By Albert Nikolaevich Shiriaev;V. G. Spokoiny
ISBN-10: 9810241011
ISBN-13: 9789810241018
The booklet is a spotlight in smooth mathematical facts which deals loads of new ideas.
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Additional info for Statistical experiments and decisions: asymptotic theory
Sample text
Zhang, 1998] that for uniform mean flow the magnitude of this second-order correction term decreases as the domain size L increases and that it is zero for an unbounded domain. It is seen from Eqs. 61) that the head terms are recursive. That is, the higherorder terms are in terms of lower-order ones. ,(h~l)(x)h~l)(y)h~l)(z)) , (h~l)(x)h~l)(y)h~l)(z)h~l)(x)) , (h (2) ( x ) h (2) (y)),and so on). It is the formal advantage of this approach that one does not have to decide a priori at which order to stop and what terms to disregard.
90) Higher-order terms can be given similarly. 2, to first order in a I the mean head is (h(x)) = h~~ and the fluctuation is h'(x) = h~l)(x). The first order head covariance is Ch(X, y) -- (h~l)(x)h~l)(y)) in terms of a}. The mean head may be corrected with higher-order terms such as (h~2)). 91) where J(X) = -dh~~ In the case under study, J = (Ho - HL)/L. Integrating the right-hand side of Eq. 92) Substitution of Eq. 80) into Eq. 93) 26 CHAPTER I I N T R O D U C T I O N which can be used to obtain the expressions for the head covariance Ch (x, y) and the cross-covariance C fh(X, y) between f (x) and h(y), Ch(x, y) -- j2 Cf(X - rl)dx do Lxf y J2x L L L Y L j2y -[- f x~foL C f ( x - n ) d x dn J2xy L L L L Cf(X -- rl)dx do + L2.....
103) becomes trivial with the solution of gr (0)(x) being a constant, and the head covariance depends on the separation vector r -- x - y rather than the actual locations of the two points. Hence the head field (and other flow variables) is stationary in space. 121) Solutions for these equations may be derived analytically. , Cf~(r) = 0), the cross-covariances 34 CHAPTER I INTRODUCTION Cf,o(r) and C~o(r) read as Cfv'(r) = 1 - (u)2,k2. 123) which are derived with the condition of Cf~v(-oo) = Cfv,(oo) = O.
Statistical experiments and decisions: asymptotic theory by Albert Nikolaevich Shiriaev;V. G. Spokoiny
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