Conditional Maximum-Likelihood Estimation, from Singly Censored Samples, of the Shape Parameters of Pareto and Limited Distributions

Use of the functional relationship between the exponential and the Pareto and limited distributions enables one to obtain conditional maximum-likelihood (ML) estimators, from singly censored samples, of the shape parameters of the Pareto distribution F
₁
(y,ϵ,K) = 1 - (y - ϵ)
^-K
and the limited distribution F
₂
(x,ω,K) = 1 - (ω - x)K by a simple transformation of the corresponding estimator of the scale parameter of the exponential distribution θ
_mn
, based on the first m order statistics of a sample of size n. Use is made of the fact that K
_mn
|ϵ = 1/θ
_mn
and K
_mn
|ω = 1/θ
_mn
, where 2mθ
_mn
/ω has the x
²
distribution with 2m degrees of freedom, to set confidence bounds on the shape parameter K of the Pareto and limited distributions. The probability densities of K
_mn
|ϵ and Kϵ
_mn
|ω, which for a given m are the same for any n ⩾ m, are obtained by a simple transformation of that of θ
_mn
. The expected values of K
_mn
|ϵ and K
_mn
|ω are determined and from them the unbiasing factors by which the ML estimators must be multiplied to obtain unbiased estimators K
_mnmn
|ω. Expressions for the variances of the estimators and for the Cramer-Rao lower bound are found. A section on numerical examples is included.