POINT INTERVAL ESTIMATION, FROM ONE-ORDER STATISTIC, OF THE LOCATION PARAMETER OF AN EXTREME-VALUE DISTRIBUTION WITH KNOWN SCALE PARAMETER AND OF THE SCALE PARAMETER OF A WEIBULL DISTRIBUTION WITH KNOWN SHAPE PARAMETER
Abstract
This paper derives a one-order statistic estimator Umnb for the location parameter of the (first) extreme-value distribution of smallest values with cumulative distribution function F(x;u,b) = 1 exp (-exp((x-u)/b)) using the minimum-variance unbiased one-order statistic estimator for the scale parameter of an exponential distribution. It is shown that exact confidence bounds, based on one-order statistic, can be easily derived for the location parameter of the extreme-value distribution and for the scale parameter of the Weibull distribution, using exact confidence bounds for the scale parameter of the exponential distribution. The estimator for u is shown to be b ln cmn + xmn where xmn is the mth order statistic from an ordered sample of size n from the extreme-value distribution with scale parameter b and cmn is the coefficient for a one-order statistic estimator of the scale parameter of an exponential distribution. Values of the factor cmn, which have previously been tabulated for n = 1(1)20, are given for n = 21(1)40. The ratios of the mean-square-errors of the maximum-likelihood estimators based on m order statistics to those of the one-order statistic estimators for the location parameter of the extreme-value distribution and the scale parameter of the Weibull distribution are investigated by Monte Carlo methods. The use of the table and related tables is discussed and illustrated by numerical examples.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1967
- Accession Number
- AD0653701
Entities
People
- Albert H. Moore
Organizations
- Air Force Research Laboratory