Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

Optimal rates of convergence in the Weibull model based on kernel-type estimators

Let F be a distribution function in the maximal domain of attraction of the Gumbel distribution such that −log(1−F(x))=x^1/θL(x) for a positive real number θ, called the Weibull tail index, and a slowly varying function L. It is well known that the estimators of θ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when L is treated as an infinite dimensional nuisance parameter belonging to some functional class. We also establish the rate optimal asymptotic property of a data-driven choice of the sample fraction that is used for estimation.     

Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξ in semiparametric models of the form Y=G(ξ^TX,ε) where G(⋅) is an unknown link function, ε is an independent error, and ξ is a p_n×1 vector. To recover the direction of ξ, we first propose an inverse regression approach regardless of the link function G(⋅); to construct a data-driven confidence region for the direction of ξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G(⋅) or its derivative. When p_n remains fixed, the empirical likelihood ratio without bias correlation can be asymptotically standard chi-square. Moreover, the asymptotic normality of the empirical likelihood ratio holds true even when the dimension p_n follows the rate of p_n=o(n^1/4) where n is the sample size. Simulation studies are carried out to assess the performance of our proposal, and a real data set is analyzed for further illustration.     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

Asymptotic confidence intervals for Poisson regression

Let (X,Y) be a R^d×N₀-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L₁ distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.     

Single-index quantile regression

Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the “curse of dimensionality”. To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a single-index function , where a univariate link function g₀(⋅) is applied to a linear combination of covariates , often called the single-index. We introduce a practical algorithm where the unknown link function g₀(⋅) is estimated by local linear quantile regression and the parametric index is estimated through linear quantile regression. Large sample properties of estimators are studied, which facilitate further inference. Both the modeling and estimation approaches are demonstrated by simulation studies and real data applications.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

Estimation of the regression operator from functional fixed-design with correlated errors

We consider the estimation of the regression operator r in the functional model: Y=r(x)+ε, where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process ε is a second order stationary process. We construct the kernel type estimate of r from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein-Uhlenbeck error processes.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Location estimation in nonparametric regression with censored data

Consider the heteroscedastic model Y=m(X)+σ(X)?, where ? and X are independent, Y is subject to right censoring, m(·) is an unknown but smooth location function (like e.g. conditional mean, median, trimmed mean…) and σ(·) an unknown but smooth scale function. In this paper we consider the estimation of m(·) under this model. The estimator we propose is a Nadaraya-Watson type estimator, for which the censored observations are replaced by ‘synthetic’ data points estimated under the above model. The estimator offers an alternative for the completely nonparametric estimator of m(·), which cannot be estimated consistently in a completely nonparametric way, whenever high quantiles of the conditional distribution of Y given X=x are involved.We obtain the asymptotic properties of the proposed estimator of m(x) and study its finite sample behaviour in a simulation study. The method is also applied to a study of quasars in astronomy.     

Minimum Hellinger distance estimation in a two-sample semiparametric model

We investigate the estimation problem of parameters in a two-sample semiparametric model. Specifically, let X₁,…,X_n be a sample from a population with distribution function G and density function g. Independent of the X_i’s, let Z₁,…,Z_m be another random sample with distribution function H and density function h(x)=exp[α+r(x)β]g(x), where α and β are unknown parameters of interest and g is an unknown density. This model has wide applications in logistic discriminant analysis, case-control studies, and analysis of receiver operating characteristic curves. Furthermore, it can be considered as a biased sampling model with weight function depending on unknown parameters. In this paper, we construct minimum Hellinger distance estimators of α and β. The proposed estimators are chosen to minimize the Hellinger distance between a semiparametric model and a nonparametric density estimator. Theoretical properties such as the existence, strong consistency and asymptotic normality are investigated. Robustness of proposed estimators is also examined using a Monte Carlo study.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences

Consider the nonparametric regression model Y_ni=g(x_ni)+ε_ni for i=1,…,n, where g is unknown, x_ni are fixed design points, and ε_ni are negatively associated random errors. Nonparametric estimator g_n(x) of g(x) will be introduced and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g_n(x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors (e.g. see J. Multivariate Anal. 25(1) (1988) 100).     

Estimation for the change point of volatility in a stochastic differential equation

We consider a multidimensional Itô process Y=(Y_t)_t∈[0,T] with some unknown drift coefficient process b_t and volatility coefficient σ(X_t,θ) with covariate process X=(X_t)_t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t^∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.     

Fragmentability of groups and metric-valued function spaces

Let (X,τ) be a topological space and let ρ be a metric defined on X. We shall say that (X,τ) is fragmented by ρ if whenever ε>0 and A is a nonempty subset of X there is a τ-open set U such that U∩A≠∅ and ρ−diam(U∩A)<ε. In this paper we consider the notion of fragmentability, and its generalisation σ-fragmentability, in the setting of topological groups and metric-valued function spaces. We show that in the presence of Baireness fragmentability of a topological group is very close to metrizability of that group. We also show that for a compact Hausdorff space X, σ-fragmentability of (C(X),‖⋅_‖∞) implies that the space Cp(X;M) of all continuous functions from X into a metric space M, endowed with the topology of pointwise convergence on X, is fragmented by a metric whose topology is at least as strong as the uniform topology on C(X;M). The primary tool used is that of topological games.     

The product of two dependent random variables with regularly varying or rapidly varying tails

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie-Gumbel-Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY. We quantify the impact as the limit, as x→∞, of the quotient of Pr(XY>x) and Pr(X^∗Y^∗>x), where X^∗ and Y^∗ are independent random variables identically distributed as X and Y, respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.