Robust empirical Bayes tests for continuous distributions

In this paper, we study the empirical Bayes two-action problem under linear loss function. Upper bounds on the regret of empirical Bayes testing rules are investigated. Previous results on this problem construct empirical Bayes tests using kernel type estimators of nonparametric functionals. Further, they have assumed specific forms, such as the continuous one-parameter exponential family for {F_θ:θΩ}, for the family of distributions of the observations. In this paper, we present a new general approach of establishing upper bounds (in terms of rate of convergence) of empirical Bayes tests for this problem. Our results are given for any family of continuous distributions and apply to empirical Bayes tests based on any type of nonparametric method of functional estimation. We show that our bounds are very sharp in the sense that they reduce to existing optimal or nearly optimal rates of convergence when applied to specific families of distributions.     

Partitioned difference families and almost difference sets

Partitioned difference families (PDFs) were first studied by Ding and Yin in conjunction with the construction of constant composition codes (CCCs). In 2008, Yin et al. presented the constructions of a number of infinite classes of PDFs based on known difference sets in GF(q). In this paper, we further investigate the constructions of PDFs by using known almost difference sets in GF(q), and establish some recursive constructions of PDFs. As their applications, we also get a number of perfect difference systems of sets (DSSs) over Z_q² with q odd prime.     

Effect of adding regressors on the equality of the BLUEs under two linear models

In this paper we consider the estimation of regression coefficients in two partitioned linear models, shortly denoted as , and , which differ only in their covariance matrices. We call and full models, and correspondingly, and small models. We give a necessary and sufficient condition for the equality between the best linear unbiased estimators (BLUEs) of X₁β₁ under and . In particular, we consider the equality of the BLUEs under the full models assuming that they are equal under the small models.     

Remarks on an integral functional driven by sub-fractional Brownian motion

This paper studies the functionals where is a one-dimension sub-fractional Brownian motion with index H∈(0,1). It shows that there exists a constant p_H∈(1,2) such that p-variation of the process (j=1,2) is equal to 0 if p>p_H, where ?_j, j=1,2, are the local time and weighted local time of S^H, respectively. This extends the classical results for Brownian motion.     

Common singular spectrum analysis of several time series

Singular spectrum analysis (SSA) is a non-parametric time series modelling technique where an observed time series is unfolded into the column vectors of a Hankel structured matrix, known as a trajectory matrix. For noise-free signals the column vectors of the trajectory matrix lie on a single R-flat. Singular value decomposition (SVD) can be used to find the orthonormal base vectors of the linear subspace parallel to this R-flat. SSA can essentially handle functions that are governed by a linear recurrent formula (LRF) and include the broad class of functions that was proposed by Buchstaber [1994. Time series analysis and Grassmannians. Amer. Math. Soc. Transl. 162 (2), 1–17]. SSA is useful to model time series with complex cyclical patterns that increase over time.Various methods have been studied to extend SSA for application to several time series, see Golyandina et al. [2003. Variants of the Caterpillar SSA-method for analysis of multidimensional time series (in Russian) hhttp://www.gistatgroup.com/cat/i]. Prior to that Von Storch and Zwiers (1999) and Allen and Robertson (1996) (see Ghil et al. [2002. Advanced spectral methods for climatic time series. Rev. Geophys. 40 (1), 3.1–3.41]) used multi-channel SSA (M-SSA), to apply SSA to “grand” block matrices. Our approach is different from all of these by using the common principal components approaches introduced by Flury [1988. Common Principal Components and Related Multivariate Models. Wiley, New York]. In this paper SSA is extended to several time series which are similar in some respects, like cointegrated, i.e. sharing a common R-flat. By using the common principal component (CPC) approach of Flury [1988. Common Principal Components and Related Multivariate Models. Wiley, New York] the SSA method is extended to common singular spectrum analysis (CSSA) where common features of several time series can be studied. CSSA decomposes the different original time series into the sum of a common small number of components which are related to common trend and oscillatory components and noise. The determination of the most likely dimension of the supporting linear subspace is studied using a heuristic approach and a hierarchical selection procedure.     

A note on zenga concentration index

Summary The Zenga index, , is shown to be a concentration index, in the sense that, ifX andY are non negative random variables with 0<E(X), E(Y)<+∞, then (X)⩾ (Y) whenever the Lorenz curves satisfyL _x(p)≤L _y(p) for all p. Research partially supported by: M.U.R.S.T. 40% ?Inferenza statistica: basi probabilistiche e sviluppi metodologici?.     

A maximum-likelihood estimator with infinite error

The standard error of the maximum-likelihood estimator for 1/μ based on a random sample of size N from the normal distribution N(μ,σ²) is infinite. This could be considered to be a disadvantage.Another disadvantage is that the bias of the estimator is undefined if the integral is interpreted in the usual sense as a Lebesgue integral. It is shown here that the integral expression for the bias can be interpreted in the sense given by the Schwartz theory of generalized functions. Furthermore, an explicit closed form expression in terms of the complex error function is derived. It is also proven that unbiased estimation of 1/μ is impossible.Further results on the maximum-likelihood estimator are investigated, including closed form expressions for the generalized moments and corresponding complete asymptotic expansions. It is observed that the problem can be reduced to a one-parameter problem depending only on , and this holds also for more general location-scale problems. The parameter can be interpreted as a shape parameter for the distribution of the maximum-likelihood estimator.An alternative estimator is suggested motivated by the asymptotic expansion for the bias, and it is argued that the suggested estimator is an improvement. The method used for the construction of the estimator is simple and generalizes to other parametric families.The problem leads to a rediscovery of a generalized mathematical expectation introduced originally by Kolmogorov [1933. Foundations of the Theory of Probability, second ed. Chelsea Publishing Company (1956)]. A brief discussion of this, and some related integrals, is provided. It is in particular argued that the principal value expectation provides a reasonable location parameter in cases where it exists. This does not hold generally for expectations interpreted in the sense given by the Schwartz theory of generalized functions.     

A Class of Multivariate Distributions

In this paper we introduce a class of multivariate distributions, known as the generalized Liouville distribution and defined by the functional form (0 ≤x_i < ∞, αi > 0, β_i > 0, q_i > 0). It is shown that such distributions can be used to derive both the Dirichlet distribution and the beta distribution of the second kind.     

Optimal robust influence functions in semiparametric regression

Robust statistics allows the distribution of the observations to be any member of a suitable neighborhood about an ideal model distribution. In this paper, the ideal models are semiparametric with finite-dimensional parameter of interest and a possibly infinite-dimensional nuisance parameter.In the asymptotic setup of shrinking neighborhoods, we derive and study the Hampel-type problem and the minmax MSE-problem. We show that, for all common types of neighborhood systems, the optimal influence function can be approximated by the optimal influence functions for certain parametric models.For general semiparametric regression models, we determine in case of error-in-variables and in case of error-free-variables.Finally, the results are applied to Cox regression where we compare our approach to that of Bednarski [1993. Robust estimation in Cox's regression model. Scand. J. Statist. 20, 213–225] in a small simulation study and on a real data set.     

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy _k . This may correspond to the case of a regression model, where one observesy _k =f(θ,x _k )+ε _k , with ε _k some random error, or to the Bernoulli case wherey _k ∈{0, 1}, with Pr[y _k =1|θ,x _k |=f(θ,x _k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x₁, y₁),...,(x _k ,y _k ) andd _k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ _i=1 ^N w _i f(θ, x _i ) with {w _i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Almost sure convergence of vertex degree densities in the vertex splitting model

We study the limiting degree distribution of the vertex splitting model introduced in Ref.^[3 David, F.; Dukes, M.; Jonsson, T.; Stefansson, S.Ö. Random tree growth by vertex splitting. J. Statist. Mech. Theory Exp. 2009, 04. doi:10.1088/1742-5468/2009/04/P04009. [Google Scholar]^]. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.     

The Uniform Asymptotics of the Overshoot of a Random Walk with Light-Tailed Increments

Let {S _n : n ≥ 0} be a random walk with light-tailed increments and negative drift, and let τ(x) be the first time when the random walk crosses a given level x ≥ 0. Tang (2007 Tang , Q. ( 2007 ). The overshoot of a random walk with negative drift . Statist. Probab. Lett. 77 : 158 – 165 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained the asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, which is uniform for y ≥ f(x) for any positive function f(x) → ∞ as x → ∞. In this article, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for 0 ≤ y ≤ N for any positive number N will be given. Using the above two results, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for y ≥ 0, is presented.