Inference for the Two Parameter Reduced Kies Distribution under Progressive Type-II Censoring

: In this paper, we obtained several recurrence relations for the single and product moments under progressively Type-II right censored order statistics and then use these results to compute the means and variances of two parameter reduced Kies distribution. Besides, these moments are then utilized to derived best linear unbiased estimators of the scale and location parameters of two parameter reduced Kies distribution. The parameters of the two parameter reduced Kies distribution are estimated under progressive type-II censoring scheme. The model parameters are estimated using the maximum likelihood estimation method. Further, we explore the asymptotic conﬁdence intervals for the model parameters. Monte Carlo simulations are performed to compare between the proposed estimation methods under progressive type-II censoring scheme. Based on our study, we can conclude that maximum likelihood estimators is decreasing with respect to an increase of the schemes and comparing the three censoring schemes, it is clear that the mean sum of squares, conﬁdence interval lengths are smaller for scheme 1 than schemes 2 and 3.


Introduction
The one parameter reduced Kies (RK) distribution was introduced by Kumar and Dharmaja [1] for modeling data and a generalization of Kies distribution. The two parameter RK distribution is a flexible model which provides left-skewed, symmetrical, right-skewed, and reversed-J shaped densities (see Figure 1). Its hazard rate function (HRF) can provide decreasing, increasing, upside-down bathtub, bathtub, and reversed-J shaped hazard rates (see Figure 2). It is noted that the bathtub and modified bathtub hazard rates are very important in the reliability engineering context. John et al. [2] investigated modified-bathtub hazard rate shape is widely used in industrial and medical applications. For example, thermal stress screening is an assembly-level electronics manufacturing process that evolved from the burn-in processes used in NASA and DoD programs. While burn-in subjects the product to expected field extremes to expose infant mortalities (patent failures), thermal stress screening briefly exposes a product to fast temperature rate-of-change and out-of-spec temperatures to trigger failures that would otherwise occur during the useful life of the product. Also Xie and Lai [3], Lai et al. [4], Chakherloo et al. [5] and Al abbasi et al. [6] pointed out bathtub hazard rate shape is widely used in reliability engineering. The motivation for using this distribution here is that it has many applications in several areas of life such as accelerated life testing, survival analysis, reliability, biology, material science, engineering, physics, chemistry, economics, business administration, meteorology, hydrology, medicine, psychology and pharmacy. For a detailed account in this regard see Murthy et al. [7] or Rinne [8] and references therein. RK distribution can be viewed as a functional form of the Weibull distribution with shape parameter λ and it can be useful for modeling data sets with increasing and bathtub shaped hazard rate functions. Simple probability distributions generally do not exhibit bathtub-shaped failure rate, including Weibull, gamma, and log-normal. In most cases, bathtub shaped hazard functions have at least two parameters, whereas reduced Kies distribution has two parameter which exhibit both increasing and bathtub shaped hazard rate. In Engineering and Medical situations, Kumar and Dharmaja [1] observed that RK distribution is a better model compared to the Weibull as well as its extended models such as beta Weibull distribution, beta generalised Weibull distribution etc. in terms of hazard function is decreasing, increasing and bathtub shaped where Weibull models are inappropriate. Kumar and Dharmaja [1] studied the estimation of the parameters by using maximum likelihood estimation method of the reduced Kies distribution. Kumar and Dharmaja [9] considered the estimation of the Kies parameters under maximum likelihood estimation method. Dey et al. [10] studied the estimation of the reduced Kies parameter under progressive type-II censoring. They compared the performance of these estimators, for small and large samples, using extensive simulations. The only paper we were able to find on progressive type-II censoring of the one parameter RK distribution is Dey et al. [10]. This paper gives recurrence relations for single moments and product moments of progressive type-II censoring order statistics based on one parameter RK distribution. It did not consider two parameter RK distribution.
Let Y 1 , Y 2 , . . . , Y n be a random variable come from a two parameter RK distribution, then its non negative probability density function (pdf) and cumulative distribution function (cdf) are given as follows: and Here λ is a shape parameter and µ is a location parameter. From Equations (1) and (2), we obtain where (ξ) c = ξ(ξ + 1), · · · , (ξ + c − 1) denotes the ascending factorial. In Figures 1 and 2, various graphs of the pdf and the hazard rate function for the two parameter RK distribution for different parameters values. These plots show that the pdf is uni-modal, positively skewed and approximately symmetric. The plots in Figure 2 indicate that the hazard rate function for the two parameter RK distribution is very flexible. It can have increasing (IFR), decreasing (DFR), upside down bathtub (UBT) or bathtub (BT) failure rate functions.
Let the experimenter decides to carry out the life-test for a pre-fixed length of time, say T. Then the data arising from such a time-constrained life-test would be of the form Y 1:s ≤ · · · ≤ Y r:s with the remaining s − s lifetimes being more than T; here, r is random (0 ≤ r ≤ s) and has a binomial distribution with parameters (s, F(T)). This situation is referred to as Type-I censoring. Suppose the experimenter decides to carry out the life-test until the time of the r th failure, then the data arising from such a life-test would be of the form Y 1:s ≤ · · · ≤ Y r:s with the remaining s − r lifetimes being more than Y r:s . This situation is referred to as Type-II censoring see Balakrishnan [11].
In life testing experiments, it is common to come across incomplete or censored data. This happens particularly when the experimenter does not observe the failure times of all units placed on the life test and this may be intentional or unintentional or may be due to time constraints or owing to the structure of a technical system. Obviously, in such a situation, the probabilistic structure of the resulting incomplete data affects the censoring mechanism and therefore suitable inferential procedures become necessary. In literature, there are various censoring schemes which include right, left and interval  censoring, single or multiple censoring and type-I or type-II censoring. However, classical Type-I  and Type-II censoring schemes are not flexibile as they do not allow removal of units at point other  than the terminal point of the experiment. A mixture of type-I and type-II schemes is known as the hybrid censoring scheme. For this reason, we consider here a more general censoring scheme called progressive type-II censoring scheme.  If the failure times are based continuous cdf F(y) with pdf f (y), the joint pdf of the progressively censored failure times Y 1:r:s , Y 2:r:s , · · · , Y r:r:s , is given by Balakrishnan and Aggarwala [12]. f Y 1:r:s ,Y 2:r:s ,··· ,Y r:r:s (y 1 , y 2 , · · · , y r ) = ∆(s, r − 1) where with ∆(s, 0) = s. Here T is the progressive censoring scheme, T 1 , T 2 , . . . , T r are numbers which are prefixed, s is the number of units we put on the life testing experiment and r is the predetermined number of failures at which experiment will be terminated. Let y 1 , y 2 , · · · , y s be a random sample of size s from the two parameter RK distribution with pdf and cdf given in (1) and (2) respectively. The corresponding progressive Type-II right censored order statistics with censoring scheme (T 1 , T 2 , · · · T r ), r ≤ s will be · · · 0<y 1 <y 2 <···<y r <∞ In the last few decades, researchers have focused their attention to recurrence relation for moments of progressive type-II censoring. Many researchers considered moments of progressive type-II censoring in their studies. For example, Aggarwala and Balakrishnan [13] studied censored order statistics of a exponential and truncated exponential distribution. Balakrishnan et al. [14] discussed the inference under progressive type-II censoring of extreme value distribution. Fernandez [15] discussed the information of estimate the parameter of exponential distribution. With regard to progressive type-II censoring order statistics, readers may refer to the works of Cohen [16] discussed in progressively censored samples in life testing experiments. Viveros and Balakrishnan [17] obtained the interval estimation of life characteristics under progressively censored data. Balakrishnan and Aggarwala [18] discussed in details the progressive Censoring including theory, method and applications. Mahmoud et al. [19] studied the parameters estimation of linear exponential distribution under Progressively censored data. Sultan et al. [20] discussed the moments and estimation of parameters of the half logistic distribution based on progressively censored data, Balakrishnan et al. [21] obtained relations for moments of progressively censored order statistics from logistic distribution. Balakrishnan and Saleh [22] discussed relations for single and product moments of progressively Type-II censored order statistics from a generalized half logistic distribution. Dey et al. [23] discussed the estimation of parameters of Rayleigh distribution under progressively Type-II censored data. Kumar et al. [24] obtained the moments of extended exponential distribution under order statistics. Malik and Kumar [25] studied moments of progressively type-II Right censored order statistics from Erlang-truncated exponential distribution. Hu and Gui [26] discussed Bayesian and Non-Bayesian inference for the generalized Pareto distribution based on Progressive Type II Censored Sample. Malik and Kumar [27] obtained the moments of exponential-Weibull distribution based on progressively censored data. Singh and Khan [28] discussed the moments of progressively type-II right censored order statistics from additive Weibull distribution. Kumar et al. [29] studied the moments and estimation of parameters of extended exponential distribution based on progressive type-II right censored order statistics and Kumar et al. [30] considered estimation of the location and scale parameters of generalized Pareto distribution based on progressively type-II censored order statistics.
The key role of this article is two fold: first, we derive recurrence relations for the single and product moments of the RK distribution based on progressive type-II right censored order statistics. The so-obtained relationships enable us to compute all these moments for all sample sizes and all possible censoring schemes, using some mathematical softwares (Mathematica, Maple), second, we discuss the maximum likelihood estimators (MLEs) and BLUEs of the scale and location-scale parameters and compare them on the basis of bias and mean squared errors.
The rest of the paper is organized as follows. Relations for single moments is presented in Section 2. The relations for double (product) moments are given in Section 3. Parameter estimation along with approximate confidence intervals are computed in Section 4. In Section 5, the potentiality of the estimation approaches is assessed via simulation results. Finally, some remarks are offered in Section 7.

Relations for Single Moments
Here, we obtain some relations for the moments of progressive type-II right censored order statistics from the two parameters reduced Kies distribution.
Proof. We have, from Equations (3) and (6) α where Integrating (9) by parts, we obtain Using Equations (6) and (10) the Equation (8) can be rewritten as Proof. Similar to the proof of Theorem 1.
Proof. Similar to the proof of Theorem 1.
Proof. Similar to the proof of Theorem 1.

Relations for Product Moments
Here, we present the relations for product moments of the progressive type-II right censored order statistics from the two parameters reduced Kies distribution. The (i, j) th product moment of the progressive type-II right censored order statistics can be written as ...
Proof. We have, from (3) and (6), Integrating by parts, we get which, when substituted into Equation (18) and using (16), we have and hence the result.
Proof. Similar to the proof of Theorem 5.
In Tables 1-4, we have presented the values of means and variances of the progressive Type-II right censored order statistics for µ = 2, 3, λ = 1.0, 2.0 and different values of r and s.

Estimation of the Parameters
In this section, we obtain the best linear unbiased estimators (BLUEs) of the location and scale parameters and the maximum likelihood estimators (MLEs) of the two parameter RK distribution using progressive type-II censored samples.

BLUEs of Location and Scale Parameters
Let Y 1:r:s , Y 2:r:s , ..., Y r:r:s be a progressively type-II right censored samples from the location-scale two parameter RK distribution with the following probability density function (20) where µ is the location parameter and σ is the scale parameter. We use the single and product moments obtained in the previous section to derive the BLUEs of the location and scale parameters µ and σ. Let and The coefficients p i and q i given by (20) and (21), respectively, satisfy the conditions ∑ r i=1 p i = 1 and ∑ r i=1 q i = 0, which are used to check the computations accuracy. Tables 5 and 6 display the coefficients p i and q i for λ = 1, 2 and, respectively. These coefficients are obtained for various sample sizes, some selected progressive censoring (T 1 , . . . , T r ), and different number of failures r.

Maximum Likelihood Method
Let Y 1:r:s , Y 2:r:s , . . . , Y r:r:s be a progressively Type-II censored sample from two parameter RK distribution with (T 1 , T 2 , . . . , T r ) being the progressive censoring scheme. The likelihood function is given by f Y 1:r:s ,Y 2:r:s ,...,Y r:r:s (y 1 , y 2 , . . . , y r ) = ∆ (s, where f (y) and F(y) are given respectively by Equations (1) and (2). Substituting Equations (1) and (2) into Equation (23), the likelihood function is The log of likelihood function is Differentiating (25) with respect to λ and µ and equating to zero, we get and It is noted that the maximum likelihood estimates (MLEs) of the parameter λ and µ cannot be obtained in closed form, therefore, a numerical techniques can be used to solve (25) to obtain the MLEs of λ and µ.
To construct the 100(1 − ξ)% two-sided asymptotic confidence intervals for the unknown parameters λ and µ, the Fisher's information matrix must be obtained. Asymptotic variance-covariance (V-C) matrix of the MLEsΘ = (∆,μ) T can be obtained by inverting Fisher information matrix, I(Θ) in the form ractically, by dropping the expectation operator E and replacing ∆ by their MLEs∆, we get the approximate asymptotic V-C matrix for the MLEs, see Cohen [31], as Fisher's elements are given by the following , and Under some regularity conditions, the asymptotic normality of MLEs∆ = (λ,μ) T is approximately bivariate normal as∆ ∼ N(∆, I −1 (∆)). Hence, using the large sample theory, the 100(1 − ξ)% two-sided ACIs for λ and µ can be obtained, respectively, bŷ λ ∓ z ξ/2; Pλλ andμ ∓ z ξ/2; Pμμ , wherePλλ andPμμ are the main diagonal elements of (25), respectively, and z ξ/2 is the percentile of the standard normal distribution with upper probability (ξ/2 ) th .
We use the algorithm introduced by Balakrishnan and Sandhu [32] to generate progressively censored two parameter RK samples. The average values of the estimates of λ and the corresponding MSEs, Average confidence interval and coverage probabilities are displayed in Table 7 for (λ, µ) = (1.5, 0.5) and (λ, µ) = (3.0, 2.0). The average values of the estimates of µ and the corresponding mean sum of squares (MSEs), Average confidence interval and coverage probabilities are displayed in Table 8 for (λ, µ) = (1.5, 0.5) and (λ, µ) = (3.0, 2.0).

Discussion
This study examined the recurrence relations for single and product moments of progressively type-II censored samples from two parameter RK distribution. We have presented the values of means and variances of the progressive Type-II right censored order statistics for µ = 2, 3, λ = 1.0, 2.0 and different values of r and s. We observe that the means and variances are decreasing with respect to r, s, µ and λ. From our study it is to be noted that the MLEs is decreasing with respect to increase the Schemes. For fixed s, when the number of observed failure r increases, the MSEs and the confidence interval lengths decreases in all cases. Comparing the three censoring schemes, it is clear that the MSEs, confidence interval lengths are smaller for Scheme 1 than Schemes 2 and 3.

Conclusions
Based on our study, we can conclude that the the MLEs is decreasing with respect to increase the Schemes. For fixed s, when the number of observed failure r increases, the MSEs and the confidence interval lengths decreases in all cases. Comparing the three censoring schemes, it is clear that the MSEs, confidence interval lengths are smaller for Scheme 1 than Schemes 2 and 3. A future work may be to derive estimation procedures for the two parameter RK distribution based on order statistics, generalized order statistics and dual generalized order statistics. Another future work may be to characterize the two parameter RK distribution based on order statistics, generalized order statistics and dual generalized order statistics.