ESTIMATION OF AN EXPONENTIAL-DISTRIBUTION

Authors
Citation
M. Brown, ESTIMATION OF AN EXPONENTIAL-DISTRIBUTION, Probability in the engineering and informational sciences, 11(3), 1997, pp. 341-359
Citations number
12
Categorie Soggetti
Operatione Research & Management Science","Engineering, Industrial","Statistic & Probability","Operatione Research & Management Science
ISSN journal
02699648
Volume
11
Issue
3
Year of publication
1997
Pages
341 - 359
Database
ISI
SICI code
0269-9648(1997)11:3<341:EOAE>2.0.ZU;2-D
Abstract
Consider a sample X-1,...,X-n from an exponential distribution with un known parameter theta. From the sample, we wish to estimate the entire distribution, {epsilon(theta)(A) = integral(A) theta e(-theta x) dx, A is an element of B(Borel sets)}. If an estimator epsilon(<(theta)ove r cap>) is used for epsilon(<(theta)over cap>) we are concerned with p roximity of epsilon(<(theta)over cap>) to epsilon(theta), under total variation distance, D(epsilon(<(theta)over cap>),epsilon(theta)) = sup (A is an element of B)\epsilon(theta)(A) - epsilon(<(theta)over cap>)( A)\. For <(theta)over cap (n)> = ((X) over bar)(-1), the maximum likel ihood estimate of theta, define d(n,1-alpha) = e(-1)((Z/n(1/2)) + [(Z (5) - Z(3) + 6Z)/72n(3/2)]), where Z = Z(1-(alpha/2)), the 1 - (alpha/ 2) percentile from the standard normal. Then, P(D(epsilon(<(theta)over cap>n),epsilon(theta)) less than or equal to d(n,1-alpha)) = (1 - al pha) + o(n(-3/2)). The preceding approximation to the 1 - alpha percen tile of D (epsilon(<(theta)over cap>n), epsilon(theta)) is very accura te even for small n. We also consider the problem of obtaining a confi dence band for the survival function, possessing minimal maximum width . Finally, a class of estimators of epsilon(theta) is compared to the maximum likelihood estimator from the viewpoint of total variation dis tance loss function.