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.     

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.     

ε-Repetitions of the Maximum Residuals in an AR(1) Model

This paper considers record values of residuals or prediction errors in a one-parameter autoregressive process and the statistic Z _n = number of ε -repetitions of this record. When the parameter of the autoregression is unknown, the prediction errors, and therefore Z _n , are unobservable. Here an observable analogue ̂ n of Z _n is considered. It is proved that under special conditions the difference Z _n − unobservable. Here an observable analogue ̂ n converges to zero in probability and therefore that unobservable. Here an observable analogue ̂ n has the same asymptotic behaviour as Z _n .     

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.     

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.     

Ordering of Sequentially Sampled Exponential Experiments

Let X ₁, X ₂, ... be a sequence of i.i.d. random variables, X _i∼ F _θ, θ∈Θ. Let N ₁ and N ₂ be two stopping rules. For a class of exponential families { F _θ: θ∈Θ} we show that the experiment Y ₁ = ( X ₁, ..., X _N1) carries more statistical information than Y ₂ = ( X ₁, ..., x _N2) only if N ₁ is stochastically larger then N 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.     

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.     

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 ).     

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.     

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.     

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 smooth monotone functions

Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D  ² f  = w Df , where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C ₀ + C ₁  D ⁻¹{exp( D ⁻¹ w )}, where C ₀ and C ₁ are arbitrary constants and D ⁻¹ is the partial integration operator. A basis for expanding w is suggested that permits explicit integration in the expression of f . In fitting data, it is also useful to regularize f by penalizing the integral of w ² since this is a measure of the relative curvature in f . Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.     

Estimating the Upper Support Point in Deconvolution

Abstract.  We consider estimation of the upper boundary point F ⁻¹ (1) of a distribution function F with finite upper boundary or 'frontier' in deconvolution problems, primarily focusing on deconvolution models where the noise density is decreasing on the positive halfline. Our estimates are based on the (non-parametric) maximum likelihood estimator (MLE) of F . We show that (1) is asymptotically never too small. If the convolution kernel has bounded support the estimator (1) can generally be expected to be consistent. In this case, we establish a relation between the extreme value index of F and the rate of convergence of (1) to the upper support point for the 'boxcar' deconvolution model. If the convolution density has unbounded support, (1) can be expected to overestimate the upper support point. We define consistent estimators , for appropriately chosen vanishing sequences ( β _n ) and study these in a particular case.     

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.     

Nonparametric multistep-ahead prediction in time series analysis

Summary.  We consider the problem of multistep-ahead prediction in time series analysis by using nonparametric smoothing techniques. Forecasting is always one of the main objectives in time series analysis. Research has shown that non-linear time series models have certain advantages in multistep-ahead forecasting. Traditionally, nonparametric k -step-ahead least squares prediction for non-linear autoregressive AR( d ) models is done by estimating E ( X _{t + k}  | X _t , …,  X _{t − d +1}) via nonparametric smoothing of X _{t + k} on ( X _t , …,  X _{t − d +1}) directly. We propose a multistage nonparametric predictor. We show that the new predictor has smaller asymptotic mean-squared error than the direct smoother, though the convergence rate is the same. Hence, the predictor proposed is more efficient. Some simulation results, advice for practical bandwidth selection and a real data example are provided.     

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.     

Computation of the Generalized F Distribution

Exact expressions for the cumulative distribution function of a random variable of the form ( α ₁ X ₁+ α ₂ X ₂)/ Y are given where X ₁, X ₂ and Y are independent chi-squared random variables. The expressions are applied to the detection of joint outliers and Hotelling's mis-specified T ² distribution.     

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).