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.     

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 ).     

A Non-parametric Test for Generalized First-order Autoregressive Models

We derive a non-parametric test for testing the presence of V(X_i,ε_i) in the non-parametric first-order autoregressive model X_i+1=T(X_i)+V(X_i,ε_i)+U(X_i)ε_i+1, where the function T(x) is assumed known. The test is constructed as a functional of a basic process for which we establish a weak invariance principle, under the null hypothesis and under stationarity and mixing assumptions. Bounds for the local and non-local powers are provided under a condition which ensures that the power tends to one as the sample size tends to infinity.The testing procedure can be applied, e.g. to bilinear models, ARCH models, EXPAR models and to some other uncommon models. Our results confirm the robustness of the test constructed in Ngatchou Wandji (1995) and in Diebolt & Ngatchou Wandji (1995).     

Characterizations of Competing Risks in Terms of Independent-Risks Proxy Models

In competing risks a failure time T and a cause C , one of p possible, are observed. A traditional representation is via a vector ( T ₁, ..., T_p ) of latent failure times such that T = min( T ₁, ..., T_p ); C is defined by T = T_C in the basic situation of failure from a single cause. There are several results in the literature to the effect that a joint distribution for ( T ₁, ..., T_p ), in which the T_j are independent, can always be constructed to yield any given bivariate distribution for ( C , T ). For this reason the prevailing wisdom is that independence cannot be assessed from competing risks data, not even with arbitrarily large sample sizes (e.g. Prentice et al. , 1978). A result was given by Crowder (1996) which shows that, under certain circumstances, independence can be assessed. The various results will be drawn together and a complete characterization can now be given in terms of independent-risks proxy models.     

A Problem of Dimensionality in Normal Mixture Analysis

Suppose the p -variate random vector W , partitioned into q variables W₁ and p - q variables W₂, follows a multivariate normal mixture distribution. If the investigator is mainly interested in estimation of the parameters of the distribution of W₁, there are two possibilities: (1) use only the data on W₁ for estimation, and (2) estimate the parameters of the p -variate mixture distribution, and then extract the estimates of the marginal distribution of W₁. In this article we study the choice between these two possibilities mainly for the case of two mixture components with identical covariance matrices. We find the asymptotic distribution of the linear discriminant function coefficients using the work of Efron (1975 ) and O'Neill (1978 ), and give a Wald–test for redundancy of W₂. A simulation study gives further insights into conditions under which W₂ should be used in the analysis: in summary, the inclusion of W₂ seems justified if Δ 2.1, the Mahalanobis distance between the two component distributions based on the conditional distribution of W₂ given W₁, is at least 2.     

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.     

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.     

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.     

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.     

Estimating the proportion of true null hypotheses, with application to DNA microarray data

Summary.  We consider the problem of estimating the proportion of true null hypotheses, π ₀, in a multiple-hypothesis set-up. The tests are based on observed p -values. We first review published estimators based on the estimator that was suggested by Schweder and Spjøtvoll. Then we derive new estimators based on nonparametric maximum likelihood estimation of the p -value density, restricting to decreasing and convex decreasing densities. The estimators of π ₀ are all derived under the assumption of independent test statistics. Their performance under dependence is investigated in a simulation study. We find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence.     

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.     

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.     

Goodness of Fit via Non-parametric Likelihood Ratios

Abstract.  To test if a density f is equal to a specified f ₀, one knows by the Neyman–Pearson lemma the form of the optimal test at a specified alternative f ₁. Any non-parametric density estimation scheme allows an estimate of f . This leads to estimated likelihood ratios. Properties are studied of tests which for the density estimation ingredient use log-linear expansions. Such expansions are either coupled with subset selectors like the Akaike information criterion and the Bayesian information criterion regimes, or use order growing with sample size. Our tests are generalized to testing the adequacy of general parametric models, and to work also in higher dimensions. The tests are related to, but are different from, the 'smooth tests' that go back to Neyman [Skandinavisk Aktuarietidsskrift 20(1937) 149] and that have been studied extensively in recent literature. Our tests are large-sample equivalent to such smooth tests under local alternative conditions, but different from the smooth tests and often better under non-local conditions.     

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.     

Effectiveness of potent antiretroviral therapy on progression of human immunodeficiency virus: Bayesian modelling and model checking via counterfactual replicates

Summary.  We analyse data from a seroincident cohort of 457 homosexual men who were infected with the human immunodeficiency virus, followed within the multicentre Italian Seroconversion Study. These data include onset times to acquired immune deficiency syndrome (AIDS), longitudinal measurements of CD4⁺ T-cell counts taken on each subject during the AIDS-free period of observation and the period of administration of a highly active antiretro- viral therapy (HAART), for the subset of individuals who received it. The aim of the study is to assess the effect of HAART on the course of the disease. We analyse the data by a Bayesian model in which the sequence of longitudinal CD4⁺ cell count observations and the associated time to AIDS are jointly modelled at an individual subject's level as depending on the treatment. We discuss the inferences obtained about the efficacy of HAART, as well as modelling and computation difficulties that were encountered in the analysis. These latter motivate a model criticism stage of the analysis, in which the model specification of CD4⁺ cell count progression and of the effect of treatment are checked. Our approach to model criticism is based on the notion of a counterfactual replicate data set Z ^c . This is a data set with the same shape and size as the observed data, which we might have observed by rerunning the study in exactly the same conditions as the actual study if the treated patients had not been treated at all. We draw samples of Z ^c from a null model M ₀, which assumes absence of treatment effect, conditioning on data collected in each subject before initiation of treatment. Model checking is performed by comparing the observed data with a set of samples of Z ^c drawn from M ₀.     

Stepwise likelihood ratio statistics in sequential studies

Summary.  It is well known that in a sequential study the probability that the likelihood ratio for a simple alternative hypothesis H ₁ versus a simple null hypothesis H ₀ will ever be greater than a positive constant c will not exceed 1/ c under H ₀. However, for a composite alternative hypothesis, this bound of 1/ c will no longer hold when a generalized likelihood ratio statistic is used. We consider a stepwise likelihood ratio statistic which, for each new observation, is updated by cumulatively multiplying the ratio of the conditional likelihoods for the composite alternative hypothesis evaluated at an estimate of the parameter obtained from the preceding observations versus the simple null hypothesis. We show that, under the null hypothesis, the probability that this stepwise likelihood ratio will ever be greater than c will not exceed 1/ c . In contrast, under the composite alternative hypothesis, this ratio will generally converge in probability to ∞. These results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis. For illustration, we conduct two simulation studies, one for a normal response and one for an exponential response, to compare the performance of a sequential test based on a stepwise likelihood ratio statistic with a constant boundary versus some existing approaches.     

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.     

Moderate Deviations for Bayes Posteriors

Let ( X_k ) k be a sequence of i.i.d. random variables taking values in a set , and consider the problem of estimating the law of X₁ in a Bayesian framework. We prove, under mild conditions on the prior, that the sequence of posterior distributions satisfies a moderate deviation principle.     

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.