Abstract
One of the scientific objectives of the Tropical Rainfall Measuring Mission is to estimate monthly mean rainfall between approximately ±37°(latitude) and over 5°×5°(latitude-longitude) regions.The present study is motivated by the need to determine a relationship between radar reflectivity Z and rain rate R. The relationship, Z=ARB, where A(>0) and B are constants, is known as the Z-R relationship. It reduces to a linear relation between log Z and log R. The regression is widely used to determine the relationship. Other methods are the method of probability matching, the method of incomplete first-moment ratio matching, and the method of Lorenz curve matching.
The paper addresses the structural model approach to the determination of the relationship. It studies the maximum likelihood (ML) estimation under a normal assumption when some observations on either of the variables are missing. As a side condition for identifiability, it is assumed that the variance of the unobservable variable log R is known. This condition is adopted because it is known that a lognormal distribution provides a close fit to the frequency distribution for area-average rain rate, conditional on rain, measured in the tropical region and the shape parameter tends to be a constant if the rain is appropriately stratified by area, time and type. The paper gives an algorithm for having ML estimates and provides asymptotic variances and covariances of the ML estimators.
