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| In 1946 Paul Halmos studied unbiased estimators of minimum variance, and planted the seed from which the subject matter of the present monograph sprang. The author has undertaken to provide experts and advanced students...
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人気のある引用132 ページ 14 ページ 70 ページ この書籍を購入
この書籍内を検索 ページ 52: Practical Statistical Sampling for Auditors, Arthur J. Wilburn Vol. 53: Statistical Signal Processing, edited by Edward J. Wegman and James G. Smith Vol ... ページ 73: Assignment Methods in Combinatorial Data Analysis, Lawrence J. Hubert Vol. 74 : Econometrics and Structural Change, Lyle D. Broemeling and Hiroki ... 8 ページ Let f be the set of all distributions whose means exist, so that J- contains all distributions having finite support on R. Then the mean functional is r ... 21 ページ Theorem 1. Suppose that £|V>(^i, . . . ,Xk)\r < oo, where r>2. Then Proof. Let p = [j] , the greatest integer < j, ... 24 ページ The proof is by induction on the number of elements in U^j5,- n Sj. First assume that | U^j 5, n Sj] = 1, so that there is an element x with x 6 5, ... 26 ページ v " (nj) the identity and the relationship (1) for c = fc, we can write (n,*) j=i where f^n is the [/-statistic of degree j based on kernel h^. ... 30 ページ (i) Because of Theorem 2 (ii), we need only prove that Since j < j>, there is an element of £2 that is not in Si and hence there is a rv Xj> say, ... 34 ページ The equation (1) splits up LI into the direct sum of two orthogonal subspaces If we define C^ to be the Hilbert space of all [/-statistics of degree j with ... 37 ページ Such generalisations are explored in Sections 2.6-2.7. Brief bibliographic comments are in Section 2.8. 2.2 Generalised [/-statistics For j = 1, 2, ... 39 ページ The mean of Uni ,nj is just ^(J^, Yi) = Pr(Jfi < yi), while by Theorem 2 the ... 69 ページ If Fn denotes the empirical distribution function of the X >5, ie if then Xi:n = F~l(i/n) and (2) can be written n «/n ' £ j J(t)F-\i/n)dt ... 70 ページ J —00 These functionals may be expanded as a Taylor series about t = 0: - t)F + tG) = T(F) + tja(x)dG + Rc(t). ... 71 ページ J — OO For functionals of the form (4) the asymptotic variance is iin(x,2/)) - F(x)F(y))J(F(x))J(F(y))dy, J — I — OO provided the integral is positive. ... 76 ページ Let [/„ , j = 1, . . . , m be U-statistics having expectations 6j and kernels 4>^ of degrees kj. Also let £ = (•'.ij) where aij ... 114 ページ Note that Ahmad (1980) has a similar result for J/-statistics. A stopping time is an integer-valued rv T such that {T = n} is in fn for each n. ... 136 ページ <j<n ft[0,l] distribution in the light of the above remarks to sup W(t) where W(t) is "[0,i] a Brownian motion. Theorem 1 does not require the X,s to have ... 142 ページ Let Uni\...,nm, j = 1, . . . ,m be generalised U -statistics, based on kernels and a common set Xj j, j = 1, . . . , n,, i = 1, . . . , m of independent ... 158 ページ Replacing £i and £2 by sample medians -Y and Y gives TN(X, Y) = ~ ]T f] A-(X,- - X, Yi - Y). ,=:l J^l We show below that this substitution does not effect ... 160 ページ Let n(t;j) = Eg(X, t; 7) and suppose that /z(<; A) = 0 which will typically be the case for degenerate kernels. Denote the vector of partial derivatives of ... 167 ページ ... (j - I)Cn-ui + (n - j)Cn-ij, 2 < j < n - 1, and Thus a necessary and sufficient condition for an L-statistic to be a [/- statistic is n,i — (n ... 218 ページ Rather, the usefulness of the jackknife in the context of J/- statistics is that (1) furnishes a consistent estimate of Var Un. The explicit application of ... 220 ページ ... If hf-j) are the kernels of the component [/-statistics Hn , Theorem 3 of Section 1.6 tells us that Eh(j)h(i>) = 0 for j ^ j> so that taking ... 227 ページ ... J) where fc(i, j) - Cov(^(Xl,. . . ,Xk,)^(Xk, ... 229 ページ In a similar manner, we can estimate the ratio A = a^/a2 which in this case is of the form g(6i,6i) = (0i/02) - J~l . Provided a2 > 0, (1) once again holds ... 232 ページ Let Un'3 ,j = 1,2, ...,2k — c be [/-statistics based on kernels $(C>J) derived from the symmetric kernels "f *c) as in Theorem 1 of Section 4.2. ... 281 ページ J. Amer. Statist. Assoc. 63, 573-580. Boos, DD and Serfling, RJ (1980). A note on differentials, and the CLT and LIL for statistical functions, ... 282 ページ Actuarial J. 107-111. Csenki, A. (1981). A theorem on the departure of randomly indexed U- statistics from normality with an application in fixed width ... 283 ページ The functional law of the iterated logarithm for von Mises functionals and multiple Wiener integrals. J. Multivariate Anal. 28, 177-189. ... 285 ページ Fisher, NI and Lee, AJ (1986). Correlation coefficients for random variables on the sphere and hypersphere. Biometrika 73, 159-164. Fraser, DAS (1954). ... 288 ページ « 隠すLee, AJ (1982). On incomplete [/-statistics having minimum variance. Austral. J. Statist. 24, 275-282. Lee, AJ (1985). On estimating the variance of a ... 出版社の情報
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