Exact Slopes of Test Statistics for the Multivariate Exponential Family

The objective of this paper is to investigate exact slopes of test statistics { T_n } when the random vectors X ₁, ..., X_n are distributed according to an unknown member of an exponential family { P _θ; θ∈Ω. Here Ω is a parameter set. We will be concerned with the hypothesis testing problem of H ₀θ∈Ω₀ vs H ₁: θ∉Ω₀ where Ω₀ is a subset of Ω. It will be shown that for an important class of problems and test statistics the exact slope of { T_n } at η in Ω−Ω₀ is determined by the shortest Kullback–Leibler distance from {θ: T_n (λ(θ)) = T_n (λ(π))} to Ω₀, λ_θ = E _θ)( X ).     

IMPROVING UPON THE BEST INVARIANT ESTIMATOR IN MULTIVARIATE LOCATION PROBLEMS

We are concerned with estimators which improve upon the best invariant estimator, in estimating a location parameter θ. If the loss function is L(θ - a) with L convex, we give sufficient conditions for the inadmissibility of δ₀(X) = X. If the loss is a weighted sum of squared errors, we find various classes of estimators δ which are better than δ₀. In general, δ is the convolution of δ₁ (an estimator which improves upon δ₀ outside of a compact set) with a suitable probability density in R^p. The critical dimension of inadmissibility depends on the estimator δ₁ We also give several examples of estimators δ obtained in this way and state some open problems.     

Empirical Likelihood for Non-Smooth Criterion Functions

Abstract.  Suppose that X ₁ ,…,  X _n is a sequence of independent random vectors, identically distributed as a d -dimensional random vector X . Let     be a parameter of interest and     be some nuisance parameter. The unknown, true parameters ( μ ₀ , ν ₀ ) are uniquely determined by the system of equations E { g ( X , μ ₀ , ν ₀ )} =   0 , where g  =  ( g ₁ ,…, g _{p + q} ) is a vector of p + q functions. In this paper we develop an empirical likelihood (EL) method to do inference for the parameter μ ₀ . The results in this paper are valid under very mild conditions on the vector of criterion functions g . In particular, we do not require that g ₁ ,…, g _{p + q} are smooth in μ or ν . This offers the advantage that the criterion function may involve indicators, which are encountered when considering, e.g. differences of quantiles, copulas, ROC curves, to mention just a few examples. We prove the asymptotic limit of the empirical log-likelihood ratio, and carry out a small simulation study to test the performance of the proposed EL method for small samples.     

Likelihood Ratio Tests Under Local and Fixed Alternatives in Monotone Function Problems

Abstract.  We focus on a class of non-standard problems involving non-parametric estimation of a monotone function that is characterized by n ^1/3 rate of convergence of the maximum likelihood estimator, non-Gaussian limit distributions and the non-existence of     -regular estimators. We have shown elsewhere that under a null hypothesis of the type ψ ( z ₀) =  θ ₀ ( ψ being the monotone function of interest) in non-standard problems of the above kind, the likelihood ratio statistic has a 'universal' limit distribution that is free of the underlying parameters in the model. In this paper, we illustrate its limiting behaviour under local alternatives of the form ψ _n ( z ), where ψ _n (·) and ψ (·) vary in O ( n ^−1/3) neighbourhoods around z ₀ and ψ _n converges to ψ at rate n ^1/3 in an appropriate metric. Apart from local alternatives, we also consider the behaviour of the likelihood ratio statistic under fixed alternatives and establish the convergence in probability of an appropriately scaled version of the same to a constant involving a Kullback–Leibler distance.     

Rejoinder to 'Ahmed, M.S. (1998). A note on regression-type estimators using multiple auxiliary information.'

In the estimators t ₃ , t ₄ , t ₅ of Mukerjee, Rao & Vijayan (1987), b _{y x} and b _{y z} are partial regression coefficients of y on x and z , respectively, based on the smaller sample. With the above interpretation of b _{y x} and b _{y z} in t ₃ , t ₄ , t ₅ , all the calculations in Mukerjee at al. (1987) are correct. In this connection, we also wish to make it explicit that b _{x z} in t ₅ is an ordinary and not a partial regression coefficient. The 'corrected' MSEs of t ₃ , t ₄ , t ₅ , as given in Ahmed (1998 Section 3) are computed assuming that our b _{y x} and b _{y z} are ordinary and not partial regression coefficients. Indeed, we had no intention of giving estimators using the corresponding ordinary regression coefficients which would lead to estimators inferior to those given by Kiregyera (1984). We accept responsibility for any notational confusion created by us and express regret to readers who have been confused by our notation. Finally, in consideration of the above, it may be noted that Tripathi & Ahmed's (1995) estimator t ₀ , quoted also in Ahmed (1998), is no better than t ₅ of Mukerjee at al. (1987).     

Goodness-of-fit Tests for Semi-Markov and Markov Survival Models with One Intermediate State

Survival data with one intermediate state are described by semi-Markov and Markov models for counting processes whose intensities are defined in terms of two stopping times T ₁< T ₂. Problems of goodness-of-fit for these models are studied. The test statistics are proposed by comparing Nelson–Aalen estimators for data stratified according to T ₁. Asymptotic distributions of these statistics are established in terms of the weak convergence of some random fields. Asymptotic consistency of these test statistics is also established. Simulation studies are included to indicate their numerical performance.     

An Analysis of Swendsen–Wang and Related Sampling Methods

Convergence rates, statistical efficiency and sampling costs are studied for the original and extended Swendsen–Wang methods of generating a sample path { S _j , j ≥1} with equilibrium distribution π , with r distinct elements, on a finite state space X of size N ₁. Given S _{j -1}, each method uses auxiliary random variables to identify the subset of X from which S _j is to be randomly sampled. Let π_min and π_max denote respectively the smallest and largest elements in π and let N_r denote the number of elements in π with value π_max. For a single auxiliary variable, uniform sampling from the subset and ( N ₁− N_r )π_min+ N_r π_max≈1, our results show rapid convergence and high statistical efficiency for large π_min/π_max or N_r / N ₁ and slow convergence and poor statistical efficiency for small π_min/π_max and N_r / N₁ . Other examples provide additional insight. For extended Swendsen–Wang methods with non-uniform subset sampling, the analysis identifies the properties of a decomposition of π( x ) that favour fast convergence and high statistical efficiency. In the absence of exploitable special structure, subset sampling can be costly regardless of which of these methods is employed.     

L p –Estimators as Estimates of a Parameter of Location for a Sharp–pointed Symmetric Density

We study the asymptotics of L _p estimators, p > 0, over a sample having a symmetric density with a sharp–point at the centre of symmetry of the distribution. The rates of convergence of the L _p estimators in this situation depend on p and on the shape of the density. To obtain some of the limit distributions, we present new results in the asymptotics of M–estimators. We extend the delta method to the case when the Euclidean norm of the conveniently normalized M–estimators converge to a power of the Euclidean norm of a (possibly Gaussian) stable distribution.     

Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models

We use Owen's (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z ₁, ..., z _n which is observed directly from a distribution F ( z ). The other one is x ₁, ..., x _m which is observed indirectly from F ( z ), where the x _i s have density ∫ p ( x | z ) dF ( z ) and p ( x | z ) is a conditional density function. We are interested in testing H ₀: p ( x | z ) = p ( x | z ; θ ), for some specified smooth density function. A semiparametric likelihood ratio based statistic is proposed and it is shown that it converges to a chi-squared distribution. This is a simple method for doing goodness of fit tests, especially when x is a discrete variable with finitely many values. In addition, we discuss estimation of θ and F ( z ) when H ₀ is true. The connection between upgraded mixture models and general estimating equations is pointed out.     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

Detecting changes in the mean of functional observations

Summary.  Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as X _i ( t )= μ ( t )+Σ_{1≤ l <∞} η _{i ,  l} +  v _l ( t ), where μ is the common mean, v _l are the eigenfunctions of the covariance operator and the η _{i ,  l} are the scores. Inferential procedures assume that the mean function μ ( t ) is the same for all values of i . If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England.     

On the effect of overdispersion on exact conditional tests

Let Y ₁, . . ., Y_n denote independent random variables such that Y_j has a one-parameter exponential family distribution with canonical parameter θ _j =λ+ψ X_j ; here X ₁, . . ., X_n are known constants. Consider a test of the null hypothesis ψ=0. Under the null hypothesis, A =Σ Y_j is sufficient for λ and, hence, a test of ψ=0 may be based on the conditional distribution of T =Σ X_j Y_j given A , which is independent of λ. In this paper, the effects of overdispersion due to a mixture model on the conditional distribution of T given A are considered.     

On Statistical Models for d-Dimensional Stable Processes, and Some Generalizations

In statistical models where jumps of a d -dimensional stable process ( S _t ) _{t ≥0} are observed in windows with certain asymptotic properties, and where parameters appearing in the Levy measure of S are to be estimated, we have asymptotically efficient estimators. If Poisson random measure μ on (0, ∞) × ( R ^d \{0}) with intensity dt Λ( dx ) replaces the jump measure of S , where Λ is a ε-finite measure on R ^d \{0} admitting tail parameters in a suitable sense, we specify a notion of neighbourhood which allows to treat efficiency in statistical experiments of the second type by switching to accompanying sequences of the stable process type considered first.     

Non-parametric Regression with Dependent Censored Data

Abstract.  Let ( X _i , Y _i ) ( i = 1 ,…, n ) be n replications of a random vector ( X , Y  ), where Y is supposed to be subject to random right censoring. The data ( X _i , Y _i ) are assumed to come from a stationary α -mixing process. We consider the problem of estimating the function m ( x ) = E ( φ ( Y ) |  X = x ), for some known transformation φ . This problem is approached in the following way: first, we introduce a transformed variable     , that is not subject to censoring and satisfies the relation     , and then we estimate m ( x ) by applying local linear regression techniques. As a by-product, we obtain a general result on the uniform rate of convergence of kernel type estimators of functionals of an unknown distribution function, under strong mixing assumptions.     

On Minimum Distance Estimation Using Kolmogorov-Lévy Type Metrics

Let X ₁, . . ., X_n be independent identically distributed random variables with a common continuous (cumulative) distribution function (d.f.) F , and F^_n the empirical d.f. (e.d.f.) based on X ₁, . . ., X_n . Let G be a smooth d.f. and Gθ = G (·–θ) its translation through θ∈ R . Using a Kolmogorov-Lévy type metric ρ_α defined on the space of d.f.s. on R , the paper derives both null and non-null limiting distributions of √ n [ ρ_α ( F_n , Gθ_n ) – ρ_α ( F, G_θ )], √ n (θ _n –θ) and √ nρ_α ( Gθ , Gθ ), where θ _n and θ are the minimum ρ_α -distance parameters for F_n and F from G , respectively. These distributions are known explicitly in important particular cases; with some complementary Monte Carlo simulations, they help us clarify our understanding of estimation using minimum distance methods and supremum type metrics. We advocate use of the minimum distance method with supremum type metrics in cases of non-null models. The resulting functionals are Hadamard differentiable and efficient. For small scale parameters the minimum distance functionals are close to medians of the parent distributions. The optimal small scale models result in minimum distance estimators having asymptotic variances very competitive and comparable with best known robust estimators.     

A NOTE ON THE MULTIVARIATE LINEAR MODEL WITH CONSTRAINTS ON THE DEPENDENT VECTOR

It is shown that the least squares estimators of B and Σ in the multivariate linear model {E Y _i= X ₁ B , D ( Y _i) =Σ, 1 ≤ i ≤ n , Y ₁ Y _n uncorrelated} subject to the constraints Y _i M = X _i N are just the usual least squares estimators = ( X'X )-¹ X'Y and ΣC = 1/n( Y-X )( Y-X ) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Y _i has a multivariate normal distribution, and examples of the applicability of the model reviewed.     

Penalized Projection Estimator for Volatility Density

Abstract.  In this paper, we consider a stochastic volatility model ( Y _t , V _t ), where the volatility (V _t ) is a positive stationary Markov process. We assume that ( ln V _t ) admits a stationary density f that we want to estimate. Only the price process Y _t is observed at n discrete times with regular sampling interval Δ . We propose a non-parametric estimator for f obtained by a penalized projection method. Under mixing assumptions on ( V _t ), we derive bounds for the quadratic risk of the estimator. Assuming that Δ=Δ _n tends to 0 while the number of observations and the length of the observation time tend to infinity, we discuss the rate of convergence of the risk. Examples of models included in this framework are given.     

The Cusum of Squares Test for Scale Changes in Infinite Order Moving Average Processes

In this paper we consider the problem of testing for a scale change in the infinite order moving average process X _j = _{i =0} a _{i j i} , where _j are i.i.d. r.v.s with E ₁ < for some > 0. In performing the test, a cusum of squares test statistic analogous to Inclan & Tiao's (1994) statistic is considered. It is well-known from the literature that outliers affect test procedures leading to false conclusions. In order to remedy this, a cusum of squares test based on trimmed observations is considered. It is demonstrated that this test is robust against outliers, is valid for infinite variance processes as well. Simulation results are given for illustration.     

Consistency of the Semi-parametric MLE in Linear Regression Models with Interval-censored Data

Abstract.  Consider the model Y = β ^' X + ε . Let F ₀ be the unknown cumulative distribution function of the random variable ε . Consistency of the semi-parametric Maximum likelihood estimator of ( β , F ₀), denoted by     , has not been established under any interval censorship (IC) model. We prove in this paper that     is consistent under the mixed case IC model and some mild assumptions.     

Bayesian model selection for join point regression with application to age-adjusted cancer rates

Summary.  The method of Bayesian model selection for join point regression models is developed. Given a set of K +1 join point models M ₀,  M ₁, …,  M _K with 0, 1, …,  K join points respec-tively, the posterior distributions of the parameters and competing models M _k are computed by Markov chain Monte Carlo simulations. The Bayes information criterion BIC is used to select the model M _k with the smallest value of BIC as the best model. Another approach based on the Bayes factor selects the model M _k with the largest posterior probability as the best model when the prior distribution of M _k is discrete uniform. Both methods are applied to analyse the observed US cancer incidence rates for some selected cancer sites. The graphs of the join point models fitted to the data are produced by using the methods proposed and compared with the method of Kim and co-workers that is based on a series of permutation tests. The analyses show that the Bayes factor is sensitive to the prior specification of the variance σ ², and that the model which is selected by BIC fits the data as well as the model that is selected by the permutation test and has the advantage of producing the posterior distribution for the join points. The Bayesian join point model and model selection method that are presented here will be integrated in the National Cancer Institute's join point software ( http://www.srab.cancer.gov/joinpoint/ ) and will be available to the public.