Improved estimation for a model arising in reliability and competing risks

Edsel A. Peña

Journal of Multivariate Analysis, 1991, vol. 36, issue 1, 18-34

Abstract: Let (Z1,M1),..., (Zn,Mn) be independent and identically distributed 1 - (p + 1) random vectors from the exponential-multinomial distribution which has density function f(z,m[theta]) = [lambda] exp(-[lambda]z) [Pi]j=1p([theta]j/[lambda])mj for z > 0 and m = (m1,...,mp) with mj [set membership, variant] {0,1} and m1p = 1, and where 1k denotes a k - 1 vector of 1's. The parameter [theta] = ([theta]1,...,[theta]p) has [theta]j > 0 and [lambda] = [theta]1p. This density function arises by observing a series system or a competing risks model with p sources of failure with the lifetime of the ith component or source of failure being exponential with mean 1/[theta]i, and where the random variable Z denotes system lifetime, while the ith component of M is a binary random variable denoting whether the ith component failed. It can also arise from the Marshall-Olkin multivariate exponential distribution. The problem of estimating [theta] with respect to the quadratic loss function L(a, [theta]) = ||a - [theta]||2/||[theta]||2, where ||v||2 = vv' for any 1 - k vector v, is considered. Equivariant estimators are characterized and it is shown that any estimator of form cN/T, where T = [Sigma]i=1nZi and N = [Sigma]i=1nMi, is inadmissible whenever c (n - 2)/n. Since the maximum likelihood and uniformly minimum variance unbiased estimators correspond to cN/T with c = 1 and c = (n - 1)/n, respectively, then they are inadmissible. An adaptive estimator, which possesses a self-consistent property, is developed and a second-order approximation to its risk function derived. It is shown that this adaptive estimator is preferable to the estimators cN/T with c = (n - 2)/(n + p - 1) and c = (n - 2)/n. The applicability of the results to the Marshall-Olkin distribution is also indicated.

Keywords: equivariant; estimator; exponential-multinomial; distribution; inadmissibility; invariance; Marshall-Olkin; multivariate; exponential; self-consistent; estimator (search for similar items in EconPapers)
Date: 1991
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/0047-259X(91)90088-J
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:jmvana:v:36:y:1991:i:1:p:18-34

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Journal of Multivariate Analysis is currently edited by de Leeuw, J.

More articles in Journal of Multivariate Analysis from Elsevier
Bibliographic data for series maintained by Catherine Liu ().