is a linear combination of the ys Thus we have for some constants a 1 a 2 a

Is a linear combination of the ys thus we have for

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is a linear combination of the y/s. Thus we have for some constants a 1 , a 2 , ..• ,a •. Since the y;'s are assumed to be observed values of independent normal variates, it follows by (6.6. 7) that the sampling distribution of a is normal. It can be shown that £(&)=ex, and (5.5.7) gives var(&)= "f.a'f var(Y;) = a 2 c where c = "f.a'f. It follows that &-ex Z ==. CT: - N(O, I). v aic (13.2.6) If a 2 is known,.inferences about ex are based on (13.2. 6) . To test H: ex= ex 0 , we compute the observed value of Z when ex= ex 0 , and then find SL= P{ IZI IZob•I} = P{xf1> z;b.}· It can be shown that the likelihood ratio statistic for testing H: ex= exa is Z 2 , and so the procedure just described is the likelihood ratio test. To construct a 95% confidence interval for ex when a is known, we note from Table Bl or B2 that P{ - l.96:;; Z:;; l.96} = 0.95. Substituting for Z and solving gives a - 1.96#c $ex $a + l.96#c as the 95% confidence interval. This is also a 5% significance interval: it consists of all parameter values ex 0 such that a likelihood ratio test of H: cx = ex 0 gives SL 0.05. It is also a likelihood or maximum likelihood interval for ex . I 13.2 . Statistical Methods 203 Inferences for o:, /J, ... : <J 2 Unknown Usually a 1 is unknown, and is estimated by s 2 as defined in (13.2.5). By (13.2.4) we have Since "f.er is distributed independently of&, it follows that v is distributed independently of Z in (I 3.2.6). When a 2 is unknown, we consider the quantity a - ex T=-- Fc which we get by replacing rJ 2 by s 2 in (13.2.6). Note that where Z - N(O, I) and V- xfn-q» independently of Z . It follows by (6 . 10 . 5) that T has Student's distribution with n - q degrees of freedom: (13.2.7) Note that T has the same degrees of freedom as the variance estimate s 2 . To test H: ex= ex 0 , we compute the observed value of T when ex= ex 0 , and then use Table BJ to find SL= P{lt(n-q)' 17;,bsl}. Alternatively, we have SL= P{tfn-qJ r;bs} = P{Fi .• -q r;b.} by (6.10.7), which can be evaluated from Table BS for the F distribution. It can be shown that the likelihood ratio statistic for testing H: ex= ex 0 is an increasing function of T 2 , and hence the procedure just described is the likelihood ratio test. To construct a 95% confidence interval for ex, we use Table B3 to find the value t such that P{ -t :s; t<n-qJ S-t} = 0.95. Substituting from (13.2.7) and solving gives &-tFc :s;ex:.:;& + tFc as the 95% confidence interval. This is also a 5% significance interval and a maximum likelihood interval for oc.
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204 13 . Analysis of Normal Measurements Inferences for a It can be argued that, when the parameters a, p, .. . are unknown, the residual sum of squares r.e? carries all of the information from the y/s concerning a. Inferences about a will therefore be based on the marginal distribution of r.ef. By (13 . 2.4) we have By (6.9.1), the p.d.f. of Vis f(v) = k.vl•/2)- te -v/2 for v > 0, where v = n - q and kv is a constant. If we now change variables using (6.1.11), we find that the p.d.f. of 'f.ef is f(v)· -- =k vM2J - le - •/ 2,_ I dv I 1 dr.ef v a2 [ vs2J(v/2)- l { vs2} 1 =k - exp -- ·- al 2 a2 a2 Based on this distribution, the log likelihood function of a is vs 2 /(a)= - v log a - - for a> 0.
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