By Denis Bosq

This publication deals a predominantly theoretical insurance of statistical prediction, with a few power purposes mentioned, whilst info and/ or parameters belong to a wide or endless dimensional house. It develops the idea of statistical prediction, non-parametric estimation by means of adaptive projection – with functions to checks of healthy and prediction, and thought of linear strategies in functionality areas with functions to prediction of constant time tactics.

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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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. In fact, its variance is 1 1 þ ð1 À eÀ2uT Þ 2u3 T 4u2 T 2 when the Crame´r–Rao bound is 1=ð2u3 TÞ.

E. ðlj xj Þ and ðuj Þ are collinear. ^ Notes As far as we know a systematic exposition of the theory of statistical prediction is not available in the literature. In this Chapter we have tried to give some elements of this topic. Presentation of the prediction model is inspired by Yatracos (1992). 1 belongs to folklore but it is fundamental since it shows that the statistician may only predict Pu g, rather than g. NOTES 39 Definition of P-sufficient statistics appear in Takeuchi and Akahira (1975).

P ; g 1 ; . . ; g q Þ :¼ pj ðuÞ, j ! 1 and ðpj ; j ! 1Þ decreases at an exponential rate. Thus we clearly have à 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 .

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