ABSTRACT
EXAMPLE4.1(Testingforspectralmoments).LetK(x)=.·(i((3>0), thenthetestingproblembecomes
H:r"f(>-) 3d,\=rrrg(,\) 3d,\, against(4.32)
TheteststatisticforEq.(4.32)is
(4.33)
TheefficacyofT11isgivenby
Wecanconstructanothertest:
(4.35)
where\II((J)={f(2(3+I)jf((J+I)2-I}.Inviewofestimationfor .Crrf(,\)'1d,\- f][g(>-)'1d.\,itisshownthatT~is,underH,asymptotically normalwithmean0andvarianceI,seeTaniguchi(1980).Thentheefficacy
Therefore,
EXAMPLE 4.2 (Testing for interpolation error). Let K(x) = x-1• then the testing problem becomes
against (4.36)
A: f/(A)- 1 dA =J r" g(A)- 1 dA. The quantity [1/27r .C"{2·n:f(A)}- 1 dAr 1 is known to be the interpolation error of X0 by the best linear interpolation based on X,, t = ±I, ±2, ... , see Hannan (1970. p. 165). The test statistic for Eq. (4.36) is
(4.37)
(4.38)
(4.39)
where
IJrfJ fJ ' M1 = -rrDB log/6 (>.) [)(),logf0(>.)d>., f;r fJ D Mg = -,:;-logg1,(>.) -;=;Jlogg1,(>.) d>..
