New estimators of discriminant coefficients as the gradient of log-odds

We consider the problem of estimating the discriminant coefficients, η=∑^1-(θ⁽¹⁾-θ⁽²⁾) based on two independent normal samples fromN _p (θ⁽¹⁾,∑) andN _p (θ⁽²⁾,∑). We are concerned with the estimation of η as the gradient of log-odds between two extreme situations. A decision theoretic approach is taken with the quadratic loss function. We derive the unbiased estimator of the essential part of the risk which is applicable for general estimators. We propose two types of new estimators and prove their dominance over the traditional estimator using this unbiased estimator.     

NS Condition of Admissibility for the Linear Estimator of Normal Mean with Unknown Variance

Suppose Y - N（β, σ^2 In）, where β ∈ R^n and σ^2 〉 0 are unknown. We study the admissibility of linear estimators of mean vector under a quadratic loss function. A necessary and sufficient condition of the admissible linear estimator is given.     

Approximation in Learning Theory

This paper addresses some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs x_i and outputs y_i, i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator f_z on the base of given data z := ((x₁,y₁),...,(x_m,y_m)) that approximates well the regression function f_ρ (or its projection) of an unknown Borel probability measure ρ defined on Z = X × Y. We assume that (x_i,y_i), i = 1,...,m, are independent and distributed according to ρ. We discuss the following two problems: I. the projection learning problem (improper function learning problem); II. universal (adaptive) estimators in the proper function learning problem. In the first problem we do not impose any restrictions on a Borel measure ρ except our standard assumption that |y|≤ M a.e. with respect to ρ. In this case we use the data z to estimate (approximate) the L₂(ρ_X) projection (f_ρ)_W of f_ρ onto a function class W of our choice. Here, ρ_X is the marginal probability measure. In [KT1,2] this problem has been studied for W satisfying the decay condition ε_n(W,B) ≤ Dn^-r of the entropy numbers ε_n(W,B) of W in a Banach space B in the case B = C(X) or B = L₂(\rho_X). In this paper we obtain the upper estimates in the case ε_n(W,L₁(ρ_X)) ≤ Dn^-r with an extra assumption that W is convex. In the second problem we assume that an unknown measure ρ satisfies some conditions. Following the standard way from nonparametric statistics we formulate these conditions of the form f_ρ ∈ Θ. Next, we assume that the only a priori information available is that f_ρ belongs to a class Θ (unknown) from a known collection {Θ} of classes. We want to build an estimator that provides approximation of f_ρ close to the optimal for the class Θ. Along with standard penalized least squares estimators we consider a new method of construction of universal estimators. This method is based on a combination of two powerful ideas in building universal estimators. The first one is the use of penalized least squares estimators. This idea works well in the case of general setting with rather abstract methods of approximation. The second one is the idea of thresholding that works very well when we use wavelets expansions as an approximation tool. A new estimator that we call the big jump estimator uses the least squares estimators and chooses a right model by a thresholding criteria instead of the penalization. In this paper we illustrate how ideas and methods of approximation theory can be used in learning theory both in formulating a problem and in solving it.     

Consistency and normality of Huber-Dutter estimators for partial linear model

For partial linear model Y = Xτβ0 g0(T) with unknown β0 ∈ Rd and an unknown smooth function g0, this paper considers the Huber-Dutter estimators of β0, scale σ for the errors and the function g0 approximated by the smoothing B-spline functions, respectively. Under some regularity conditions, the Huber-Dutter estimators of β0 and σ are shown to be asymptotically normal with the rate of convergence n-1/2 and the B-spline Huber-Dutter estimator of g0 achieves the optimal rate of convergence in nonparametric regression. A simulation study and two examples demonstrate that the Huber-Dutter estimator of β0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator.     

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

For partial linear model Y=X~τβ_0 _(g0)(T) εwith unknown β_0∈R~d and an unknown smooth function go, this paper considers the Huber-Dutter estimators of β_0, scale σfor the errors and the function go respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β_0 and σare asymptotically normal with convergence rate n~((-1)/2) and the B-spline Huber-Dutter estimator of go achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β_0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.     

Parametric estimation for the standard and geometric telegraph process observed at discrete times

The telegraph process X(t), t ≥ 0, (Goldstein, Q J Mech Appl Math 4:129–156, 1951) and the geometric telegraph process with μ a known real constant and σ > 0 a parameter are supposed to be observed at n + 1 equidistant time points t _i  = iΔ _n ,i = 0,1,..., n. For both models λ, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also σ > 0 has to be estimated. We propose different estimators of the parameters and we investigate their performance under the asymptotics, i.e. Δ _n → 0, nΔ _n =  T < ∞ as n → ∞, with T > 0 fixed. The process X(t) in non markovian, non stationary and not ergodic thus we build a contrast function to derive an estimator. Given the complexity of the equations involved only estimators on the first model can be studied analytically. Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size n.      

Inference concerning the population correlation coefficient from bivariate normal samples based on minimal observations

Summary Let (X, Y) be bivariate normally distributed with means (μ ₁,μ ₂), variances (σ ₁ ² ,σ ₂ ² ) and correlation betweenX andY equal to ρ. Let (X _i ,Y _i ) be independent observations on (X,Y) fori=1,2,...,n. Because of practical considerations onlyZ _i =min (X _i ,Y _i) is observed. In this paper, as in certain routine applications, assuming the means and the variances to be known in advance, an unbiased consistent estimator of the unknown distribution parameter ρ is proposed. A comparison between the traditional maximum likelihood estimator and the unbiased estimator is made. Finally, the problem is extended to multivariate normal populations with common mean, common variance and common non-negative correlation coefficient.     

Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval

For location families with densitiesf ₀(x−θ), we study the problem of estimating θ for location invariant lossL(θ,d)=ρ(d−θ), and under a lower-bound constraint of the form θ≥a. We show, that for quite general (f ₀, ρ), the Bayes estimator δ _U with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa'sIERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ _U . Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ∈[a, b]. Research supported by NSERC of Canada.     

Full likelihood inferences in the Cox model: an empirical likelihood approach

For the regression parameter β ₀ in the Cox model, there have been several estimators constructed based on various types of approximated likelihood, but none of them has demonstrated small-sample advantage over Cox’s partial likelihood estimator. In this article, we derive the full likelihood function for (β ₀, F ₀), where F ₀ is the baseline distribution in the Cox model. Using the empirical likelihood parameterization, we explicitly profile out nuisance parameter F ₀ to obtain the full-profile likelihood function for β ₀ and the maximum likelihood estimator (MLE) for (β ₀, F ₀). The relation between the MLE and Cox’s partial likelihood estimator for β ₀ is made clear by showing that Taylor’s expansion gives Cox’s partial likelihood estimating function as the leading term of the full-profile likelihood estimating function. We show that the log full-likelihood ratio has an asymptotic chi-squared distribution, while the simulation studies indicate that for small or moderate sample sizes, the MLE performs favorably over Cox’s partial likelihood estimator. In a real dataset example, our full likelihood ratio test and Cox’s partial likelihood ratio test lead to statistically different conclusions.     

On universal estimators in learning theory

This paper addresses the problem of constructing and analyzing estimators for the regression problem in supervised learning. Recently, there has been great interest in studying universal estimators. The term “universal” means that, on the one hand, the estimator does not depend on the a priori assumption that the regression function f _ρ belongs to some class F from a collection of classes F and, on the other hand, the estimation error for f _ρ is close to the optimal error for the class F. This paper is an illustration of how the general technique of constructing universal estimators, developed in the author’s previous paper, can be applied in concrete situations. The setting of the problem studied in the paper has been motivated by a recent paper by Smale and Zhou. The starting point for us is a kernel K(x, u) defined on X × Ω. On the base of this kernel, we build an estimator that is universal for classes defined in terms of nonlinear approximations with regard to the system {K(·, u)} _uεΩ. To construct an easily implementable estimator, we apply the relaxed greedy algorithm. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 256–272.     

A minimum discrimination information estimator of preliminary conjectured normal variance

For the problem of estimating the normal variance ² based on random sampleX ₁,...,X _n when a preliminary conjectured interval [C ₀ ^–1 ₀ ² ,C _{0 0} ² ] is available, the minimum discrimination information (MDI) approach is presented. This provides a simple way of specifying the prior information, and also allows to consider a shrinkage type estimator. MDI estimator and its mean square error are derived. The estimator compares favorably with the previously proposed estimators in terms of mean square error efficiency.     

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y ₁, ..., Y _n are modeled in dependence of 1-periodic, second order stationary random functions X ₁, ...,X _n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill-posedness to be known. Then we generalize the procedure to a random set of admissible m’s and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in terms of a general weighted L ²-risk. This means that we provide adaptive estimators of both β and its derivatives.     

On quartic half-arc-transitive metacirculants

Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1, n≥2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ ^m has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed. Both authors were supported in part by “ARRS – Agencija za znanost Republike Slovenije”, program no. P1-0285.     

On the exact non-null distribution of Wilks'L VC criterion and power studies

Letx ₁, x₂, ..., x_Nbep×1 random vectors distributed independently asN(u, Σ), Σ>0;u and Σ are unknown. In this paper, we derive the exact non-null distribution of Wilks' likelihood ratio criterion,L _VC, for testingH:∑=σ ²[(1−ρ)I+ρee′], σ>0 and ρ are unknown against the alternativeA≠H,e′=(1, 1, …, 1): 1×p. The distribution has been derived in three series forms: (1) a series of Meijer'sG-functions through Mellin transform, (2) an, alternate series using contour, intergration and (3) a series of chi square distributions. Powers have been computed based on these forms of the distribution forp=2 and 3.     

A functional characterization of certain mixed volumes

A function over the convex coneK{inn}, of convex bodiesK in Euclideann-space (where addition is vector addition, positive scalar multiplication is dilatation), which is linear overK{inn}, increasing with respect to set inclusion, and zero at point bodies must be a mixed volumeV(K; đ, p−1;σ ₁, …,σ _n−p). Heređ, takenp−1 times, is inK{inn} andσ ₁, …,σ _n−pare pairwise orthogonal unit segments spanning the orthogonal complement of the affine hull ofđ.     

Estimation of a Normal Covariance Matrix with Incomplete Data under Stein′s Loss

Suppose that we have (n − a) independent observations from N_p(0, Σ) and that, in addition, we have a independent observations available on the last (p − c) coordinates. Assuming that both observations are independent, we consider the problem of estimating Σ under the Stein′s loss function, and show that some estimators invariant under the permutation of the last (p − c) coordinates as well as under those of the first c coordinates are better than the minimax estimators of Eaten. The estimators considered outperform the maximum likelihood estimator (MLE) under the Stein′s loss function as well. The method involved here is computation of an unbiased estimate of the risk of an invariant estimator considered in this article. In addition we discuss its application to the problem of estimating a covariance matrix in a GMANOVA model since the estimation problem of the covariance matrix with extra data can be regarded as its canonical form.     

On a complete description of the class of functions without zeros analytic in a disk and having given orders

For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A _σ ^ρ of functions g(z) analytic in the unit disk = {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρ_T[g] = σ, and ρ_M[g] = ρ, where

On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Some remarks on estimation of parameters of uniform distribution

The best (in the sense of quadratic risk) unbiased estimators are constructed for the function f(x)=σ(2x/(n+1)−1)+μ from a sample of size n from the uniform distribution over [μ−σ, μ+σ] with unknown μ and σ. The best unbiased estimator for σ with μ being known is also presented. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 36–39, Perm, 1991.     

List Point Arboricity of Dense Graphs

Let G be a simple graph. The point arboricity ρ(G) of G is defined as the minimum number of subsets in a partition of the point set of G so that each subset induces an acyclic subgraph. The list point arboricity ρ _l (G) is the minimum k so that there is an acyclic L-coloring for any list assignment L of G which |L(v)| ≥ k. So ρ(G) ≤ ρ _l (G) for any graph G. Xue and Wu proved that the list point arboricity of bipartite graphs can be arbitrarily large. As an analogue to the well-known theorem of Ohba for list chromatic number, we obtain ρ _l (G + K _n ) = ρ(G + K _n ) for any fixed graph G when n is sufficiently large. As a consequence, if ρ(G) is close enough to half of the number of vertices in G, then ρ _l (G) = ρ(G). Particularly, we determine that , where K _2(n) is the complete n-partite graph with each partite set containing exactly two vertices. We also conjecture that for a graph G with n vertices, if then ρ _l (G) = ρ(G). Research supported by NSFC (No.10601044) and XJEDU2006S05.