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.     

The NPMLE for Doubly Censored Current Status Data

In biostatistical applications interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, but with its distribution over an observed interval ' A, B ' known to be uniformly distributed; the data is referred to as doubly censored current status data. These authors used this model to handle application in AIDS partner studies focusing on the NPMLE of the distribution G of T . The model is a submodel of the current status model, but the distribution G is essentially the derivative of the distribution of interest F in the current status model. In this paper we establish that the NPMLE of G is uniformly consistent and that the resulting estimators for the n ^1/2-estimable parameters are efficient. We propose an iterative weighted pool-adjacent-violator-algorithm to compute the estimator. It is also shown that, without smoothness assumptions, the NPMLE of F converges at rate n ^−2/5 in L ²-norm while the NPMLE of F in the non-parametric current status data model converges at rate n ^−1/3 in L ²-norm, which shows that there is a substantial gain in using the submodel information.     

MINIMAX ESTIMATION OF A MULTIVARIATE NORMAL MEAN UNDER A CONVEX LOSS FUNCTION

Let X = (X₁, - X_p)prime; ˜ N_p (μ, Σ) where μ= (μ₁, -, μ_p)' and Σ= diag (Σ²₁, -, Σ²_p) are both unknown and p3. Let (n_i - 2) w_i/Σ²_i! X²_ni, independent. of w_i (I ≠ j = 1, -, p). Assume that (w₁, -, w_p) and X are independent. Define W = diag (w₁, -, w_p) and ¶ X ¶²_w= X'W^-1Q^-1W^-1X where Q = diag (q₁, -,n q_p), q_i > 0, i = 1, -, p. In this paper, the minimax estimator of Berger & Bock (1976), given by δ (X, W) = [I_p - r(X, W) ¶ X ¶^-2_w Q^-1W^-1] X, is shown to be minimax relative to the convex loss (δ - μ)'[αQ + (1 - α) Σ^-1] δ - μ)/C, where C =α tr (Σ) + (1 - α)p and 0 α 1, under certain conditions on r(X, W). This generalizes the above mentioned result of Berger & Bock.     

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.     

Some Bounds on the Distribution of Certain Quadratic Forms in Normal Random Variables

The paper derives bounds on the distribution of the quadratic forms Z = y ^H( X Γ X ^H)⁻¹ y and W = y ^H(σ² I + X Γ X ^H)⁻¹ y , where the elements of the M × 1 vector y and the M × N matrix X are independent identically distributed (i.i.d.) complex zero mean Normal variables, Γ is some N × N diagonal matrix with positive diagonal elements, I , is the identity, σ² is a constant and ^H denotes the Hermitian transpose. The bounds are convenient for numerical work and appear to be tight for small values of M . This work has applications in digital mobile radio for a specific channel where M antennas are used to receive a signal with N interferers. Some of these applications in radio communication systems are discussed.     

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.     

Testing a Regression Model When We Have Smooth Alternatives in Mind

Goodness-of-fit tests based on residual sums of squares are standard procedures used when fitting regression models. Often we have a smooth alternative in mind, a qualitative feature that the χ²-test does not take into account. We show that the power of detecting a smooth alternative increases when we smooth the current model as well. The proposed test is shown to be able to detect any continuous local alternative tending to zero slower than n ^−1/2. Theoretical results also address minimax non-parametric hypothesis testing in Sobolev spaces. A simulation study is presented, and the procedure is applied to expenditure curve estimation.     

Efficient Bandwidth Selection in Non-parametric Regression

In this paper we use non-parametric local polynomial methods to estimate the regression function, m ( x ). Y may be a binary or continuous response variable, and X is continuous with non-uniform density. The main contributions of this paper are the weak convergence of a bandwidth process for kernels of order (0, k ), k =2 ^j , j ≥1 and the proposal of a local data-driven bandwidth selection method which is particularly beneficial for the case when X is not distributed uniformly. This selection method minimizes estimates of the asymptotic MSE and estimates the bias portion in an innovative way which relies on the order of the kernel and not estimation of m ²( x ) directly. We show that utilization of this method results in the achievement of the optimal asymptotic MSE by the estimator, i.e. the method is efficient. Simulation studies are provided which illustrate the method for both binary and continuous response cases.     

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.     

On a Posterior Predictive Density Sample Size Criterion

Abstract.  Let Ω be a space of densities with respect to some σ -finite measure μ and let Π be a prior distribution having support Ω with respect to some suitable topology. Conditional on f , let X ⁿ  = ( X ₁ ,…, X _n ) be an independent and identically distributed sample of size n from f . This paper introduces a Bayesian non-parametric criterion for sample size determination which is based on the integrated squared distance between posterior predictive densities. An expression for the sample size is obtained when the prior is a Dirichlet mixture of normal densities.     

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.     

Simple Transformation Techniques for Improved Non-parametric Regression

We propose and investigate two new methods for achieving less bias in non- parametric regression. We show that the new methods have bias of order h ⁴, where h is a smoothing parameter, in contrast to the basic kernel estimator's order h ². The methods are conceptually very simple. At the first stage, perform an ordinary non-parametric regression on { x_i , Y_i } to obtain m^ ( x_i ) (we use local linear fitting). In the first method, at the second stage, repeat the non-parametric regression but on the transformed dataset { m^ ( x_i , Y_i )}, taking the estimator at x to be this second stage estimator at m^ ( x ). In the second, and more appealing, method, again perform non-parametric regression on { m^ ( x_i , Y_i )}, but this time make the kernel weights depend on the original x scale rather than using the m^ ( x ) scale. We concentrate more of our effort in this paper on the latter because of its advantages over the former. Our emphasis is largely theoretical, but we also show that the latter method has practical potential through some simulated examples.     

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.     

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.     

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.     

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     

Tests for Bivariate Symmetry Against Ordered Alternatives in Square Contingency Tables

Let X and Y denote two ordinal response variables, each having I levels. When subjects are classified on both variables, there are I 2 possible combinations of classifications. Let pij = Pr (X = i, Y = j) . This paper introduces a family of tests based on φ –divergence measures for testing H₀: pij = pji against H1: pij ≥ pji (I≥ j) ; and for testing H1 against H2: pij unrestricted. A simulation study assesses some of the family of tests introduced in this paper in comparison to the likelihood ratio test.     

Testing Hypotheses in the Functional Linear Model

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of R and the response is scalar. The response is modelled as Y =Ψ( X )+ ɛ , where Ψ is some linear continuous operator defined on the space of square integrable functions and valued in R . The random input X is independent from the noise ɛ . In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of Ψ restricted to the Hilbert space generated by the random variable X . We introduce two test statistics based on the norm of the empirical cross-covariance operator of ( X , Y ). The first test statistic relies on a χ ² approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X . The test procedures can be applied to check a given relationship between X and Y . The method is illustrated through a simulation study.     

Non-parametric Kernel Estimation of the Coefficient of a Diffusion

In this work we exhibit a non-parametric estimator of kernel type, for the diffusion coefficient when one observes a one-dimensional diffusion process at times i / n for i = , ..., n and study its asymptotics as n ←∞. When the diffusion coefficient has regularity r ≥ 1, we obtain a rate 1/ n ^{r /(1+2 r )}, both for pointwise estimation and for estimation on a compact subset of R: this is the same rate as for non-parametric estimation of a density with i.i.d. observations.     

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.