Smoothing for discrete-valued time series

We deal with smoothed estimators for conditional probability functions of discrete-valued time series { Y_t } under two different settings. When the conditional distribution of Y_t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya–Watson estimator stands out as it shares the advantages of both Nadaraya–Watson and local linear regression estimators. The asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics, which shows that the smoothed methods proposed outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to this assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994–1995. An ad hoc model selection procedure based on a local Akaike information criterion is proposed to select the significant pollutant indices.     

Third-Order Asymptotic Distributions of Classic Test Statistics for One-Parameter Exponential Family Models

ABSTRACT

In this article we derive third-order asymptotic expansions for the non null distribution functions of four classic statistics under a sequence of local alternatives in one-parameter exponential family models. Our results are quite general and cover a wide range of important distributions.     

A semiparametric maximum likelihood ratio test for the change point in copula models

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. We also discuss the Gaussian-type approximations for the semiparametric likelihood ratio. The asymptotic distribution of the proposed statistic under specified alternatives is shown to be normal, and an approximation to the power function is given. Simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the double exponential and Gaussian-type approximations.     

On the Distribution of Likelihood Ratio Test Statistics for Cointegration Rank

This paper analyses the likelihood ratio test for the hypothesis of reduced cointegration rank in a Gaussian vector autoregressive model. The usual asymptotic distribution typically gives rather large size distortions. This is explained by the fact that the asymptotic distribution of the likelihood ratio test statistic varies across the parameter space. A much improved distribution approximation can be obtained using local asymptotic theory. The idea is discussed for some low dimensional examples.     

Bayesian inference with misspecified models

This article reviews Bayesian inference from the perspective that the designated model is misspecified. This misspecification has implications in interpretation of objects, such as the prior distribution, which has been the cause of recent questioning of the appropriateness of Bayesian inference in this scenario. The main focus of this article is to establish the suitability of applying the Bayes update to a misspecified model, and relies on representation theorems for sequences of symmetric distributions; the identification of parameter values of interest; and the construction of sequences of distributions which act as the guesses as to where the next observation is coming from. A conclusion is that a clear identification of the fundamental starting point for the Bayesian is described.     

SUFFICIENT CONDITIONS FOR A GEOMETRIC TAIL IN A QBD PROCESS WITH MANY COUNTABLE LEVELS AND PHASES

Abstract

In this paper, we present sufficient conditions, under which the stationary probability vector of a QBD process with both infinite levels and phases decays geometrically, characterized by the convergence norm η and the 1/η-left-invariant vector x of the rate matrix R. We also present a method to compute η and x based on spectral properties of the censored matrix of a matrix function constructed with the repeating blocks of the transition matrix of the QBD process. What makes this method attractive is its simplicity; finding η reduces to determining the zeros of a polynomial. We demonstrate the application of our method through a few interesting examples.     

Second order tail behaviour of randomly weighted heavy-tailed sums and their maxima

Abstract

In this paper the second order asymptotics of the tail probabilities of randomly weighted sums and their maxima are established in the case that the underlying primary random variables are subexponential. No any assumption is made on the dependence structure between the random weights, but we assume these weights are bounded away from zero and infinity.     

Is statistics too difficult?

By means of several historical examples, it is shown that it does not appear to be easy to build bridges between rigorous mathematics and reasonable data-analytic procedures for scientific measurements. After mentioning both some positive and some negative aspects of statistics, a formal framework for statistics is presented which contains the concept formation, derivation of results and interpretation of mathematical statistics as three essential steps. The difficulties especially of interpretation are shown for examples in several areas of statistics, such as asymptotics and robustness. Some problems of statistics in two subject-matter sciences are discussed, and a summary and outlook are given.     

Asymptotic Distributions of Quasi‐Maximum Likelihood Estimators for Spatial Autoregressive Models

This paper investigates asymptotic properties of the maximum likelihood estimator and the quasi‐maximum likelihood estimator for the spatial autoregressive model. The rates of convergence of those estimators may depend on some general features of the spatial weights matrix of the model. It is important to make the distinction with different spatial scenarios. Under the scenario that each unit will be influenced by only a few neighboring units, the estimators may have ‐rate of convergence and be asymptotically normal. When each unit can be influenced by many neighbors, irregularity of the information matrix may occur and various components of the estimators may have different rates of convergence.