Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

Extremely weakly unconditionally convergent series

It is proved that if Σ _i=1 ^∞ X _i is a non-convergent series in a Banach spaceX such that Σ _i=1 ^∞ |f(X _i )|<∞ for all extreme pointsf of the unit ball ofX*, thenX contains a subspace isomorphic toc ₀, improving a result of Bessaga and Pelczynski. The proof uses Fonf’s result that Lindenstrauss-Phelps spaces contain isomorphs ofc ₀. Supported in part by NSF-MCS-8002393.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Vector measures: where are their integrals?

Let ν be a vector measure with values in a Banach space Z. The integration map $I_\nu: L^1(\nu)\to Z$ , given by $f\mapsto \int f\,d\nu$ for f ∈ L ¹(ν), always has a formal extension to its bidual operator $I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}$ . So, we may consider the “integral” of any element f ^** of L ¹(ν)^** as I _ν ^** (f ^**). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z ^**. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X ^** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X′^* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I _ν ^** for the particular vector measure ν defined by ν(A) := T(χ _A ).     

Using Stepanov's method for exponential sums involving rational functions

For any additive character ψ and multiplicative character χ on a finite field F_q, and rational functions f,g in F_q(x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑_x∈Fq?Sχ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ω_i of modulus q^1/2 and the number of modulus 1.     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Identifying variable points on a smooth curve

LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x ₁, …,x _n ] an unorderedn-tuple of not necessarily distinct points onX. Byf _x :X →Y _x we denote the normalization which identifies thex ₁, …,x _n and maps them to the only and universal singularity of a complex curveY _x . Thenf _x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ _X of all complex-analytic quotients ofX and construct a morphismS ⁿ (X) →Q _X such that each and everyf _x corresponds to the image of the pointx on then-fold symmetric powerS ⁿ (X). For everyn ≥ 2 the mappingS ⁿ (X) →Q _X is a closed embedding; the points of its image have embedding dimensionn(n ? 1) inQ _X . HenceS ²(X) is a smooth connected component ofQ _X . On the other hand, a deformation argument yields thatS ⁿ (X) is part of the singular locus of the complex spaceQ _X provided thatn ≥ 3.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

Constrained best approximation in Hilbert space III. Applications ton-convex functions

This paper continues the study of best approximation in a Hilbert spaceX from a subsetK which is the intersection of a closed convex coneC and a closed linear variety, with special emphasis on application to then-convex functions. A subtle separation theorem is utilized to significantly extend the results in [4] and to obtain new results even for the “classical” cone of nonnegative functions. It was shown in [4] that finding best approximations inK to anyf inX can be reduced to the (generally much simpler) problem of finding best approximations to a certain perturbation off from either the coneC or a certain subconeC _F. We will show how to determine this subconeC _F, give the precise condition characterizing whenC _F=C, and apply and strengthen these general results in the practically important case whenC is the cone ofn-convex functions inL ₂ (a,b),     

Uniqueness properties in finite-continuous additive measurement

Let ? be a binary relation on A×X, and suppose that there are real valued functions f on A and g on X such that, for all ax, by ∈ A×X, ax ? by if and only if f (a)+g(x) ? f(b)+g(y). This paper establishes uniqueness properties for f and g when A is a finite set, X is a real interval with g increasing on X, and for any a, b and x there is a y for which f(a)+g(x)=f(b)+g(y). The resultant uniqueness properties occupy an intermediate position among uniqueness properties for other structural cases of two-factor additive measurement.It is shown that f is unique up to a positive affine transformation (αf+β₁ with α > 0), but that g is unique up to a similar positive affine transformation (αg+β₂) if and only if the ratio [f(a)?f(b)]/[f(a)?f(c)] is irrational for some a, b, c ∈ A. When the f ratios are rational for all cases where they are defined, there will be a half-open interval (x₀, x₁) in X such that the restriction of g on (x₀, x₁) can be any increasing function for which sup {g(x)?g(x₀): x₀ ? x < x₁} does not exceed a specified bound, and, when g is thus defines on (x₀, x₁), it will be uniquely determined on the rest of X. In general, g must be continuous only in the ‘irrational’ case.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Pettis integrability of Stone transforms

Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x ^* fx^*B(X^*)}, whereB(X^*) is the unit ball inX ^*.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

Estimation of the Location of the Maximum of a Regression Function Using Extreme Order Statistics

In this paper, we consider the problem of approximating the location,x₀∈C, of a maximum of a regresion function,θ(x), under certain weak assumptions onθ. HereCis a bounded interval inR. A specific algorithm considered in this paper is as follows. Taking a random sampleX₁, …, X_nfrom a distribution overC, we have (X_i, Y_i), whereY_iis the outcome of noisy measurement ofθ(X_i). Arrange theY_i's in nondecreasing order and take the average of ther X_i's which are associated with therlargest order statistics ofY_i. This average,x₀, will then be used as an estimate ofx₀. The utility of such an algorithm with fixed r is evaluated in this paper. To be specific, the convergence rates ofx₀tox₀are derived. Those rates will depend on the right tail of the noise distribution and the shape ofθ(·) nearx₀.     

Analytic functions whose range is dense in a ball

Denote by Δ(resp. $\text{Δ}$) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on $\text{}\text{?}\text{gD ? E}$, analytic on Δ and satisfying sup$\text{{6}\text{f}\text{?}\text{(z)6:zε}\text{δ}\text{?E}$. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.