Asymptotic variance and MSE for P[Y>X] in sampling from two one-truncation parameter families

In this research note we have presented some of the asymptotic theorems related to two one-truncation parameter families of distributions and use them to obtained explicit expression of asymptotic variance for U-estimable parametric function or asymptotic MSE of biased estimator. Comparison of different estimators and other inferential problem such as asymptotic minimum mean square error equivariant estimator and mean square error stabilization transformation have been tacked. Asymptotic MSE and variance of well-known stress-strength model, P[Y>X] have been derived and performance of MLE and the UMVU estimator in terms of ARE and LRE have been obtained with illustrative example.Abbreviations: UMVU; Uniformly Minimum Variance Unbiased, MLE; Maximum Likelihood Estimator, MSE; Mean Square Error, LRE; Limiting Risk Efficiency, ARE; Asymptotic Relative. _____. This work is supported by Extra Mural Research (EMR Individual Centric) Science and Engineering Research Board (SERB), New Delhi, India. File No. EMR/2016/005269. Efficiency, GUD; Generalized Uniform Distribution, pdf; probability density function, CDF; cumulative density function.

Illustrations explain the paper; they do not show study data.

AI-generated illustration

↗ Enlarge

LRE Efficiency MappingClose ×

AI-generated conceptual illustration (Abstract mathematical probability distributions). It explains the idea and does not show study data.

Open full-size image ↗

Introduction

One-truncation parameter family of distributions mean that left or right extremity of range of distribution involve parameter. Rohatgi (1989) [14] consider probability density function (pdf) of two independent right-truncation distributions as 𝑓 1 (𝑥; 𝜃 1 ) = 𝑞 1 (𝜃 1 )ℎ 1 (𝑥); 𝑎 < 𝑥 < 𝜃 Respectively, where-∞ < 𝑎 < 𝑏 < ∞ are known constants, 𝑞 𝑖 ; (1, 2, 3, 4) are everywhere differentiable functions, ℎ 𝑖 ; (1, 2, 3, 4) are absolutely continuous functions for two sample problem and obtained explicit expression for the UMVU estimator of any U-estimable parametric function 𝑔 = 𝑔(𝜃 𝑖 , 𝜃 𝑗 ); 𝑖 ≠ 𝑗 = 1, 2, 3, 4 by virtue of explicit expression for the Uniformly Minimum Variance Unbiased (UMVU) estimator of reliability functional, 𝑃[𝑌 > 𝑋] have obtained. Amongst many others Beg and Singh (1979) [2] , Beg (1980a, 1980b, 1980c, 1983) [3,4,5,6] , Dixit, Ali and Woo (2003) [9] , Dixit and Phal (2009) [10] , obtained the UMVU estimator of 𝑃[𝑌 > 𝑋] when the independent random variables 𝑋and 𝑌 follows truncation parameter families. References to previous work and other technical aspect of this problem [see Simonoff et al. (1986) [16] , Samuel et al. (2002)] [12] . Problem of obtaining explicit expression of variance or MSE was not properly or optimally addressed due to intractability of exact distributions to be needed.

Asymptotic distributions [main results]

Let 𝑋 1 , 𝑋 2 , … 𝑋 𝑛 and 𝑍 1 , 𝑍 2 , … , 𝑍 𝑛 be random samples with common pdf 𝑓 1 and 𝑓 3 where 𝑓 1 and 𝑓 3 defined in (1.1/1) and (1.7/7) respectively. We assume that 𝑋 𝑖 ′ s and 𝑍 𝑖 ′ s are independent. Further 𝜃 1 and 𝜃 3 are not functionally related. Let 𝑋 1:𝑛 < 𝑋 2:𝑛 < ⋯ < 𝑋 𝑛:𝑛 and 𝑍 1:𝑛 < 𝑍 2:𝑛 < ⋯ < 𝑍 𝑛:𝑛 be corresponding sets of order statistics. The complete sufficient statistics of parameters 𝜃 = (𝜃 1 , 𝜃 3 ) is 𝑋 = (𝑋 𝑛:𝑛 , 𝑍 1:𝑛 ) and it's joint pdf at 𝑥 = (𝑥 1 , 𝑧 1 ) is given by

Proof: Consider Young's form of Taylor series expansion of 𝑔(𝑋) around 𝜃 such as

ARE and LRE of MLE and the UMVU Estimator

In the set-up of theorem 2.1, theorem 2.2 and theorem 2.3, Rohatgi (1989) [14] Respectively, respectively. In order to compare the performance of MLE and the UMVU estimator, the following theorems have been established. At the order 𝑜(𝑛 -3 ).

Taking 𝐺(𝑋 𝑛:𝑛 , 𝑌 𝑛:𝑛 ) = 𝑔 ̂(𝜃 1 , 𝜃 2 ) derived by Rohatgi (1989) [14] ,

We Respectively. Let 𝑔 = 𝑔(𝜃 1 , 𝜃 2 ) = 𝑒 -𝜃 1 𝜃 2 then using theorem 3.1, one can obtain following results

Minimum mean square error equivariant estimator

Let 𝔵 be sample space for two samples, one sample taken from 𝑓 1 defined 1.

Let 𝐺 * be induced group of transformation on parametric space, Θ defined as

Where 𝐵 ∈ 𝔅 and 𝑔 * ∈ 𝐺 * . Assume that 𝐺 * is one to one and 𝑔 * Θ = Θ for all 𝑔 * ∈ 𝐺 * , such that

Further the square error loss function be invariant with respect to 𝐺 * . Let induced group of transformation on space of complete sufficient statistics

Theorem 4.1 In the set-up of theorem 2.1 and with respect to above transformation 𝐺 at the order 𝑜(𝑛 -2 ) the asymptotic minimum MSE equivariant estimator, 𝑔̅ (𝜃

MSE stabilized transformation

The asymptotic inference regarding the parameters 𝜃 1 and 𝜃 2 require the asymptotic distributions of (𝜃 1 -𝑋 𝑛:𝑛 ) and (𝜃 2 -𝑌 𝑛:𝑛 ) respectively but as MSE depends on parameters and the purpose will not be served until suitable MSE stabilization transformation is applied. The following theorem serve the purpose of suitable MSE stabilization transformation. And using theorem 3.1, theorem 4.1 follow. In the set-up of theorem 3.2 and theorem 3.3 it is easy to obtain MSE stabilized transformation by analogues arguments.

Asymptotic MSE and variance 𝑷[𝒀 > 𝑿]

Statistical inference concerning the reliability functional; 𝑃[𝑌 > 𝑋] has been of great interest to researchers who are doing research in reliability theory, survival analysis, engineering and industrial statistics. Asymptotic MSE of MLE and asymptotic variance of the UMVU estimator of reliability functional has been obtained and compared their performance in terms of LRE. In the set-up of theorem 2.1, 𝑔 ̃= 𝑔 ̃(𝜃 1 , 𝜃 2 ) = 𝑔(𝑋 𝑛:𝑛 , 𝑌 𝑛:𝑛 ) is MLE of And the UMVU estimator of 𝑔 derived by Rohatgi (1989) [14] is Plant develops into the reproductive phase of growth, a mat of smaller roots grows near the surface to a depth of approximately 1/6-th of maximum depth achieve [See G. Ooms and K.L. Moore (1991) [13] ]. Dixit (2003 and 2009) [9,10] assumed that a set of random variables 𝑋 1 , 𝑋 2 , … , 𝑋 𝑛 represents the masses of roots where out of n random variables some of these roots (say n) have different masses therefore, those masses have different uniform distribution with unknown parameters and these k observations are distributed with Generalized Uniform Distribution (GUD). the 𝐿𝑅𝐸[𝑔 ̂, 𝑔 ̃] remain same and is 1, because 𝛼 = -𝛽 which agreed to 3.2/18. In this case asymptotically 𝑔 ̂ is not more efficient then 𝑔 ̃.

1. Beg MA, Singh N. Estimation of 𝑃[𝑌 < 𝑋] for two Pareto distribution, IEEE Transaction on Reliability 1979;28(5):411-414.
2. Beg MA. Unbiased estimators for truncation and scale parameters, American Journal of Mathematical and Management Sciences 1983;3:251-274.
3. Beg MA, Singh N. Estimation of 𝑃[𝑌 < 𝑋] for two Pareto distribution, IEEE Transaction on Reliability 1979;28(5):411-414.
4. David H. Order Statistics, Wiley, New York 1969.
5. Dixit UJ, Masoom A, Woo J. Efficient estimation of parameters of a uniform distribution in the presence of outliers, IEEE Transaction on Reliability 2003;29(4):363-369.
6. Dixit UJ, Phal KD. Estimation of 𝑃[𝑌 < 𝑋] for the uniform distribution in the Presence of Outliers, Journal of Probability and Statistical Science 2009;7(1):11-18.
7. Johnson NL, Kotz S, Pensky M. Distributions in Statistics Continuous Univariate Distributions-1, John Wiley & Sons Inc., 1970.
8. Kotz Samuel, Yan L, Pensky M. The Stress-Strength Model and its Generalizations Theory and Applications, World Scientific, London 2002.
9. Ooms G, Moore KL. A model assay for genetic and environmental changes in the architecture of intact roots systems of plants grown in vitro, Plant Cell, Tissue and Organ Culture 1991;27:129-139.
10. Rohatgi VK. Unbiased estimation of parametric function in sampling from two one-truncation parameter families, Australian Journal of Statistics 1989;31(2):327-332.
11. Serfling RF. Approximation theorems of Mathematical Statistics, John Wiley, New York 1980.
12. Simonoff JS, Hochberg Y, Reiser B. Alternative estimation procedures for 𝑃[𝑋 < 𝑌] in categorized data, Biometrics 1986;42:895-907.
13. Tate RF. Unbiased estimation; function of location and scale parameters, Annals of Mathematical Statistics 1959;30:341-366.