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.     

Logistic regression with a partially observed covariate

We present results of a Monte Carlo study comparing four methods of estimating the parameters of the logistic model logit (pr (Y = 1 | X, Z)) = α₀ + α ₁ X + α ₂ Z where X and Z are continuous covariates and X is always observed but Z is sometimes missing. The four methods examined are 1) logistic regression using complete cases, 2) logistic regression with filled-in values of Z obtained from the regression of Z on X and Y, 3) logistic regression with filled-in values of Z and random error added, and 4) maximum likelihood estimation assuming the distribution of Z given X and Y is normal. Effects of different percent missing for Z and different missing value mechanisms on the bias and mean absolute deviation of the estimators are examined for data sets of N = 200 and N = 400.     

Test of Association Between Two Ordinal Variables While Adjusting for Covariates

We propose a new set of test statistics to examine the association between two ordinal categorical variables X and Y after adjusting for continuous and/or categorical covariates Z. Our approach first fits multinomial (e.g., proportional odds) models of X and Y, separately, on Z. For each subject, we then compute the conditional distributions of X and Y given Z. If there is no relationship between X and Y after adjusting for Z, then these conditional distributions will be independent, and the observed value of (X, Y) for a subject is expected to follow the product distribution of these conditional distributions. We consider two simple ways of testing the null of conditional independence, both of which treat X and Y equally, in the sense that they do not require specifying an outcome and a predictor variable. The first approach adds these product distributions across all subjects to obtain the expected distribution of (X, Y) under the null and then contrasts it with the observed unconditional distribution of (X, Y). Our second approach computes "residuals" from the two multinomial models and then tests for correlation between these residuals; we define a new individual-level residual for models with ordinal outcomes. We present methods for computing p-values using either the empirical or asymptotic distributions of our test statistics. Through simulations, we demonstrate that our test statistics perform well in terms of power and Type I error rate when compared to proportional odds models which treat X as either a continuous or categorical predictor. We apply our methods to data from a study of visual impairment in children and to a study of cervical abnormalities in human immunodeficiency virus (HIV)-infected women. Supplemental materials for the article are available online.     

Attaining specified reliability in component design: a stochastic approximation approach

In the design, manufacture and maintenance of components, particular attention is paid to component reliability R, the probability that the strength X of a component will exceed a stress Y to which it will be subjected. The problem addressed here is the design (or redesign) of a compoFent to meet a specified reliability R*. While certain characteristics of the random variables X and Y are assumed (symmetry of X about a unique median for example) it is not assumed that the form of the distribution of (X,Y) is known, nor that X and Y are independent. A design is recomnended based on a variation of the stochastic approximation procedure due to Dupac and Kral (1972) which in general estimates recursively the root of a regression curve assuming both independent and dependent regression variables are subject to experimental error.     

Measurement error in regression analysis

Consider the linear regression model Y = Xθ+ ε where Y denotes a vector of n observations on the dependent variable, X is a known matrix, θ is a vector of parameters to be estimated and e is a random vector of uncorrelated errors. If X'X is nearly singular, that is if the smallest characteristic root of X'X s small then a small perurbation in the elements of X, such as due to measurement errors, induces considerable variation in the least squares estimate of θ. In this paper we examine for the asymptotic case when n is large the effect of perturbation with regard to the bias and mean squared error of the estimate.     

Moments for a ratio of correlated gamma variates

This paper considers the finite integral moments for the ratio, R = X/Y, where X and Y re correlated gamma distributed variables. An analytical and numerical comparison is given for two classes of underlying bivariate gamma distributions. It is shown that the two bivariate gamma structures provide indentical experessions for the mth unadjussted moment, E(R^m), if and only if either of the following conditions hold : 1) X and Y are uncorrelated of 2) m=1. A numerical evaluation is performed to determine the extent that the two methods differ whenever the variables are correlated     

A property of the yule distribution and its applications

The relationship Y = RX between two random variables X and Y, where R is distributed independently of X in (0, l), is known to have important consequences in different fields such as income distribution analysis, Inventory decision models, etc.

In this paper it is shown that when X and Y are discrete random variables, relationships of similar nature lead to Yule-type distributions. The implications of the results are studied in connection with problems of income underreporting and inventory decision making.     

Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data

Let ( X , Y ) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a non-parametric likelihood ratio approach. This approach was introduced by Thomas & Grunkemeier (1975 ), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.     

A closed form solution for tre least squares regression problem with linear inequality constraints

In this paper we consider a linear model Y = Xβ+e with linear inequality constraints R'β≥r, where X and R are known and full column rank matrices. The closed form of the inequality constrained least squares (ICLS) estimator is given. We provide two examples which illustrate the use of this closed form in the computation of estimates.     

Exponential models,brownian motion,and independence

Some examples of steep, reproductive exponential models are considered. These models are shown to possess a τ-parallel foliation in the terminology of Barndorff-Nielsen and Blaesild. The independence of certain functions follows directly from the foliation. Suppose X(t) is a Wiener process with drift where X(t) = W(t) + ct, 0 < t < T. Furthermore let Y = max [X(s), 0 < s < T]. The joint density of Y and X = X(T), the end value, is studied within the framework of an exponential model, and it is shown that Y(Y – X) is independent of X. It is further shown that Y(Y – X) suitably scaled has an exponential distribution. Further examples are considered by randomizing on T.     

Linear model estimation errors with estimated design parameters

The problem of error estimation of parameters b in a linear model,Y = Xb+ e, is considered when the elements of the design matrix X are functions of an unknown ‘design’ parameter vector c. An estimated value c is substituted in X to obtain a derived design matrix [Xtilde]. Even though the usual linear model conditions are not satisfied with [Xtilde], there are situations in physical applications where the least squares solution to the parameters is used without concern for the magnitude of the resulting error. Such a solution can suffer from serious errors.

This paper examines bias and covariance errors of such estimators. Using a first-order Taylor series expansion, we derive approximations to the bias and covariance matrix of the estimated parameters. The bias approximation is a sum of two terms:One is due to the dependence between ? and Y; the other is due to the estimation errors of ? and is proportional to b, the parameter being estimated. The covariance matrix approximation, on the other hand, is composed of three omponents:One component is due to the dependence between ? and Y; the second is the covariance matrix ∑_b corresponding to the minimum variance unbiased b, as if the design parameters were known without error; and the third is an additional component due to the errors in the design parameters. It is shown that the third error component is directly proportional to bb'. Thus, estimation of large parameters with wrong design matrix [Xtilde] will have larger errors of estimation. The results are illustrated with a simple linear example.     

Double sampling for exact values in the normal discriminant model with application to binary regression

Increasing attention is being given to problems involving binary outcomes with covariates subject to measurement error. Here, we consider the two group normal discriminant model where a subset of the continuous variates are subject to error and will typically be replaced by a vector of surrogates, perhaps of different dimension. Correcting for the measurement error is made possible by a double sampling scheme in which the surrogates are collected on all units and true values are obtained on a random subset of units. Such a scheme allows us to consider a rich set of measurement error models which extend the traditional additive error model. Maximum likelihood estimators and their asymptotic properties are derived under a variety of models for the relationship between true values and the surrogates. Specific attention is given to the coefficients in the resulting logistic regression model. Optimal allocations are derived which minimize the variance of the estimated slope subject to cost constraints for the case where there is a univariate covariate but a possibly multivariate surrogate.     

Non-parametric Estimation of the Residual Distribution

Consider a heteroscedastic regression model Y = m ( X ) +σ( X )ε, where the functions m and σ are "smooth", and ε is independent of X . An estimator of the distribution of ε based on non-parametric regression residuals is proposed and its weak convergence is obtained. Applications to prediction intervals and goodness-of-fit tests are discussed.     

Finite-population prediction under error-in-variables superpopulation models

The prediction problem in finite populations is considered under error-in-variables super population models. The models considered are the usual regression models involving at most two variables, x and y, where both may be measured with error. Properties of some classical predictors are investigated. A Bayesian approach is proposed.     

Joint distribution of simultaneous exposures to several carcinogens in a case–control study: sample size determination

Consider a population of individuals who are free of a disease under study, and who are exposed simultaneously at random exposure levels, say X,Y,Z,… to several risk factors which are suspected to cause the disease in the populationm. At any specified levels X=x, Y=y, Z=z, …, the incidence rate of the disease in the population ot risk is given by the exposure–response relationship r(x,y,z,…) = P(disease|x,y,z,…). The present paper examines the relationship between the joint distribution of the exposure variables X,Y,Z, … in the population at risk and the joint distribution of the exposure variables U,V,W,… among cases under the linear and the exponential risk models. It is proven that under the exponential risk model, these two joint distributions belong to the same family of multivariate probability distributions, possibly with different parameters values. For example, if the exposure variables in the population at risk have jointly a multivariate normal distribution, so do the exposure variables among cases; if the former variables have jointly a multinomial distribution, so do the latter. More generally, it is demonstrated that if the joint distribution of the exposure variables in the population at risk belongs to the exponential family of multivariate probability distributions, so does the joint distribution of exposure variables among cases. If the epidemiologist can specify the differnce among the mean exposure levels in the case and control groups which are considered to be clinically or etiologically important in the study, the results of the present paper may be used to make sample size determinations for the case–control study, corresponding to specified protection levels, i.e., size α and 1–β of a statistical test. The multivariate normal, the multinomial, the negative multinomial and Fisher's multivariate logarithmic series exposure distributions are used to illustrate our results.     

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.     

Bayesian Estimation of Survival Functions under Stochastic Precedence

When estimating the distributions of two random variables, X and Y, investigators often have prior information that Y tends to be bigger than X. To formalize this prior belief, one could potentially assume stochastic ordering between X and Y, which implies Pr(X < or = z) > or = Pr(Y < or = z) for all z in the domain of X and Y. Stochastic ordering is quite restrictive, though, and this article focuses instead on Bayesian estimation of the distribution functions of X and Y under the weaker stochastic precedence constraint, Pr(X < or = Y) > or = 0.5. We consider the case where both X and Y are categorical variables with common support and develop a Gibbs sampling algorithm for posterior computation. The method is then generalized to the case where X and Y are survival times. The proposed approach is illustrated using data on survival after tumor removal for patients with malignant melanoma.     

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.     

Computation of the central and noncentral f distributions

Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ²(λ, m) and Y is independently χ²(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.     

Quasi-likelihood Estimation of Non-invertible Moving Average Processes

Classical methods based on Gaussian likelihood or least-squares cannot identify non-invertible moving average processes, while recent non-Gaussian results are based on full likelihood consideration. Since the error distribution is rarely known a quasi-likelihood approach is desirable, but its consistency properties are yet unknown. In this paper we study the quasi-likelihood associated with the Laplacian model, a convenient non-Gaussian model that yields a modified L ₁ procedure. We show that consistency holds for all standard heavy tailed errors, but not for light tailed errors, showing that a quasi-likelihood procedure cannot be applied blindly to estimate non-invertible models. This is an interesting contrast to the standard results of the quasi-likelihood in regression models, where consistency usually holds much more generally. Similar results hold for estimation of non-causal non-invertible ARMA processes. Various simulation studies are presented to validate the theory and to show the effect of the error distribution, and an analysis of the US unemployment series is given as an illustration.