Local-Maximum-Likelihood Estimation of the Parameters of Three-Parameter Lognormal Populations from Complete and Censored Samples

The natural logarithm of the likelihood function is written down for the m - r order statistics remaining after censoring the n - m largest and the r smallest observations of a sample of size n(0<r<m<n) from a three-parameter lognormal population. Its first partial derivatives with respect to the parameters, when equated to zero, yield the likelihood equations, and the negatives of its second partial derivatives with respect to the parameters are the elements of the information matrix. Algebraic solution of the likelihood equations is impossible, so it is necessary to resort to iteration on an electronic computer. The iterative procedure proposed is applicable to special cases in which one or two of the parameters are known as well as to the most general case in which all three parameters are unknown. A modification of the procedure allows circumvention of a certain anomaly which sometimes occurs in maximum-likelihood estimation of the parameters of a three-parameter lognormal population from small samples. The information matrix is inverted to obtain the asymptotic variances and covariances of the local-maximum-likelihood estimators, which are tabulated for various values of the censoring proportions q sub 1 = r/ n from below and q sub 2 = (n - m)/n from above. Results are reported of a Monte Carlo study conducted to check the validity of the asymptotic variances and covariances and their applicability to samples of moderate size.