Estimating the Parameters of a Linear Function of a Random Variable

Abstract

LetWbe a random variable andXbe a linear function ofW:X= αW+ β. Given a sample of independent observations distributed asX, we wish to find maximum-likelihood estimators for the parameters α and β of the linear function. If the support ofWis the whole real axis, the maximum-likelihood estimators can be found by the usual methods; in other cases restricted maximization techniques must be used. Here we restrict attention to cases in which the support is the interval [a, +∞) or the lattice {a + kh, k = 0, 1, 2, …} and the density ofWis decreasing over the support. Special attention is given to the exponential and geometric distributions. If the parameter of the distribution is also unknown, its maximum-likelihood estimator can also be obtained when the support ofWis the lattice.