Download Inference and prediction in large dimensions by Denis Bosq PDF
By Denis Bosq
This ebook bargains a predominantly theoretical insurance of statistical prediction, with a few strength purposes mentioned, whilst information and/ or parameters belong to a wide or countless dimensional house. It develops the speculation of statistical prediction, non-parametric estimation through adaptive projection – with functions to checks of healthy and prediction, and concept of linear procedures in functionality areas with purposes to prediction of continuing time approaches.
This paintings is within the Wiley-Dunod sequence co-published among Dunod (www.dunod.com) and John Wiley and Sons, Ltd.
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Additional resources for Inference and prediction in large dimensions
1(ii) entails jPu ðUT 2 A; VT 2 BÞ À Pu ðUT 2 AÞPu ðVT 2 BÞj sup A2BRd ;B2BRd au ð’ðTÞ À cðTÞÞ ! 0; then, by using classical properties of ? weak convergence? (see Billingsley 1968) one deduces that w LðhUT ; VT iÞ ! LðhU; ViÞ and since w LðhUT ; dT iÞ ! dð0Þ ; p and cðTÞhT;h ðuÞ ! 0, the desired result follows. 4 Predicting some common processes We now apply the previous asymptotic results to some common situations. 15 (continued) Let ðXt ; t 2 ZÞ be an AR(1) process such that Varu X0 ¼ 1 and Eu jX0 jm < 1 where m > 4.
7) has a stationary solution Xt ¼ 1 X pj XtÀj þ "t ; t2Z j¼1 where pj ¼ pj ð’1 ; . . ; ’p ; g 1 ; . . ; g q Þ :¼ pj ðuÞ, j ! 1 and ðpj ; j ! 1Þ decreases at an exponential rate. Thus we clearly have Ã XTþ1 ¼ 1 X pj ðuÞXTþ1Àj j¼1 and if b uT is the classical empirical estimator of u, see Brockwell and Davis (1991), one may apply the above results. Details are left to the reader. 14 (continued) Consider the Ornstein–Uhlenbeck process Xt ¼ Z t eÀuðtÀsÞ dWðsÞ; t 2 R; ðu > 0Þ; À1 in order to study quadratic error of the predictor it is convenient to choose the parameter b¼ 1 ¼ Varu X0 ; 2u and its natural estimator bT ¼ 1 b T Z 0 T Xt2 dt: Here h rT;h ðb; YT Þ ¼ eÀ2b XT ; and since @rT;h ðb; YT Þ @h 2 jXT j; he2 50 ASYMPTOTIC PREDICTION one may take fðTÞ ¼ T and ZT ¼ ð2=hÞeÀ2 XT .
10 gives efficiency of pðXÞ for predicting u Xt2 dt; RT 0 Xt2 dt. 27 EFFICIENT PREDICTORS Taking u0 ¼ u2 as a new parameter, one obtains 0Z T pﬃﬃﬃﬃ X 2 À X 2 À T u 0 f1 ðX; u0 Þ ¼ exp À Xt2 dt À u0 T ; 2 2 0 hence À 1 2 Z 0 T Xt2 dt is efficient for predicting 1 X 2 À X02 À T 1 pﬃﬃﬃ0ﬃ T ¼ ðXT2 À X02 À TÞ: 2 4u 2 u This means that the empirical moment of order 2, Z 1 T 2 X dt T 0 t is efficient for predicting 1 X2 X2 1À T þ 0 : 2u T T It can be shown that it is not efficient for estimating Eu ðX02 Þ ¼ 1=2u.