Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin

Abstract

A general method of estimating parameters in continuous univariate distributions is proposed. It is especially suited to cases where one of the parameters is an unknown shifted origin. This occurs, for example, in the three-parameter lognormal, gamma and Weibull models. For such distributions it is known that maximum likelihood (ML) estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators. Properties of the proposed method are described. In particular it is shown to give consistent estimators with asymptotic efficiency equal to ML estimators when these exist. Moreover it gives consistent, asymptotically efficient estimators in situations where ML fails. Examples are given including numerical ones showing the advantages of the method.

A semi-parametric method for transforming data to normality

“…To solve this problem Atkinson et al (1991) proposed the use of grouped likelihood, which removes the unbounded maximum of the likelihood. A related procedure is the maximum product of spacings proposed by Cheng and Amin (1983), for which Titterington (1985) showed that this may be interpreted as a form of grouped likelihood. In this paper neither the likelihood nor the grouped likelihood estimation procedures are considered for the shifted Box-Cox transformation.…”

Workload modeling for resource usage analysis and simulation in cloud computing

“…This limits its utilization in the case of distributions used in this work [2,23]. As noted in Table 5, five different estimation methods were used [23]: Maximum Log-Likelihood (mle), Histogram Fitting (hist), Quantile Matching (quant), Probability Weighted Moments (pwm) [24], and Maximum Product of Spacing Estimator (mps) [25].…”

Analysis of extreme rainfall using the log logistic distribution

“…For parameter estimation of univariate distributions the MPS method of maximising the geometric mean of the sample spacings was developed by Cheng and Amin (1983) as an alternative to maximum likelihood. It was shown in Fitzgerald (1996) that the MPS method applied to a sample of size n is equivalent to using ML for a sample of size (n+1) where…”

“…The statistical tools used are: (1) the conventional bootstrap to sample the data with replacement (2) the ellipsoid of concentration (Crame`r, 1946) which gives a method of randomly varying the parameters within bounds set by their covariance matrix. (3) The maximum product of spacings solution (Cheng and Amin 1983); for a sample of size n (x 1 , x 2 , ..., x n ) it is equivalent to the ML solution of a more extreme sample of size (n+1) with x 1 ¢ < x 1 , x 2 ¢ < x 2 , ....x n ¢ < x n , x n+1…”