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     

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 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.     

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.     

A Limit Theorem for Solutions of Inequalities

Let H ( p ) be the set { x ∈ X : h ( x ) ≤ p } where h is a real-valued lower semicontinuous function on a locally compact separable metric space X . This paper presents a general limit theorem for the sequence of random sets H _n ( p ) = { x ∈ X : h _n ( x ) ≤ p } n ≥ 1, where h _n , n ≥ 1, are functions that estimate h     

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.     

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.     

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.     

Stochastic Monotonicity and Conditioning in the Limit

Suppose that {( X _n , Y _n )} is a sequence of pairs of cector-valued stochastic variables which converges weakly to ( X , Y ), and that { y _n } converges to y . Sufficient conditions for the conditional distribution of X _n given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.     

A NOTE ON CONFIDENCE BOUNDS FOR CERTAIN RATIOS OF CHARACTERISTIC ROOTS OF COVARIANCE MATRICES

Let σ₁, …, σ_k be the covariance matrices of k p -variate normal populations. Let Λ_ij be the j th largest characteristic root of σ_i (j=1, …, p; i=1, …, k). In this note we obtain simultaneous confidence intervals on (i)Λ_{i+1, p}/Λ_ipand by using methods similar to those of Khatri (1965).     

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.     

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.     

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.     

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.     

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.     

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.     

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 ).     

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.     

An Efficient Simulation Method for the Computation of a Class of Conditional Expectations

Suppose that the random vector X and the random variable Y are jointly continuous. Also suppose that an observation x of X can be easily simulated and that the probability density function of Y conditional on X = x is known. The paper presents an efficient simulation-based algorithm for estimating E{ g ( X , Y ) | h ( X , Y ) = r } where g and h are real-valued functions. This algorithm is applicable to time series problems in which X = ( X ₁, . . . , X _n−1) and Y = X_n where { x_t } is a discrete time stochastic process for which ( X₁ , . . . , X_n ) is a continuous random vector. A numerical example from time series analysis illustrates the algorithim, for prediction for an ARCH(1) process.     

Modelling and smoothing parameter estimation with multiple quadratic penalties

Penalized likelihood methods provide a range of practical modelling tools, including spline smoothing, generalized additive models and variants of ridge regression. Selecting the correct weights for penalties is a critical part of using these methods and in the single-penalty case the analyst has several well-founded techniques to choose from. However, many modelling problems suggest a formulation employing multiple penalties, and here general methodology is lacking. A wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimize || W ^1/2 ( Xp − y ) ||²ρ+Σ _{i =1} ^m  θ _i p ' S _i p ( p is a parameter vector, X a design matrix, S _i a non-negative definite coefficient matrix defining the i th penalty with associated smoothing parameter θ _i , W a diagonal weight matrix, y a vector of data or pseudodata and ρ an 'overall' smoothing parameter included for computational efficiency). This paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross-validation to this problem and how this allows non-linear, generalized linear and linear models to be fitted using multiple penalties, substantially increasing the scope of penalized modelling methods. Examples of non-linear modelling, generalized additive modelling and anisotropic smoothing are given.