Kunio Shimizu

A Bivariate Mixed Lognormal Distribution with an Analysis of Rainfall Data

Abstract The paper proposes a bivariate Mixed lognormal (Δ2) distribution as a probability model for representing rainfalls, containing zeros, measured at two monitoring sites and provides the maximum-likelihood (ML) estimates or ML estimating equations of the parameters for the Δ2 distribution. The distribution extends the univariate mixed lognormal distribution to the bivariate case. Two procedures for model selection are proposed: the first is the use of statistical test and the second is done by minimizing the Akaikes information criterion (AIC). An illustration for an analysis of the AMeDAS (Automated Meteorological Data Acquisition System) daily rainfall dataset observed in the summer half-year (Mayx96October) of 1988 is given. A Δ2 distribution is fitted to the bivariate data obtained from Tokyo and each of the other 1048 monitoring sites. Among the 1048 cases investigated, the agreement between the models selected by the test (5% significance level) and the minimum AIC procedures is 846 cases (80.7...

Optimal thresholds for the estimation of area rain-rate moments by the threshold method

Abstract Optimization of the threshold method, achieved by determination of the threshold that maximizes the correlation between an area-average rain-rate moment and the area coverage of rain rates exceeding the threshold, is demonstrated empirically and theoretically. Empirical results for a sequence of GATE radar snapshots show optimal thresholds of 5 and 27 mm h−1 for the first and second moments, respectively. Theoretical optimization of the threshold method by the maximum-likelihood approach of Kedem and Pavlopoulos predicts optimal thresholds near 5 and 26 mm h−1 for lognormally distributed rain rates with GATE-like parameters. The agreement between theory and observations suggests that the optimal threshold can be understood as arising due to sampling variations, from snapshot to snapshot, of a parent rain-rate distribution. Optimal thresholds for gamma and inverse Gaussian distributions are also derived and compared

Uniformly minimum variance unbiased estimation in lognormal and related distributions

The uniformly mimimum variance unbiased estimators and their variances from independent samples of lognormal distributions are concisely expressed using the hypergeometric functions

Optimal thresholds for a mixture of lognormal distributions as the continuous part of the mixed distribution

Abstract The paper considers a mixture of two and three lognormal distributions as the continuous part of the mixed distribution, which consists of a positive, continuous, skewed distribution with discrete mass at the origin. A mixture of two lognormal distributions with constant shape parameter is unimodal. A mixture of three lognormal distributions with constant shape parameter is not always unimodal. A sufficient condition for unimodality is provided. Expressions produced by fitting the first and second moments of a single lognormal distribution to those of lognormal mixtures are treated. Empirical optimal thresholds for estimating area rain-rate first and second moments are compared with theoretical optimal thresholds chosen by the maximum likelihood approach and by the distance function approach. Theoretical optimal thresholds for estimating the area rain-rate first moment are insensitive to distribution variations relative to those for the second moment.

On umvu estimators for the multivariate lognormal distribution and their variances

Uniformly minimum variance unbiased estimators of several parameters of the multivariate lognormal distribution are expressed by using the hypergeometric functions of matrix argument. And the variances are given in special cases.

Umvu estimation for covariances and first product moments of transformed variables

Suppose that a pair of random variables has a bivariate normal distribution with mean vector and covariance matrix denoted by N2:(μ, ∑), and the functions f and g are of recursive type. Then the UMVU estimators for Cov[f(X), g(Y)] and E[f(X)g(Y)] are given from independent samples having N2:(μ, ∑) for all i = 1,…n (n ≧ 3).

Unbiased estimation of the autocovariance function in a stationary generalized lognormal process

Let [Yt], t = 0, ±1, ±2, …, be a stationary sequence of normally distributed random variables with means, μ variances σ2 and autocorrelation coefficients ρh, h = 0, ±1, ±2, …, and let f be a recursive-type function. A stationary generalized lognormal porcess is defined as the sequence of {f(Yt)], t = 0, ±1, ±2, …. The paper provides unbiased estimators for the autocovariance function of a stationary generalized lognormal process with known μ and unknown σ2 and ρh Thier variances are also given.

Covariances between umvu estimators for means of transformed variables

Let be a random vector having a bivariate normal distribution with mean and covariance matrix denoted and let f and g be recursive type functions . the UMVU estimator for θ1 = E[f(X)] and θ2 = E[g(Y)], respectively, which are given from independent samples Then the covariance between is given