1. Introduction

Ordered random variables such as record values have gained a lot of importance during the past years. Record values are widely used in many situations of daily life as well as in statistical modelling and inference; to describe random variables arranged in order of magnitude.

Inverse Rayleigh distribution (IRD) is used in the area of acceptance sampling plan in which life time of a product follows the IRD. The software developed by Wood [1] is used for fitting data to this distribution. The IRD is considered as a model for a life time random variable, and has been assumed by Rosiah and Kantam [2]. The probability density function of two parameter IRD with location parameter and scale parameter λ is

(1)

Hirai [3] estimated the location and scale parameters of Rayleigh distribution by quadratic coefficients method. Dyer and Whisenand [4] derived the best linear unbiased estimator (BLUE) of the parameter of the Rayleigh distribution using order statistics in a Type II censored sample. The concept of record values from normal distribution was discussed by Balakrishnan and Chan [5]. They computed mean, variance and covariance’s for the record values from normal distribution and gave the BLUEs of location and scale parameter for the first n record values. Aleem [6] found concomitants of order statistics for bivariate IRD and their moments. Hirai and Akhter [7] discussed cubic coefficients estimates and asymptotic properties of the estimates from the two parameters Rayleigh distribution. Shawky and Bakoban [8] considered order statistics from an exponentiated Gamma distribution. Estimates of the scale parameter of exponentiated Gamma distribution under Type II censoring were obtained. Akhter and Hirai [9] estimates the scale parameter based on order statistics from Rayleigh distribution using type II singly and doubly censored samples by BLUE and Bloom’s method for sample size up to 8. Azaz and Akhter [10] discussed the estimation of location and scale parameters using upper record values of Rayleigh distribution by BLUE and alternative best linear unbiased estimates (ABLUE) from Type II singly and doubly censored samples. Tables of coefficients for BLUE and ABLUE of location and scale parameters were presented for various choices of censoring for . Feroze and Aslam [11] discussed the Bayesian analysis of IRD under singly and doubly Type II censored data. The Bayes estimators and corresponding risks have been derived under the assumption of non-informative priors and using symmetric and asymmetric loss functions. Jabeen et al. [12] discussed the estimation of location and scale parameters of Weibull distribution using concept of generalized order statistics by the BLUE and ABLUE from Type II singly and doubly censored samples.

This paper is designed for estimation of parameters of IRD for Type II singly and doubly censored data based on lower record values for a sample size up to 15. Moments of Record values are discussed in section 2. The estimation of location and scale parameters of IRD using Type II singly and doubly censored data under lower record values is described in section 3. Estimation of location and scale parameters of IRD under censoring with 10 missing observations are presented in section 4. Finally, conclusions are reported in Section 5.

2. Moments of Record Values Relating IRD

The i^th moment of is given by

(2)

For i =1 and 2.

The product moment of is (r < s)

(3)

Variance of the r^th lower record value is given as:

(4)

and covariance is:

(5)

Tables 1 and 2 provides the means and the variance-covariance of the r^th lower record values for the IRD with λ=1 for r up to 15.

Table 1. Means of lower record values for the Inverse Rayleigh distribution for 2 ≤ r ≤ 15

Table 2. Variance-Covariance of Lower Record values from the Inverse Rayleigh distribution 2 ≤ r ≤ s ≤ 15

3. Type II Singly and Doubly Censored Data under Lower Record Values

Lloyd’s method using lower record values is used to the censored samples. The BLUE’s of θ and λ are given by

(6)

(7)

where

and is the variance covariance matrix of r-m₁-m₂ record values. Where m₁ and m₂ indicate the number of missing observations from left and right side respectively. Furthermore, the variance and covariance of these BLUE’s are given by

(8)

(9)

(10)

The values of coefficients a_i and b_i of Lloyd’s estimators of θ and λ are obtained when record numbers are 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 for all possible choice of censoring. Variances and covariance’s of the these estimators are provided in Appendix-I.

4. Estimation of Location and Scale Parameters of IRD under Censoring with 10 Missing Observations

We consider the estimation of scale and location parameters of IRD when a total of 10 observations are missing on left or on right side for the case r =15. The purpose is to investigate the impact of different combinations of m₁ and m₂ on variances of and such that m₁+ m₂=10.

Table 3. Variances of scale and location parameters for censored data with 10 missing observations

Figure 1. Variance of θ (10 missing observations)

Figure 2. Variance of λ (10 missing observations)

From figures 1 and 2 when Type II singly and doubly censoring done by Lloyd’s method we may conclude:

• Left censoring provides much more precise estimators of λ and θ than the right censoring when the same number of observations is not available in the sample.

• For the same sample size r when m₁ increases the estimator of λ and θ gradually loses their efficiency.

• Also, for the same r when m₂ increases the estimator of λ and θ again loses their efficiency.

• When m₁ and m₂ both increase within the same sample size r efficiency of these estimators decreases.

So the variance of the estimators of the scale and location parameters under censoring depends on whether the censoring is left, or right, or double censoring. Also, it depends on the number of observations censored. For example, if only 5 observations are to be included in a sample of size 7 we can have two ways of doing it when m₁=1, m₂=1 and m₁=2, m₂=0. The corresponding censored estimators have the variances 0.576777 and 0.489284 respectively.

For different combinations of m₁ and m₂ with m₁+m₂ remaining fixed within sample size r the variances of the estimators change e.g. with m₁=1, m₂=1 and m₁=2, m₂=0 the range of variances of scale parameter is 0.087493. We have developed below a table that provides such information based on different sample sizes.

Thus a censored sample including the same number of observations may yield a large range. The range decreases when the number of available observations increases for the same sample size. But it increases when the sample size increases including the same number of available observations. The above information is useful for planning a sampling scheme for the estimation of location and scale parameters.

Table 4. Variance range for censored estimators of the scale parameter

5. Conclusions

It is interesting to discover that left censoring provides much more precise estimators of λ and θ than the right censoring when the same number of observations is missing in the sample. Furthermore, as the sample size increases but the number of censored observations remains constant, the precision of these estimates increases exponentially. Covariance’s of the above estimators also display a noteworthy behaviour. Firstly, these covariance’s are negative. Secondly, right censoring provides larger covariance’s of Lloyd’s estimators of λ and θ than the left censoring. Also as the sample size increases but the number of censored observations remains constant, these covariance’s approach zero rapidly. A censored sample including the same number of observations may yield a large range. The range decreases when the number of available observations increases for the same sample size. But it increases when the sample size increases including the same number of available observations. This aspect therefore must not be ignore when planning a censored sampling scheme for the estimation of location and scale parameters.

Appendix-I

Appendix-I. Variances and covariance’s of BLUEs for location and scale based on singly and doubly censored samples