Progressively Type-II Right Censored Order Statistics from Hjorth Distribution and Related Inference

In this paper some recurrence relations satisfied by single and product moments of progressively Type-II right censored order statistics from Hjorth distribution have been obtained. Then we use these results to compute the moments for all sample sizes and all censoring schemes (R1, R2, ..., Rm),m ≤ n, which allow us to obtain BLUEs of location and scale parameters based on progressively Type-II right censored samples. The best linear unbiased predictors of censored failure times are then discussed briefly. Finally, a numerical example with real data is presented to illustrate the inferential method developed here.


Introduction
The scheme of progressive Type-II censoring is of importance in reliability and life-testing experiments. It allows the experimenter to remove units from a life test at various stages during the experiment which may lead to a saving of costs and of time (see Cohen [12] and Sen [29]). In such a random experiment, a group of n independent and identical experimental units is put on a life test at time zero with continuous, identically distributed failure times X 1 , X 2 , ..., X n . After the j th failure, a prespecified number R j ≥ 0 of the n − j − ∑ j−1 i=0 R i remaining (or surviving) units are randomly withdrawn from the experiment, 1  , and these are called progressively Type-II right censored order statistics of size m from a sample of size n with progressive censoring scheme (R 1 , R 2 , ..., R m ). Thus, in this type of sampling, m failures are observed, ∑ m j=1 R j units are progressively censored and n = m + ∑ m j=1 R j denotes the number of units in the life test. The withdrawal of units may be seen as a model describing drop-outs of units due to failures which have causes other than the specific one under study. In this sense, progressive censoring schemes are applied in clinical trials as well. Here, the drop-outs of patients may be caused by migration, lack of interest or by personal or ethical decisions, and they are regarded as random withdrawals. For a detailed discussion of progressive censoring and the relevant developments in this area, one may refer to Sen [29] and Balakrishnan and Aggarwala [4]. The situation with no censoring corresponds to the special case with m = n and R 1 = R 2 = ... = R m = 0, whereas the situation with ordinary Type-II right censoring at a given order statistic corresponds to the special case with , X (R1,R2,...,Rm) 2:m:n , ..., X (R1,R2,...,Rm) m:m:n is given by (cf. Balakrishnan and Sandhu [11] and Saran and Pushkarna [27]) f X1:m:n,...,Xm:m:n (x 1 , x 2 , ..., x m ) = A(n, m − 1) (1) where A(n, m − 1) = n(n − R 1 Here, note that all the factors in A(n, m − 1) are positive integers. Also it may be observed that the different factors in A(n, m − 1) represent the number of units still on test immediately preceding the first, second, ..., m th observed failures, respectively. Similarly, for convenience in notation, let us define for q = 0, 1, ..., p − 1, with all the factors being positive integers. Progressive censoring and associated inferential procedures have been extensively studied in the literature for a number of distributions by several authors. Cohen ([12], [13], [14], [15] and [16]), Mann ([21], [22]), Cohen and Whitten [17], Viveros and Balakrishnan [30], Balakrishnan and Sandhu [11], Aggarwala and Balakrishnan [1] and Balakrishnan and Aggarwala [4] have derived recurrence relations for single and product moments of progressively Type-II right censored order statistics from exponential, Pareto and power function distributions and their truncated forms. Saran and Pande [26], Saran and Pushkarna ([27], [28]), Saran et al. [25] and Pushkarna et al. [24] have derived recurrence relations for single and product moments of the corresponding progressively Type-II right censored order statistics from half logistic, Burr, left truncated logistic, Frechet and a general class of doubly truncated continuous distributions. Mahmoud et al. [20] derived some new recurrence relations for single and product moments of progressively Type-II right censored order statistics from the linear exponential distribution and also obtained maximum likelihood estimators (MLEs) of the location and scale parameters. Balakrishnan et al. [5] and Balakrishnan and Saleh ([7], [8], [9], [10]) have established several recurrence relations for single and product moments of progressively Type-II right censored order statistics from logistic, half-logistic, log-logistic, generalized half logistic and generalized logistic distributions and utilized them to derive the best linear unbiased estimators of the location and scale parameters. In this paper, we derive some recurrence relations satisfied by the single and product moments of progressively Type-II right censored order statistics from Hjorth distribution. These relations enable the recursive computation of moments for all sample sizes and all possible progressive censoring schemes. They generalize the corresponding results for exponential distribution due to Aggarwala and Balakrishnan [1]. Then we use these results to compute the means, variances and covariances of progressively Type-II right censored order statistics for some specific values of the parameters, which will be utilized to derive the best linear unbiased estimators (BLUEs) of location and scale parameters of the location-scale Hjorth distribution as well as their variances and covariances. Tables of these quantities are presented for different sample sizes up to n = 8 and some selected progressive censoring schemes, corresponding to particular values of the parameters. Further, for the special case R 1 = R 2 = ... = R m = 0, the derived results would reduce to the general recurrence relations for the usual order statistics from the Hjorth distribution. Also, we briefly discuss the best linear unbiased predictors (BLUPs) of the censored failure times by making use of the results developed on the BLUEs. Finally, one numerical example on real data is presented to illustrate all the methods of inference developed here.

Hjorth distribution
Hjorth distribution is a reliability distribution with increasing, decreasing, constant and bathtub shaped failure rates as its special cases. This distribution is also known as IDB distribution (cf. Hjorth [19]  and characterizing differential equation, respectively, are given by: The failure rate of this distribution is readily seen to be Special cases of the Hjorth distribution are: θ = 0 : the Rayleigh distribution (a Weibull distribution), δ = β = 0 : the exponential distribution (a Weibull distribution), δ = 0 : decreasing failure rate, δ ≥ θβ : increasing failure rate, 0 < δ ≤ θβ : bathtub curve.
More details on this distribution can be found in Hjorth [19]. The graphs of the p.d.f. and c.d.f. of Hjorth distribution as given in (2)

Recurrence relations for single moments
In this section, we shall establish several recurrence relations for single moments of progressively Type-II right censored order statistics from Hjorth distribution satisfying the characterizing differential equation ( Proof From (7), for n = m = r = 1, we obtain using (4), we have Integrating by parts all integrals on the R.H.S. of the above equation by taking (1 − F (x 1 )) for differentiation and the rest of the integrand for integration, and then after some simplification, it leads to the required result (8).  ] .

Proof
Proceeding in a similar manner as in Theorem 1, we can easily establish the relation (9).

Proof
The relation in (10) may be proved by following exactly the same steps as those used in proving Theorem 4, which is presented next.

486
PROGRESSIVELY TYPE-II RIGHT CENSORED ORDER STATISTICS Using the characterizing differential equation (4), we have where Integration by parts yields, Upon substituting for (15) in (14) and then substituting the resultant expression for I(x r−1 , x r+1 ) in (12) and simplifying, it leads to Theorem 4.
Next, we state another result on single moments which can easily be established on similar lines.

Theorem 5
there is no censoring before the time of the k th failure, then the first k progressively Type-II right censored order statistics are simply the first k usual 487 order statistics. Thus, for the special case R 1 = R 2 = ... = R m = 0, so that m = n in which case the progressively censored order statistics become the usual order statistics X 1:n , X 2:n , ..., X n:n , the recurrence relations established in Section 3 would reduce to the corresponding recurrence relations for the single moments of usual order statistics from the Hjorth distribution satisfying the characterizing differential equation (4).
x > 0, the recurrence relations in Section 3 will reduce to and verify the corresponding recurrence relations established by Aggarwala and Balakrishnan [1] for the progressively Type-II right censored order statistics from exponential distribution. It may be mentioned that one can derive similar recurrence relations for progressively Type-II right censored order statistics by taking different values of parameters as special cases of Hjorth distribution as given in Section 2.

Recurrence relations for product moments
Using (1) we can write the product moments of progressively Type-II right censored order statistics as follows: In this Section, we shall derive various recurrence relations for the product moments of progressively Type-II right censored order statistics from Hjorth distribution with p.d.f. f(x) and c.d.f. F(x) satisfying the characterizing differential equation (4).

Proof
The relation in (18) may be proved by following exactly the same steps as those used in proving Theorem 7.

Proof
The relation in (21) may be proved by following the similar steps as those used in proving (19).
Next, we state another result on product moments which can easily be established on similar lines.
Remark 3. For the special case R 1 = R 2 = ... = R m = 0, the recurrence relations established in Section 4 reduce to the corresponding recurrence relations for the product moments of usual order statistics from the Hjorth distribution satisfying the characterizing differential equation (4). Balakrishnan [1] for the product moments of progressively Type-II right censored order statistics from exponential distribution. It may be mentioned that one can derive similar recurrence relations for product moments of progressively Type-II right censored order statistics by taking different values of parameters as special cases of Hjorth distribution as given in Section 2.

Numerical Results
The recurrence relations obtained in the preceding Sections 3 and 4 allow us to evaluate the means, variances and covariances of progressively Type-II right censored order statistics from Hjorth distribution for all sample sizes 'n' and all censoring schemes (R 1 , R 2 , ..., R m ), m < n. These quantities can be used for various inferential purposes; for example, they are useful in determining BLUEs of location/scale parameters and BLUPs of censored failure times. In this section, we compute means, variances and covariances of the progressively Type-II right censored order statistics from Hjorth distribution for some specific values of parameters, viz.   Suppose we obtain a progressively Type-II censored data from the location-scale parameter Hjorth distribution with c.d.f. as given in (6).
In this section, we make use of means, variances and covariances of progressively Type-II right censored order statistics as determined by using the recurrence relations given in Sections 3 and 4 for deriving the BLUEs of the location and scale parameters µ and σ as well as the variances and covariance of these estimates. Let Then, the BLUEs of µ and σ are obtained by minimizing the generalized variance Q(δ) = (Y − Aδ) T ∑ T (Y − Aδ) with respect to δ, where δ = (µ, σ) T , A is m × 2 matrix (1, µ), 1 is m × 1 vector with components all 1's, µ is the mean vector of X, and ∑ is the variance-covariance matrix of X. The minimization leads to the expressions for the BLUES's of µ and σ as (see Anrold et al. [2] and Balakrishnan and Cohen [6]) and and the variances and covariance of these BLUEs are given by V ar(σ * ) = σ 2 The coefficients of the BLUEs in (23) and (24) satisfy the conditions ∑ m r=1 a r = 1 and ∑ m r=1 b r = 0 respectively. The coefficients of the BLUEs for µ and σ, and variances and covariance of these estimates are presented in Tables  3, 4 and 5, respectively, for various sample sizes up to n = 8 and for different choices of m and progressive censoring schemes.
and its variance is given by Also, µ i:m+1:n and σ i,j:m+1:n denote respectively the mean and covariance of the progressively Type-II right censored order statistics from the standard (µ = 0, σ = 1) distribution, and µ * and σ * are the BLUEs of µ and σ based on the progressively Type-II censoed sample Y. The BLUPs and their variances can therefore be readily computed from the means, variances and covariances of the progressively Type-II right censoed order statistics produced in Section 5. It is also illustrated in the next section with a numerical example using a real data set.

Illustrative Example
Consider the following data which represent failure times of air conditioning equipment in a boeing 720 airplane(Proschan [23]), arranged in increasing order of magnitude: 12,21,26,27,29,29,48,57,59,70,74,153,326,386,502. Take a random sample of size 7 from this data as: 21,26,27,29,29,48, 57 and assuming that this sample data follows Hjorth distribution in (6) and then producing a progressively censored data from above sample data, we have m Scheme y i:m:n 3 1,2,1 21, 26, 29 Before carrying out the inferential analysis for these data, let us verify model assumption. Specifically, the progressively Type-II censored data y i:3:7 were plotted against the values µ i:3:7 for i = 1, 2, 3 determined in Section 5 (as 0.037129, 0.09133, 0.232938) and this indicates a very high correlation (correlation coefficient is 0.922773) which suggest that the Hjorth model is a good fit model for these data.

Conclusion
In this paper, we have established several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from Hjorth distribution. With the help of these relations and using R software, we have computed all the means, variances and covariances of progressively Type-II right censored order statistics for different sample sizes and all possible censoring schemes. These moments have then been used to obtain the best linear unbiased estimators (BLUEs) of location and scale parameters of location-scale Hjorth distribution (6), as well as the best linear unbiased predictors (BLUPs) of the times to failure of the surviving units in the experiment. Finally, a numerical example has been presented to illustrate all the inferential methods developed here using a real data set.