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 .     

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.     

Sequential Dynamical Systems Over Words

In this paper we study sequential dynamical systems (SDS) over words. An SDS is a triple consisting of: (a) a graph Y with vertex set {v₁, ..., v_n}, (b) a family of Y-local functions , where K is a finite field and (c) a word w, i.e., a family (w₁, ..., w_k), where w_i is a Y-vertex. A map is called Y-local if and only if it fixes all variables and depends exclusively on the variables , for . An SDS induces an SDS- map, , obtained by the map-composition of the functions according to w. We show that an SDS induces in addition the graph G(w,Y) having vertex set {1, ..., k} where r, s are adjacent if and only if w_s, w_r are adjacent in Y. G(w, Y) is acted upon by S_k via and Fix(w) is the group of G(w, Y) graph automorphisms which fix w. We analyze G(w, Y)-automorphisms via an exact sequence involving the normalizer of Fix(w) in Aut(G(w, Y)), Fix(w) and Aut(Y). Furthermore we introduce an equivalence relation over words and prove a bijection between word equivalence classes and certain orbits of acyclic orientations of G(w, Y). Received September 12, 2004     

Strong consistency properties of nonparametric estimators for randomly censored data,I: The product-limit estimator

In reliability and survival-time studies one frequently encounters the followingrandom censorship model:X ₁,Y ₁,X ₂,Y ₂, is an independent sequence of nonnegative rv's, theX _n' s having common distributionF and theY _n' s having common distributionG, Z _n =min{X _n ,Y _n },T _n =I[X _n <-Y _n ]; ifX _n represents the (potential) time to death of then-th individual in the sample andY _n is his (potential) censoring time thenZ _n represents the actual observation time andT _n represents the type of observation (T _n =O is a censoring,T _n =1 is a death). One way to estimateF from the observationsZ ₁.T ₁,Z ₂,T ₂, (and without recourse to theX _n' s) is by means of theproduct limit estimator (Kaplan andMeier [6]). It is shown that a.s., uniformly on [0,T] ifH(T ^–)<1 wherel–H=(l–F) (l–G), uniformly onR if whereT _F =sup {x:F(x)<1}; rates of convergence are also established. These results are used in Part II of this study to establish strong consistency of some density and failure rate estimators based on .The third author's research was partly supported by National Research Council of Canada     

Limit distributions of U-statistics resampled by symmetric stable laws

Summary If (Y _i) and (V _i) are independent random sequences such thatY _i are i.i.d. random variables belonging to the normal domain of attraction of a symmetric -stable law, 0<<2, andV _i are i.i.d. random variables, then the limit distributions of U-statistics , coincide with the probability laws of multiple stochastic integralsX ^d f = ... f (t ₁, ... ,t _d)dX(t _d) with respect to a symmetric -stable processX(t).The research was originated during author's visit at ORIE, Cornell University     

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

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.     

Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

On weak asymptotic isomorphy of memoryless correlated sources

Let {(X _i,Z _i)} be an i.i.d. sequence of random pairs in a finite set × x ℒ; we will call it a discrete memoryless stationary correlated (DMSC) source with generic distribution dist(X ₁,Z ₁). Two DMSC sources {(X _i,Z _i)} and {(X _i′,Z _i′)} are called asymptotically isomorphic in the weak sense if for every ε>0 and sufficiently largen, there exists a joint distribution dist(X ⁿ,Z ⁿ,X′ ⁿ,Z′ ⁿ) ofn-length blocks of the two sources such that . For single sources of equal entropy, McMillan’s theorem implies asymptotic isomorphy in the sense suggested by this definition. For correlated sources, however, no nontrivial cases of weak asymptotic isomorphy are known. We show that some spectral properties of the generic distributions are invariant for weak asymptotic isomorphy, and these properties wholly determine the generic distribution in many cases.     

A note on the gl constant ofE \mathop \otimes \limits^ \vee F F

LetX andY be Banach spaces. TFAE (1)X andY do not contain subspaces uniformly isomorphic to (2) The local unconditional structure constant of the space of bounded operatorsL (X*_k,Y _k) tends to infinity for every increasing sequence and of finite-dimensional subspaces ofX andY respectively.     

Total Variation Distance for Poisson Subset Numbers

Let n be an integer and A₀,..., A_k random subsets of {1,..., n} of fixed sizes a₀,..., a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable , the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime. Received February 24, 2005     

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.     

Density estimation for linear processes

Summary LetX _t , ...,X _n be random variables forming a realization from a linear process where {Z _t } is a sequence of independent and identically distributed random variables with E|Z _t |<∞ for some ε>0, andg _r →0 asr→∞ at some specified rate. LetX ₁ have a probability density functionf. It is then established that for every realx, the standard kernel type estimator based onX _t (1≦t≦n) is, under some general regularity conditions, asymptotically normal and converges a.s. tof(x) asn→∞. Research was supported in part by the Air Force Office of Scientific Research Grant No. AFOSR-81-0058.     

Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

LetX ₁,X ₂, ...,X _n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY ₁,Y ₂, ...,Y _n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X _i }. We can only observeZ _i =min(X _i ,Y _i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ_(n)=1 i.o.)=0 ifF(τ _H ) < 1 andP(δ _n =0 i.o. )=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s., where {T _n } is a sequence of constants such that 1−H(T _n )=n ^−α(logn)^β(log logn)^γ.     

Designs of ϕ-optimal control for second-order processes

Consider a realization of the process on the intervalT=[0,1] for functionsf ₁(t),f ₂(t),...,f _n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f _k (t)} _k=1 ⁿ to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD _s-optimal design problems are discussed.     

On A Hypergraph Turán Problem Of Frankl

Let be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P_i∪P_j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K_r: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain can be obtained by partitioning V = V1 ∪V₂ and taking as edges all sets of size 2k that intersect each of V₁ and V₂ in an odd number of elements. Let denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no has at most as many edges as . Sidorenko has given an upper bound of for the Tur′an density of for any r, and a construction establishing a matching lower bound when r is of the form 2^p+1. In this paper we also show that when r=2^p+1, any -free hypergraph of density looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of to , where c(r) is a constant depending only on r. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials. * Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.     

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

The spectrum for overlarge sets of directed triple systems

A directed triple system of order v,denoted by DTS(v,λ),is a pair(X,B)where X is a v- set and B is a collection of transitive triples on X such that every ordered pair of X belongs toλtriples of B.An overlarge set of disjoint DTS(v,λ),denoted by OLDTS(v,λ),is a collection{(Y\{y},A_i)}_i, such that Y is a(v 1)-set,each(Y\{y},A_i)is a DTS(v,λ)and all A_i's form a partition of all transitive triples of Y.In this paper,we shall discuss the existence problem of OLDTS(v,λ)and give the following conclusion:there exists an OLDTS(v,λ)if and only if eitherλ=1 and v≡0,1(mod 3),orλ=3 and v≠2.