Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Approximation of contractible valued correspondences by functions

Let X and Y be absolute neighborhood retracts (this is a large class of spaces) with X compact, and let F:X→Y be an upper hemicontinuous correspondence whose values are compact and contractible. It is shown that any neighborhood of the graph of F contains the graph of a continuous function f:X→Y. The relevance of this result to fixed point theory is indicated. It is also shown that if X is ‘locally infinite’, then F can be approximated in the stronger sense of the graph of f being close to the graph of F and every point in the graph of F being close to some point in the graph of f. A conjectured generalization of the main result is stated.     

Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X ₁,...,X _n ) be a sample from an unknown cumulative distribution function F defined on the real line . The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

Incorrect asymptotic size of subsampling procedures based on post-consistent model selection estimators

Subsampling and the m out of n bootstrap have been suggested in the literature as methods for carrying out inference based on post-model selection estimators and shrinkage estimators. In this paper we consider a subsampling confidence interval (CI) that is based on an estimator that can be viewed either as a post-model selection estimator that employs a consistent model selection procedure or as a super-efficient estimator. We show that the subsampling CI (of nominal level 1−α for any α(0,1)) has asymptotic confidence size (defined to be the limit of finite-sample size) equal to zero in a very simple regular model. The same result holds for the m out of n bootstrap provided m²/n→0 and the observations are i.i.d. Similar zero-asymptotic-confidence-size results hold in more complicated models that are covered by the general results given in the paper and for super-efficient and shrinkage estimators that are not post-model selection estimators. Based on these results, subsampling and the m out of n bootstrap are not recommended for obtaining inference based on post-consistent model selection or shrinkage estimators.     

Iterative Density Estimation from Contaminated Observations

We introduce an iterative procedure for estimating the unknown density of a random variable X from n independent copies of Y=X+ɛ, where ɛ is normally distributed measurement error independent of X. Mean integrated squared error convergence rates are studied over function classes arising from Fourier conditions. Minimax rates are derived for these classes. It is found that the sequence of estimators defined by the iterative procedure attains the optimal rates. In addition, it is shown that the sequence of estimators converges exponentially fast to an estimator within the class of deconvoluting kernel density estimators. The iterative scheme shows how, in practice, density estimation from indirect observations may be performed by simply correcting an appropriate ordinary density estimator. This allows to assess the effect that the perturbation due to contamination by ɛ has on the density to be estimated. We also suggest a method to select the smoothing parameter required by the iterative approach and, utilizing this method, perform a simulation study.     

Continua of stochastic dominance relations for bounded probability distributions

Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), F_αG means that ∫^x_o(x−y^α−1dF(y)∫^x_o(x−y)^α−1dG(y) for all xε[0, b], and F>_αG means that F_αG and F≠G. Each _α is reflexive and transitive and each>_α is asymmetric and transitive. Both _α and>_α increase as α increases but their limits are not complete. A class U_α of utility functions is defined to give F>_αG iff ∫udF>∫udG for all uεU_α. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each _α, and their limit is essentially the constant functions on (0, b].     

Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.     

Continua of stochastic dominance relations for unbounded probability distributions

Let P = {F,G,…} be the set of all probability distribution functions with support (0, ∞). An unrestricted stochastic dominance relation>_∞ is defined on P for each real ∞ 1, where F >_∞ G means that ^x_y = 0 (x - y)^{∞ - 1} dG(y) ^x_{n = 0}(x − y)^∞−1 dG(y) for all ∞ 0, with < for some x. These relations are partial orders that increase as ∞ increases with limit relation>_∞. A class U_∞ of utility functions u on (0, ∞) is defined in such a way that F >_∞ G iff udF > udG for all u ε U_∞. The U_∞ decrease as ∞ increases and have a non-empty intersection U_∞. Each u ε U_∞ is an increasing function that has derivatives of all orders that alternate in sign. Criteria which tell when F eventually dominates G in the sense of F >_∞ G are noted. Comparisons with bounded stochastic dominance results are made in several places.     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

Distributions determined by conditioning on a pair of order statistics

LetX ₁,X ₂, ...,X _n (n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF ⁻¹ over (a,b), −∞≤a<b≤∞ the support ofF. LetX _1:n <X _2:n <...<X _n:n be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<s≤n

Three dual regression schemes for the Lukacs theorem

In the paper we study regressional versions of Lukacs' characterization of the gamma law. We consider constancy of regression instead of Lukacs' independence condition in three new schemes. Up to now the constancy of regressions of U=X/(X + Y) given V=X + Y for independent X and Y has been considered in the literature. Here we are concerned with constancy of regressions for X and Y while independence of U and V is assumed instead.     

Nonparametric identification of a binary random factor in cross section data

Suppose V and U are two independent mean zero random variables, where V has an asymmetric distribution with two mass points and U has some zero odd moments (having a symmetric distribution suffices). We show that the distributions of V and U are nonparametrically identified just from observing the sum V+U, and provide a pointwise rate root n estimator. This can permit point identification of average treatment effects when the econometrician does not observe who was treated. We extend our results to include covariates X, showing that we can nonparametrically identify and estimate cross section regression models of the form Y=g(X,D^∗)+U, where D^∗ is an unobserved binary regressor.     

On the upper and lower semicontinuity of the Aumann integral

Let (T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L₁(μ,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T,τ,μ). We show that if φ:T×P→2^X is a set-valued function such that for each fixed pεP, φ(·,p) has a measurable graph and for each fixed tεT, φ(t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫_Tφ(t,p)dμ(t)= {∫_Tx(t)dμ(t):xεS_φ(p)}, where S_φ(p)= {yεL₁(μ,X):y(t)εφ(t,p)μ−a.e.}, is either upper or lower semicontinuous in the variable p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for X=Rⁿ, and they have useful applications in general equilibrium and game theory.     

On optimal operation of a jointly owned enterprise

An enterprise is owned jointly by m agents, the ith agent's share being θ_i > 0 where ∑_iθ_i=1. The enterprise is able to produce some non-negative n-vector x of goods where x lies in some convex production set X. An operation consists of choosing a vector from X and distributing it among the agents. The problem is to find an operations such that the value of the ith agent's bundle measured in a given price system is proportional to θ_i and such that the operation is (Pareto) optimal with respect to the agent's preferences. It is shown under standard assumptions that operations which are both optimal and proportional always exist. It is also shown that these operations are unique if (a) X is given by a separable production function, and (b) when X represents production of a single good over n time periods.     

Asymptotic results and tests for the choice of approximative models in nonlinear two-phases regression models, heteroscedatic case

This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t _i)=f(t _i)+ε(t _i), i=1,2,…,n, where the (ε(t _i))_i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time we assume that the ε(t _i) have non constant and unknown variances σ²(t _i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation. Received: February 1996     

Estimating the conditional mode of a stationary stochastic process from noisy observations

Let {(X _i, Y _i,)}, i≥1, be a strictly stationary process from noisy observations. We examine the effect of the noise in the response Y and the covariates X on the nonparametric estimation of the conditional mode function. To estimate this function we are using deconvoluting kernel estimators. The asymptotic behavior of these estimators depends on the smoothness of the noise distribution, which is classified as either ordinary smooth or super smooth. Uniform convergence with almost sure convergence rates is established for strongly mixing stochastic processes, when the noise distribution is ordinary smooth. Received: April 1998     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

Shrinkage estimation of <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">X</Emphasis><<Emphasis Type="Italic">Y</Emphasis>) in the exponential case with common location parameter

We consider the problem of estimating R=P(X<Y) where X and Y have independent exponential distributions with parameters and respectively and a common location parameter . Assuming that there is a prior guess or estimate R₀, we develop various shrinkage estimators of R that incorporate this prior information. The performance of the new estimators is investigated and compared with the maximum likelihood estimator using Monte Carlo methods. It is found that some of these estimators are very successful in taking advantage of the prior estimate available.Acknowledgments. The authors are grateful to the editor and to the referees for their constructive comments that resulted in a substantial improvement of the paper.