Applied Statistics

2Point Estimation

2.1 Introduction

The theory of point estimation is described in most books about mathematical statistics, and we refer here, as in other chapters, mainly to Rasch and Schott (2018).

We describe the problem as follows. Let the distribution P_θ of a random variable y depend on a parameter (vector)  θ ∈ Ω ⊆ R^p,  p ≥ 1 . With the help of a realisation, Y, of a random sample Y = (y₁, y₂, … , y_n)^T,  n ≥ 1 we have to make a statement concerning the value of θ (or a function of it). The elements of a random sample Y are independently and identically distributed (i.i.d) like y. Obviously the statement about θ should be as precise as possible. What this really means depends on the choice of the loss function defined in section 1.4 in Rasch and Schott (2018). We define an estimator S(Y), i.e. a measurable mapping of Rⁿ onto Ω taking the value S(Y) for the realisation Y=(y₁, y₂, … , y_n)^T of Y, where S(Y) is called the estimate of θ. The estimate is thus the realisation of the estimator. In this chapter, data are assumed to be realisations (y₁, y₂, … , y_n ) of one random sample where n is called the sample size; the case of more than one sample is discussed in the following chapters. The random sample, i.e. the random variable y stems from some distribution, which is described when the method of estimation depends on the distribution – like in the maximum likelihood estimation. For this distribution the rth central moment

2.1

is assumed to exist where μ = E(y) is the expectation ...