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## A Cautionary Note on Likelihood Ratio Tests in Mixture Models

### Annals of the Institute of Statistical Mathematics (2000-09-01) 52: 481-487 , September 01, 2000

We show that iterative methods for maximizing the likelihood in a mixture of exponentials model depend strongly on their particular implementation. Different starting strategies and stopping rules yield completely different estimators of the parameters. This is demonstrated for the likelihood ratio test of homogeneity against two-component exponential mixtures, when the test statistic is calculated by the EM algorithm.

## Front Matter - Stochastic Population Models

### Stochastic Population Models (2000-01-01): 145 , January 01, 2000

## Joint Distribution of Rises and Falls

### Annals of the Institute of Statistical Mathematics (2000-09-01) 52: 415-425 , September 01, 2000

The marginal distributions of the number of rises and the number of falls have been used successfully in various areas of statistics, especially in non-parametric statistical inference. Carlitz (1972, *Duke Math. J.**39*, 268–269) showed that the generating function of the joint distribution for the numbers of rises and falls satisfies certain complex combinatorial equations, and pointed out that he had been unable to derive the explicit formula for the joint distribution from these equations. After more than two decades, this latter problem remains unsolved. In this article, the joint distribution is obtained via the probabilistic method of finite Markov chain imbedding for random permutations. A numerical example is provided to illustrate the theoretical results and the corresponding computational procedures.

## Asymptotic Theory of Estimation and Testing for Stochastic Processes

### Asymptotic Theory of Statistical Inference for Time Series (2000-01-01): 51-165 , January 01, 2000

In classical time series analysis the asymptotic estimation and testing theory was developed for linear processes, which include the AR, MA, and ARMA models. However, in the last twenty years a lot of more complicated stochastic process models have been introduced, such as, nonlinear time series models, diffusion processes, point processes, and nonergodic processes. This chapter is devoted to providing a modern asymptotic estimation and testing theory for those various stochastic process models. The approach is mainly based on the LAN results given in the previous chapter. More concretely, in Section 3.1 we discuss the asymptotic estimation and testing theory for non-Gaussian vector linear processes in view of LAN. The results are very general and grasp a lot of other works dealing with AR, MA, and ARMA models as special cases. Section 3.2 reviews some elements of nonlinear time series models and the asymptotic estimation theory based on the conditional least squares estimator and maximum likelihood estimator (MLE). We address the problem of statistical model selection in general fashion. Also the asymptotic theory for nonergodic models is mentioned. Recently much attention has been paid to continuous time processes (especially diffusion processes), which appear in finance. Hence, in Section 3.3 we describe the foundation of stochastic integrals and diffusion processes. Then the LAN-based asymptotic theory of estimation for them is studied.

## Time Series

### XploRe — Learning Guide (2000-01-01): 247-271 , January 01, 2000

The purpose of this chapter is to show how XploRe may be used by practitioners for analyzing observed time series. Some of the time series tools are standard in the literature. The more elaborated nonlinearity tests based on artificial neural networks are implemented for the nonadvanced use.

## Stochastic Convergence

### Probability via Expectation (2000-01-01): 282-289 , January 01, 2000

Probability theory is founded on an empirical limit concept, and its most characteristic conclusions take the form of limit theorems. Thus, a sequence of r.v.s. 282-1 which one suspects has some kind of limit property for large *n* is a familiar object. For example, the convergence of the sample average *X*_{n} to a common expected value *E*(*X*) (in mean square, Exercise 2.8.6; in probability, Exercise 2.9.14 or in distribution, Section 7.3) has been a recurrent theme. Other unforced examples are provided by the convergence of estimates or conditional expectations with increasing size of the observation set upon which they are based (Chapter 14), and the convergence of the standardize sum *u*_{n} to normality (Section 7.4). Any infinite sum of r.v.s which we encounter should be construed as a limit, in some sense, of a finite sum. Consider, for instance, the sum 282-2 of Section 6.1, or the formal solution 282-3 of the stochastic difference equation 282-4.

## Robust Bayesian Analysis

### Robust Bayesian Analysis (2000-01-01): 152 , January 01, 2000

## Fieller's Interval and the Bootstrap-Fieller Interval for the Incremental Cost-Effectiveness Ratio

### Health Services and Outcomes Research Methodology (2000-12-01) 1: 291-303 , December 01, 2000

## Hierarchical priors for Bayesian CART shrinkage

### Statistics and Computing (2000-01-01) 10: 17-24 , January 01, 2000

The Bayesian CART (classification and regression tree) approach proposed by Chipman, George and McCulloch (1998) entails putting a prior distribution on the set of all CART models and then using stochastic search to select a model. The main thrust of this paper is to propose a new class of hierarchical priors which enhance the potential of this Bayesian approach. These priors indicate a preference for smooth local mean structure, resulting in tree models which shrink predictions from adjacent terminal node towards each other. Past methods for tree shrinkage have searched for trees without shrinking, and applied shrinkage to the identified tree only after the search. By using hierarchical priors in the stochastic search, the proposed method searches for shrunk trees that fit well and improves the tree through shrinkage of predictions.

## Correspondence Analysis

### XploRe® — Application Guide (2000-01-01): 339-358 , January 01, 2000

*Correspondence analysis* (CA) is a descriptive method which allows us to analyze and to XploRe the structure of contingency tables (or, by extension, non-negative tables where rows and columns are the entities of interest). It is similar to *principal component analysis* (PCA) in the sense that it attempts to obtain a representation of either the *I* row items or the *J* column items in a low dimensional space, while preserving at best total variation in the table.