On Parameter Estimation for Truncated Pearson Type III Distributions

Abstract

The problem of estimating the parameters of the Pearson Type III probability density function (p.d.f.) $\phi(t, \alpha) = \alpha f(\alpha t) = \lbrack\Gamma(p)\rbrack^{-1}\alpha^pt^{p - 1}e^{-\alpha t},\begin{cases}0 \leqq t \\ 0 < \alpha \\ 0 < p \end{cases}\end{equation*}$ assuming various forms of truncation has been considered recently by A. C. Cohen, Jr. [2], Des Raj [4] and others. In this paper we obtain maximum likelihood estimates of the parameter $\alpha$ with $p$ is known apriori. Truncation is at a known point $T > 0$. Four cases are considered: truncation to the right of $T$ with the number of observations in the region of truncation (1) known, and (2) not known; and truncation to the left of $T$ with the number of observations in the region of truncation (3) known, and (4) not known. The information lost in not knowing the number of observations in the regions of truncation is measured in terms of the R. A. Fisher indices of information. The study of cases (2), and hence of case (1), is an outgrowth of the author's experience with a life testing program from which, unfortunately, data had been recorded only for those specimens which failed within 100 hours of testing. Despite this anomaly of experimental design a maximum likelihood estimate of $\alpha$ was found to exist. The analysis proceeded on the assumption of an exponential failure law $(p = 1)$. Another instance of case (2) arises naturally in connection with the distribution of population in urban communities. Colin Clark [1] has found urban population density, as a function of radial distance from city center, to be adequately described by the Pearson Type III p.d.f. with $p = 2$. The maximum likelihood estimate of the unspecific parameter, $\alpha$, for cities with circular peripheries, is contained in Section 2. Cases (3) and (4) are included for the sake of completeness. A possible area of application is to be found in the field of telemetry where frequently the result of random experiment is measured by an instrument which responds only to inputs in excess of a fixed magnitude.