Asymptotic normality of a quadratic measure of orthogonal series type density estimate

LetX ₁,...,X _n be i.i.d. random variable with a common densityf. Let be an estimate off(x) based on a complete orthonormal basis {φ _k :k≧0} ofL ₂[a, b]. A Martingale central limit theorem is used to show that , where and .     

Inequalities for a distribution with monotone hazard rate

Summary LetX be a positive random variable with the survival function and the densityf. LetX have the moments μ=E(X) and μ₂=E(X ²) and put ε=|1-μ₂/2μ²|. Put and . It is proved that the following inequalities hold: , for allx>0, ifq(x) is monotone and that , ifq ₁ (x) is monotone. It is also shown that Brown's inequality which holds wheneverq ₁ (x) is increasing is not valid in general whenq ₁ is decreasing. The Institute of Statistical Mathematics     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

A note on the median of a distribution

Summary LetF be a distribution function over the real line. DefineR _p(y)=∫|x−y|^pdF(x) forp≧1. Forp>1 there is a unique minimizer ofR _p(·), sayγ _p. Under mild conditions onF it is shown that exists and that the limit is a median. Thus, one could use this limit as a definition of a unique median. Also it is shown that whereR is the right extremity ofF andL is the left extremity ofF provided that −∞<L≦R<∞. A similar result is available ifL=−∞,R=∞, yetF has symmetric tails.     

Some convergence theorems on a supercritical Galton-Watson process

Summary LetX=(X _n; n≧0,X ₀=1) be a supercritical Galton-Watson process. The limiting distribution of ) where is the m.l.e. of the offspring mean, is derived. As an application of this result, some limit theorems leading ultimately to a parameter free result of statistical interest, are also established.     

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions

LetF andG be two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability densitiesf andg, respectively. Matusita [3] defines a measure of the closeness, affinity, betweenF andG as: . Based on two independent samples fromF andG we propose to estimate ρ by , where and are taken to be the kernel estimates off(x) andg(x), respectively, as given by Parzen [5]. In this note sufficient conditions are given such that (i) asx→∞ and (ii) with probability one, asn→∞. Research supported in part by the National Research Council of Canada and by McMaster University Science and Engineering Research Board. The author is presently with the Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152.     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

When does the union of random spherical caps become connected?

DropN random caps all of the same angular radius on a unit sphere. LetU denote the part of the surface covered by these caps. We prove that if , then the probability thatU is connected tends to 1 asN→∞, while ifc<1, then the probability thatU is connected tends to 0 asN→∞.     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Maximal deviation theory of some estimators of prior distribution functions

Summary Let be a sequence of independent identically distributed random variables withθ ₁∼G and the conditional distribution ofx ₁ givenθ ₁=θ given by . HereG is unknown andF _θ(·) is known. This paper provides estimators ofG based onx ₁, …,x _n such that the random variable sup has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF _θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF _θ and an exponential distribution forG. Research supported by the National Science Foundation under Grant #MCS77-26809.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

Asymptotic expansions of the Mehler-Fock transform

Asymptotic expansions in the two limitsx → + ∞ andx → 0+ are obtained for the Mehler-Fock transform

Rates in the empirical bayes estimation problem with non-identical components

This paper considers empirical Bayes estimation of the mean θ of the univariate normal densityf ₀ with known variance where the sample sizesm(n) may vary with the component problems but remain bounded by <∞. Let {(θ _n ,X _n =(X _n,1,...,X _{n, m(n)} ))} be a sequence of independent random vectors where theθ _n are unobservable and iidG and, givenθ _n =θ has densityf _θ ^m(n) . The first part of the paper exhibits estimators for the density of and its derivative whose mean-squared errors go to zero with rates and respectively. LetR ^m(n+1)(G) denote the Bayes risk in the squared-error loss estimation ofθ _n+1 usingX _n+1. For given 0<a<1, we exhibitt _n (X₁,...,X _n ;X _n+1) such that . forn>1 under the assumption that the support ofG is in [0, 1]. Under the weaker condition that E[|θ|^2+γ]<∞ for some γ>0, we exhibitt _n ^* (X ₁,...,X _n ;X _n+1) such that forn>1.     

Interval estimation of the critical value in a general linear model

Summary This paper concerns interval estimation of the critical value θ which satisfies under the general linear model,Y _i =μ(x _i )+ε _i (i=1,2,···), where for and the functional forms off _j ^′ s are known. From an asymptotic expansion it is shown that, under reasonable conditions, the limiting distribution of is normal. Thus in the large-sample case a confidence interval for θ can be obtained. Such a result is useful when one is interested in carrying out a retrospective analysis rather than designing the experiment (as in the Kiefer-Wolfowitz procedure). In Section 3 a sequential procedure is considered for confidence intervals with fixed width 2d. It is shown that, for a given stopping variableN, is also asymptotically normal asd→0. Thus the coverage probability converges to 1−α (preassigned) asd→0. An example of application in estimating the phase parameter in circadian rhythms is given for the purpose of illustration. Research partially supported by the NSF Grant DMS-8502346.     

Eigenvalues and power growth

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ _p (T) the point spectrum ofT and by the unit circle. We investigate how the growth of the sequence ‖T ⁿ ‖ is influenced by the size of the set (T) and by the geometry of the spaceX. We also prove analogous results forC ₀-semigroups(T _t )_t≥0. Research partially supported by grants from NSERC, FQRNT and the Canada research chairs program.     

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR _n:X→X such thatR _nR_m=R_min(_n,m) ifn≠m and lim _n→∞ R _n x=x for allx∈X. We prove that, ifR _n−R_n −1 factors uniformly through somel _p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL _Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .     

Limit theorems on a linear explosive stochastic model for time series with moving average error

Summary LetX(t) be a linear autoregressively generated explosive time series, with autoregressive coefficientsb ₁,…,b_q, and a constant termb ₀, and an error term ; a₀=1. Where ε(t),t≧1 are independent, Eε(t)=0, and Eε ²(t)=σ² is positive and finite. In this paper two categories of -consisent and asymptotically singularly normal estimators are proposed for (b ₁,…,b_q, b₀) thus settling an open problem since the publication of the paper (Venkataraman [5]). Based on these estimators several additional limit theorems based on estimated error residuals are proved. The parameter-free limit theorems of Spectral and Quenouille types of this paper serve as asymptotic goodness of fit tests for the model generatingX(t).     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).