Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.4 SOME BASIC ESTIMATORS

a 1

a 2

ˆa 1

aˆ 2

atrue atrue

aˆobs ˆaobs

(a) (b)

Figure 31.4 (a) The ellipseQ(aˆ,a)=cinaˆ-space. (b) The ellipseQ(a,aˆobs)=c

ina-space that corresponds to a confidence regionRat the level 1−α,when

csatisfies (31.39).

confidence level 1−αis given byQ(a,aˆobs)=c,wheretheconstantcsatisfies
∫c

0

P(χ^2 M)d(χ^2 M)=1−α, (31.39)

andP(χ^2 M) is the chi-squared PDF of orderM, discussed in subsection 30.9.4. This

integral may be evaluated numerically to determine the constantc. Alternatively,

some reference books tabulate the values ofccorresponding to given confidence

levels and various values ofM. A representative selection of values ofcis given

in table 31.2; there the number of degrees of freedom is denoted by the more

usualn, rather thanM.

31.4 Some basic estimators

In many cases, one does not know the functional form of the population from

which a sample is drawn. Nevertheless, in a case where the sample values

x 1 ,x 2 ,...,xNare each drawnindependentlyfrom a one-dimensional population

P(x), it is possible to construct some basic estimators for the moments and central

moments ofP(x). In this section, we investigate the estimating properties of the

common sample statistics presented in section 31.2. In fact, expectation values

and variances of these sample statistics can be calculatedwithoutprior knowledge

of the functional form of the population; they depend only on the sample sizeN

and certain moments and central moments ofP(x).

31.4.1 Population meanμ

Let us suppose that the parent populationP(x) has meanμand varianceσ^2 .An

obvious estimatorμˆof the population mean is the sample meanx ̄. Providedμ

andσ^2 are both finite, we may apply the central limit theorem directly to obtain