Mathematical Methods for Physics and Engineering - Matematica.NET

31.5 MAXIMUM-LIKELIHOOD METHODSubstituting these values into (31.50), we obtain( ) 1/2 Nˆσ x =s x ± ( ˆV [ ˆσ x ]) 1/2 =12.2 ± 6.7, (31.65)N − 1( ) 1/2 Nˆσ y =s y ± ( ˆV [ ˆσ y ]) 1/2 =11.2 ± 3.6. (31.66)N − 1Finally, we estimate the population correlation Corr[x, y], which we shall denote by ρ.From (31.62), we haveˆρ =NN − 1 r xy =0.60.Under the assumption that the sample was drawn from a two-dimensional Gaussianpopulation P (x, y), the variance of our estimator is given by (31.64). Since we do not knowthe true value of ρ, we must use our estimate ˆρ. Thus, we find that the standard error ∆ρin our estimate is given approximately by∆ρ ≈ 10 ( ) 1[1 − (0.60) 2 ] 2 =0.05. ◭9 1031.5 Maximum-likelihood methodThe population from which the sample x 1 ,x 2 ,...,x N is drawn is, in general,unknown. In the previous section, we assumed that the sample values were independentand drawn from a one-dimensional population P (x), and we consideredbasic estimators of the moments and central moments of P (x). We did not, however,assume a particular functional form for P (x). We now discuss the processof data modelling, in which a specific form is assumed for the population.In the most general case, it will not be known whether the sample values areindependent, and so let us consider the full joint population P (x), where x is thepoint in the N-dimensional data space with coordinates x 1 ,x 2 ,...,x N .Wethenadopt the hypothesis H that the probability distribution of the sample values hassome particular functional form L(x; a), dependent on the values of some set ofparameters a i , i =1, 2,...,m. Thus, we haveP (x|a,H)=L(x; a),where we make explicit the conditioning on both the assumed functional form andon the parameter values. L(x; a) is called the likelihood function. Hypotheses of thistype form the basis of data modelling and parameter estimation. One proposes aparticular model for the underlying population and then attempts to estimate fromthe sample values x 1 ,x 2 ,...,x N the values of the parameters a defining this model.◮A company measures the duration (in minutes) of the N intervals x i , i = 1, 2,...,Nbetween successive telephone calls received by its switchboard. Suppose that the samplevalues x i are drawn independently from the distribution P (x|τ) =(1/τ)exp(−x/τ), whereτis the mean interval between calls. Calculate the likelihood function L(x; τ).Since the sample values are independent and drawn from the stated distribution, the1255