Mathematical Methods for Physics and Engineering - Matematica.NET

31.5 MAXIMUM-LIKELIHOOD METHODBy substituting σ =1/u (so that dσ = −du/u 2 ) and integrating by parts either (N − 2)/2or (N − 3)/2 times, we findP (µ|x,H) ∝ [ N(¯x − µ) 2 + Ns 2] −(N−1)/2,where we have used the fact that ∑ i (x i − µ) 2 = N(¯x − µ) 2 + Ns 2 , ¯x being the sample meanand s 2 the sample variance. We may now obtain the 95% central confidence interval byfinding the values µ − and µ + for which∫ µ−P (µ|x,H) dµ =0.025 and P (µ|x,H) dµ =0.025.−∞µ +The normalisation of the posterior distribution and the values µ − and µ + are easilyobtained by numerical integration. Substituting in the appropriate values N = 10, ¯x =1.11and s =1.01, we find the required confidence interval to be [0.29, 1.97].To obtain a confidence interval on σ, we must first obtain the corresponding marginalposterior distribution. From (31.87), again using the fact that ∑ i (x i−µ) 2 = N(¯x−µ) 2 +Ns 2 ,this is given byP (σ|x,H) ∝ 1 ( ) ∫ ∞]σ exp − Ns2N(¯x − µ)2exp[− dµ.N 2σ 2 −∞ 2σ 2Noting that the integral of a one-dimensional Gaussian is proportional to σ, we concludethatP (σ|x,H) ∝ 1 ( )σ exp − Ns2 .N−1 2σ 2The 95% central confidence interval on σ can then be found in an analogous manner tothat on µ, by solving numerically the equations∫ σ−0P (σ|x,H) dσ =0.025andWe find the required interval to be [0.76, 2.16]. ◭∫ ∞∫ ∞σ +P (σ|x,H) dσ =0.025.31.5.6 Behaviour of ML estimators for large NAs mentioned in subsection 31.3.6, in the large-sample limit N →∞, the samplingdistribution of a set of (consistent) estimators â, whether ML or not, will tend,in general, to a multivariate Gaussian centred on the true values a. Thisisadirect consequence of the central limit theorem. Similarly, in the limit N →∞thelikelihood function L(x; a) also tends towards a multivariate Gaussian but onecentred on the ML estimate(s) â. Thus ML estimators are always asymptoticallyconsistent. This limiting process was illustrated for the one-dimensional case byfigure 31.5.Thus, as N becomes large, the likelihood function tends to the formwhere Q denotes the quadratic formL(x; a) =L max exp [ − 1 2 Q(a, â)] ,Q(a, â) =(a − â) T V −1 (a − â)1269