Revisiting Best Linear Unbiased Estimation of Location-Scale Parameters Based on Optimally Selected Order Statistics Using Compound Design

Abstract

We introduce here the compound optimal design strategy to determine the Best Linear Unbiased Estimates (BLUEs) of location and scale parameters based on suitably chosen few order statistics. It is shown that the linear estimates of the parameters from any location-scale distribution, based on few optimally chosen ranks, obtained from the compound optimal design criterion is indeed the BLUE. Further extension of the strategy to Asymptotically Best Linear Unbiased Estimates (ABLUEs) is also discussed. The validation of the proposed strategy is justified through a numerical study for normal, Laplace and logistic distributions. Finally, a real-life data set is analyzed to illustrate the proposed strategy.

Appendix

1.1 A1. Laplace Population

For the standard exponential population with pdf

$$g(z) = e^{-z}, ~0\leq z < \infty,$$

the following results are well-known (see Arnold et al. 1992, p.72):

$$ \begin{array}{@{}rcl@{}} \mathrm{E}[Z_{r:n}] & = & \sum\limits_{i=n-r+1}^{n} \frac{1}{i}, ~r = 1, 2,\cdots, n,\\ \text{Var}[Z_{r:n}] & = & \sum\limits_{i=n-r+1}^{n} \frac{1}{i^{2}}, ~r = 1, 2,\cdots, n,\\ \text{Cov}[Z_{r:n}, Z_{s:n}] & = & \text{Var}[Z_{r:n}] ~=~ \sum\limits_{i=n-r+1}^{n} \frac{1}{i^{2}}, ~1\leq r < s \leq n, \\ \text{Cov}[Z_{s:n}, Z_{r:n}] & = & \text{Cov}[Z_{r:n}, Z_{s:n}], ~1\leq s < r \leq n, \end{array} $$

where Z_r:n denotes the r th order statistic from a sample of size n. Now, let us denote the order statistics from the standard Laplace population with pdf

$$h(x) = \frac{1}{2}e^{-\mid x\mid},~ -\infty < x < \infty,$$

as in (2). Then, the means, variances and covariances of order statistics from the above standard Laplace density can be found readily from the expressions, based on the moments of exponential order statistics presented above, established by (Govindarajulu 1963); see also (Balakrishnan et al. 1993) for a probabilistic proof.

1.2 A2. Logistic Population

Let us denote the order statistics from the standard logistic population with pdf

$$h(x) = \frac{e^{-x}}{(1+e^{-x})^{2}}, ~-\infty < x < \infty,$$

as in (2). Further, let us denote E[X_i:n] and $\mathrm {E}[X^{2}_{i:n}]$ by β_i:n and $\beta ^{(2)}_{i:n},$ respectively, for i = 1, 2,⋯ ,n, and E[X_i:nX_j:n] by β_{i, j:n}, for 1 ≤ i < j ≤ n. Then, the means, variances and covariances of order statistics from the standard logistic population can be determined in a simple recursive manner using a set of recurrence relations starting with the values of β_1:1 = 0, $\beta ^{(2)}_{1:1} = \frac {\pi ^{2}}{3}$ and β_1,2:2 = 0; see (Shah 1966; 1970) and (Gupta and Balakrishnan 1992).