Point and Interval Estimators, Based on m Order Statistics, for the Scale Parameter of a Weibull Population with Known Shape Parameter

Abstract

A derivation is given of the maximum likelihood estimator , based on the first m out of n ordered observations, of the scale parameter θ of a Weibull population with known shape parameter K. It is shown that 2m k/θK has a chi-square distribution with 2m degrees of freedom (independent of n). Use is made of this fact to set upper confidence bounds with confidence level 1 – P (lower confidence bounds with confidence level P) on the scale parameter θ. Formulas are given for the mean squared deviations of the upper and lower confidence bounds from the true value of the parameter. From these one can obtain expressions for the efficiency of confidence bounds and confidence intervals. The expected value of is also determined, and from it the unbiasing factor / by which must be multiplied to obtain an unbiased estimator . An expression for the variance of the unbiased estimator is found. Values of the unbiasing factor and of the variance of the unbiased estimator, both of which are independent of n, are tabled fo...