Asymptotically minimax non–parametric function estimation with positivity constraints I

The spatially inhomogeneous smoothness of the non-parametric density or regression-function to be estimated by non-parametric methods is often modelled by Besov- and Triebel-type smoothness constraints. For such problems, Donoho and Johnstone [D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998), pp. 879–921.], Delyon and Juditsky [B. Delyon and A. Juditsky, On minimax wavelet estimators, Appl. Comput. Harmon. Anal. 3 (1996), pp. 215–228.] studied minimax rates of convergence for wavelet estimators with thresholding, while Lepski et al. [O.V. Lepski, E. Mammen, and V.G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors, Ann. Stat. 25 (1997), pp. 929–947.] proposed a variable bandwidth selection for kernel estimators that achieved optimal rates over Besov classes. However, a second challenge in many real applications of non-parametric curve estimation is that the function must be positive. Here, we show how to construct estimators under positivity constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class.     

Random coefficients regression models. a review 1

A review is given of random regression coefficients models. The emphasis is put on the problem of estimating the mean regression coefficients and the covariance matrix of the coefficients. Prediction of the individual random coefficients is not discussed. The main purpose of the review is to point to the practical aspects of the models and the problem of statistical inference in finite samples. Some problems for future research are indicated.     

A note on ∞manque in variance covariance components models

Stimualted by C.R. Rao's MINQUE J. Focke and G. Dewess introduced the so called r-and ∞ MINQUE. Although they developed a unique charecterization of ∞-MINQUE, they did not give explicite formulas for its computation. The goal this paper is to close this lack and to etend the concept to more general models.     

Bivariate exponential-type distributions with linear regression

Let Xbe a bivariate exponential-type random vector (BIDLIKAR, PATIL (1968)), than it is proved:

1. If P(X ≥0) = 1 is valid, then Xhas linear regression to both directions if and only if Xpossesses a symmetric Γ-distribution.

2. Xpossesses linear regression to both directions with constant regression coefficients (independent of the parameter vector ? of the exponential-type distribution (BIDLIKAR, PATIL (1968)) if and only if Xis normal distributed.     

Strong consistency of least squares estimates with i.i.d. errors with mean values not necessarily defined

We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r<2, and null mean value when r>1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.     

Variance bounds for estimators in autoregressive models with constraints

We consider nonlinear and heteroscedastic autoregressive models whose residuals are martingale increments with conditional distributions that fulfil certain constraints. We treat two classes of constraints: residuals depending on the past through some function of the past observations only, and residuals that are invariant under some finite group of transformations. We determine the efficient influence function for estimators of the autoregressive parameter in such models, calculate variance bounds, discuss information gains, and suggest how to construct efficient estimators. Without constraints, efficient estimators can be given by weighted least squares estimators. With the constraints considered here, efficient estimators are obtained differently, as one-step improvements of some initial estimator, similarly as in autoregressive models with independent increments.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.