ISSN:
1572-9230
Keywords:
Central limit theorem
;
local limit theorem
;
convergence rates
;
asymptotic expansions
;
Cramer-von Mises statistic
;
Anderson-Darling statistic
;
ω-statistics
;
trimmed ω-statistics
;
Banach spaces
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetB be a real separable Banach space and letX,X 1,X 2,...∈B denote a sequence of independent identically distributed random variables taking values inB. DenoteS n =n −1/2(X 1+...X n ). Let π:B→R be a polynomial. We consider (truncated) Edgeworth expansions and other asymptotic expansions for the distribution function of the r.v. π(S n ) with uniform and nonuniform bounds for the remainder terms. Expansions for the density of π(S n ) and its higher order derivatives are derived as well. As an application of the general results we get expansions in the integral and local limit theorems for ω-statistics $$\omega _n^p (q)\mathop { = n^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} }\limits^\Delta \smallint _{(0,1)} \{ F_n (x) - x\} ^p q(x)dx$$ and investigate smoothness properties of their distribution functions. Herep≥2 is an even number,q: [0, 1]→[0, ∞] is a measurable weight function, andF n denotes the empirical distribution function. Roughly speaking, we show that in order to get an asymptotic expansion with remainder termO(n −α), α〈p/2, for the distribution function of the ω-statistic, it is sufficient thatq is nontrivial, i.e., mes{t∈(0, 1):q(t)≠0}〉0. Expansions of arbitrary length are available provided the weight functionq is absolutely continuous and positive on an nonempty subinterval of (0, 1). Similar results hold for the density of the distribution function and its derivatives providedq satisfies certain very mild smoothness condition and is bounded away from zero. The last condition is essential since the distribution function of the ω-statistic has no density whenq is vanishing on an nonempty subinterval of (0, 1).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01049174
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