms are obtained as follows,

On Estimation and Prediction for the Inverted Kumaraswamy Distribution Based on General Progressive Censored Samples
and Using (16) where  is defined by, Pak.j.stat.oper.res. Vol.XIV No.3 2018 pp717-

On Estimation and Prediction for the Inverted Kumaraswamy Distribution Based on General Progressive Censored Samples
For  , we put    = ) , ( u and the BS ˆ is given by: Also, the Bayes estimates for  and  under the LINEX loss function 2 L can be obtained as follows: for  , we take and BL ˆ is given by, Also, the Bayes estimate for  under the LINEX loss function is given by, For the Bayes estimate of  under the generalized entropy loss function 3 L , we use and BE  is given by, , to obtain the Bayes estimate for  under 3 L , , Finally the Bayes estimate of  using Al-Bayatti loss function 4 L is obtained as follows, The integration in (55) cannot be computed analytically, so we will use Lindely method in (32) to obtain an approximate for this integration by putting ) ,  simulation study. All calculations are constructed using Wolfram Mathematica 9. We will compare the MLEs and the Bayesian estimators under four loss functions, also the asymptotic confidence intervals will constructed with confidence degree 95% . The samples are generated from the Inverted Kumaraswamy distribution with the parameters  and  , which have the 2. Scheme II: c . The process of simulation will be executed 1000 times, then the average value are calculated to be the estimate value. Also we obtain the mean of 1000 lower and upper confidence limits for the asymptotic confidence intervals of the parameters with 95% confidence limits. In Tables 2 and 3, we present the the average of estimates and mean square error (MSE) of the MLEs and the BSEs for the informative Weibull priors, while the estimates using the informative priors of the exponential special case are obtained in Tables 4 and 5. The asymptotic confidence intervals with 95% confidence degree for 0.5 =  and 1 =  are included in Table 6.

Concluding remarks
In this work, we study the estimates for the parameters of inverted Kumaraswamy distribution under the general progressive censored samples. The estimates are obtained using the maximum likelihood method and Bayesian method under four different types of loss functions. Two sample Bayesian prediction intervals are conducted for a future sample depending on the old sample units. According to these results of simulation and the introduced example, we can draw the following conclusions: • The estimators that obtained from Bayesian method are very close to the real values of parameters than the estimators of Maximum likelihood method.
• The estimators that depend on samples with large size n and large values of m are better than those with small values.
• In most cases, the smallest MSEs are obtained under Al-Bayatti loss function with 0.5 = − c while the MSE using the LINEX loss function with 1 = h is less than that obtained by 0.5 = − h . • In most cases, we noted that the Bayesian estimates using the Weibull priors are better than that obtained by exponential priors and non-informative priors. • Small differences are noted, when we use the set of hyper parameters for informative priors and those for non-informative priors.
• the asymptotic confidence bounds contains the the MLE estimates for  and  , the width of the intervals become small for large values of n and m