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.     

Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , θ) depending on an unknown parameter θ that may be multidimensional. We are interested in the estimation of θ and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times t_i = iΔ , i = 0, ..., n . We study a particular class of estimating functions of the form ∑ f (θ, X _{t i −1}) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.     

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.     

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.     

Approximations of distributions for some standardized partial sums in sequential analysis

In sequential analysis it is often necessary to determine the distributions of √t Y _t and/or √a Y _t , where t is a stopping time of the form t = inf{ n ≥ 1 : n+S_n+ξ_n> a }, Y _n is the sample mean of n independent and identically distributed random variables (iidrvs) Y_i with mean zero and variance one, S_n is the partial sum of iidrvs X_i with mean zero and a positive finite variance, and { ξ_n } is a sequence of random variables that converges in distribution to a random variable ξ as n →∞ and ξ_n is independent of ( X_n+1, Y_n+1), (X_n+2, Y_n+2), . . . for all n ≥ 1. Anscombe's (1952) central limit theorem asserts that both √t Y _t and √a Y _t are asymptotically normal for large a , but a normal approximation is not accurate enough for many applications. Refined approximations are available only for a few special cases of the general setting above and are often very complex. This paper provides some simple Edgeworth approximations that are numerically satisfactory for the problems it considers.     

Estimation in Semiparametric Marginal Shared Gamma Frailty Models

The semiparametric marginal shared frailty models in survival analysis have the non–parametric hazard functions multiplied by a random frailty in each cluster, and the survival times conditional on frailties are assumed to be independent. In addition, the marginal hazard functions have the same form as in the usual Cox proportional hazard models. In this paper, an approach based on maximum likelihood and expectation–maximization is applied to semiparametric marginal shared gamma frailty models, where the frailties are assumed to be gamma distributed with mean 1 and variance θ. The estimates of the fixed–effect parameters and their standard errors obtained using this approach are compared in terms of both bias and efficiency with those obtained using the extended marginal approach. Similarly, the standard errors of our frailty variance estimates are found to compare favourably with those obtained using other methods. The asymptotic distribution of the frailty variance estimates is shown to be a 50–50 mixture of a point mass at zero and a truncated normal random variable on the positive axis for θ₀ = 0. Simulations demonstrate that, for θ₀ < 0, it is approximately an x −(100 − x )%, 0 ≤ x ≤ 50, mixture between a point mass at zero and a truncated normal random variable on the positive axis for small samples and small values of θ₀; otherwise, it is approximately normal.     

Statistical properties of detrended fluctuation analysis

The main goal of this work is to consider the detrended fluctuation analysis (DFA), proposed by Peng et al. [Mosaic organization of DNA nucleotides, Phys. Rev. E. 49(5) (1994), 1685–1689]. This is a well-known method for analysing the long-range dependence in non-stationary time series. Here we describe the DFA method and we prove its consistency and its exact distribution, based on the usual i.i.d. assumption, as an estimator for the fractional parameter d. In the literature it is well established that the nucleotide sequences present long-range dependence property. In this work, we analyse the long dependence property in view of the autoregressive moving average fractionally integrated ARFIMA(p, d, q) processes through the analysis of four nucleotide sequences. For estimating the fractional parameter d we consider the semiparametric regression method based on the periodogram function, in both classical and robust versions; the semiparametric R/S(n) method, proposed by Hurst [Long term storage in reservoirs, Trans. Am. Soc. Civil Eng. 116 (1986), 770–779] and the maximum likelihood method (see [R. Fox and M.S. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Statist. 14 (1986), 517–532]), by considering the approximation suggested by Whittle [Hypothesis Testing in Time Series Analysis (1953), Hafner, New York].     

MINIMUM MSE ESTIMATION BY LINEAR COMBINATIONS OF ORDER STATISTICS1

In this paper we assume that in a random sample of size ndrawn from a population having the pdf f(x; θ) the smallest r₁ observations and the largest r₂ observations are censored (r₁0, r₂0). We consider the problem of estimating θ on the basis of the middle n-r₁-r₂ observations when either f(x;θ)=θ^-1f(x/θ) or f(x;θ) = (aθ)¹f(x-θ)/aθ) where f(·) is a known pdf, a (<0) is known and θ (>0) is unknown. The minimum mean square error (MSE) linear estimator of θ proposed in this paper is a “shrinkage” of the minimum variance linear unbiased estimator of θ. We obtain explicit expressions of these estimators and their mean square errors when (i) f(·) is the uniform pdf defined on an interval of length one and (ii) f(·) is the standard exponential pdf, i.e., f(x) = exp(–x), x0. Various special cases of censoring from the left (right) and no censoring are considered.     

An inclusion-consistent solution to the problem of absurd confidence statements: 2. Consistent nonexact-confidence-interval estimation

Two consistent nonexact-confidence-interval estimation methods, both derived from the consistency-equivalence theorem in Plante (1991), are suggested for estimation of problematic parametric functions with no consistent exact solution and for which standard optimal confidence procedures are inadequate or even absurd, i.e., can provide confidence statements with a 95% empty or all-inclusive confidence set. A belt C(·) from a consistent nonexact-belt family, used with two confidence coefficients (γ = inf_θ P_θ [ θ ? C(X)] and γ⁺ = sup_θ P_θ[θ ? C(X)], is shown to provide a consistent nonexact-belt solution for estimating μ₂ -μ₁ in the Behrens-Fisher problem. A rule for consistent behaviour enables any confidence belt to be used consistently by providing each sample point with best upper and lower confidence levels [δ⁺(x) ≥ γ⁺, δ(x) ≤ γ], which give least-conservative consistent confidence statements ranging from practically exact through informative to noninformative. The rule also provides a consistency correction L(x) = δ⁺(x)-δ(X) enabling alternative confidence solutions to be compared on grounds of adequacy; this is demonstrated by comparing consistent conservative sample-point-wise solutions with inconsistent standard solutions for estimating μ₂/μ₁ (Creasy-Fieller-Neyman problem) and $\sqrt {\mu _1^2 + \mu _2^2 }$, a distance-estimation problem closely related to Stein's 1959 example     

A NOTE ON ESTIMATING THE LOCATION AND SCALE PARAMETERS OF THE EXPONENTIAL DISTRIBUTION FROM A CENSORED SAMPLE

   
Replacing f (x)/F (x) by α+β(x- θ)/σ in the maximum likelihood equations ∂L/∂θ and ∂L/∂σ calculated from a censored sample, a pair of estimators θ^e and σ^e, is obtained. The variances and covariances of these estimators are calculated and compared with the corresponding values for the best linear unbiassed (BLU) estimators.     

A Characterization of the Normal Family of Distributions Based on the Bias of the Maximum Likelihood Estimator

This paper characterizes the family of Normal distributions within the class of exponential families of distributions, via the structure of the bias of the maximum likelihood estimator Θ _n of the canonical parameter Θ . More specifically, when E _θ ( Θ n ) – Θ = (1/ n ) Q ( Θ ) + o (1/ n ), the equality Q ( Θ ) = 0 proves to be a property of the Normal distribution only. The same conclusion is obtained for the one-dimensional case bt assuming that Q ( Θ ) is a polynomial of Θ .     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

Semiparametric Sieve-Type Generalized Least Squares Inference

This article considers the problem of statistical inference in linear regression models with dependent errors. A sieve-type generalized least squares (GLS) procedure is proposed based on an autoregressive approximation to the generating mechanism of the errors. The asymptotic properties of the sieve-type GLS estimator are established under general conditions, including mixingale-type conditions as well as conditions which allow for long-range dependence in the stochastic regressors and/or the errors. A Monte Carlo study examines the finite-sample properties of the method for testing regression hypotheses.     

The Distribution of Sums of I.I.D. Pareto Random Variables with Arbitrary Shape Parameter

Laplace transforms are used to derive an exact expression for the cdf of the sum of n i.i.d. Pareto random variables with common pdf f(x) = (α/β)(1 + x/β)^?α?1 for x > 0, where α > 0 and is not an integer, and β > 0. An attractive feature of this expression is that it involves an integral of non oscillating real-valued functions on the positive real line. Examples of values of cdfs are provided and are compared to those determined via simulations.     

A Non-parametric Test for Generalized First-order Autoregressive Models

We derive a non-parametric test for testing the presence of V(X_i,ε_i) in the non-parametric first-order autoregressive model X_i+1=T(X_i)+V(X_i,ε_i)+U(X_i)ε_i+1, where the function T(x) is assumed known. The test is constructed as a functional of a basic process for which we establish a weak invariance principle, under the null hypothesis and under stationarity and mixing assumptions. Bounds for the local and non-local powers are provided under a condition which ensures that the power tends to one as the sample size tends to infinity.The testing procedure can be applied, e.g. to bilinear models, ARCH models, EXPAR models and to some other uncommon models. Our results confirm the robustness of the test constructed in Ngatchou Wandji (1995) and in Diebolt & Ngatchou Wandji (1995).     

Estimating the Upper Support Point in Deconvolution

Abstract.  We consider estimation of the upper boundary point F ⁻¹ (1) of a distribution function F with finite upper boundary or 'frontier' in deconvolution problems, primarily focusing on deconvolution models where the noise density is decreasing on the positive halfline. Our estimates are based on the (non-parametric) maximum likelihood estimator (MLE) of F . We show that (1) is asymptotically never too small. If the convolution kernel has bounded support the estimator (1) can generally be expected to be consistent. In this case, we establish a relation between the extreme value index of F and the rate of convergence of (1) to the upper support point for the 'boxcar' deconvolution model. If the convolution density has unbounded support, (1) can be expected to overestimate the upper support point. We define consistent estimators , for appropriately chosen vanishing sequences ( β _n ) and study these in a particular case.     

Linear. empirical bayes estimation in the case of the wishart distribution

We consider independent pairs (X₁,∑₁), (X₂,∑₂),…,(X_n∑_n), where each Si is distributed according to some unknown density function g(∑) and, given ∑_i = ∑, X has a conditional density function g(x|∑) of the Wishart type. In each pair, the first component is observable but the second is not. After the (n + l)-th observation X_n+i is obtained, the objective is to estimate ∑ _n+i corresponding to X_n+i. This estimator is called an empirical Bayes (EB) estimator of ∑. We construct a linear EB estimator of ∑ and examine its precision.     

A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator f?_n(x) is proposed which uses kernel-density and deconvolution techniques. The estimator f?_n(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants M_n and h_n which the user may choose. The optimal choices of M_n and h_n depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for M_n and h_n which minimize upper-bound functions of the mean squared error for f?_n(x) are recommended.     

A consistent estimator for logistic mixed effect models

We propose a consistent and locally efficient method of estimating the model parameters of a logistic mixed effect model with random slopes. Our approach relaxes two typical assumptions: the random effects being normally distributed, and the covariates and random effects being independent of each other. Adhering to these assumptions is particularly difficult in health studies where, in many cases, we have limited resources to design experiments and gather data in long‐term studies, while new findings from other fields might emerge, suggesting the violation of such assumptions. So it is crucial to have an estimator that is robust to such violations; then we could make better use of current data harvested using various valuable resources. Our method generalizes the framework presented in Garcia & Ma (2016) which also deals with a logistic mixed effect model but only considers a random intercept. A simulation study reveals that our proposed estimator remains consistent even when the independence and normality assumptions are violated. This contrasts favourably with the traditional maximum likelihood estimator which is likely to be inconsistent when there is dependence between the covariates and random effects. Application of this work to a study of Huntington's disease reveals that disease diagnosis can be enhanced using assessments of cognitive performance. The Canadian Journal of Statistics 47: 140–156; 2019 © 2019 Statistical Society of Canada     

Efficiency in estimation of memory

We study the efficiency of semiparametric estimates of memory parameter. We propose a class of shift invariant tapers of order (p,q). For a fixed p, the variance inflation factor of the new tapers approaches 1 as q   goes to infinity. We show that for d∈(−1/2,p+1/2)$d\in (-1/2,p+1/2)$, the proposed tapered Gaussian semiparametric estimator has the same limiting distribution as the nontapered version for d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The new estimator is mean and polynomial trend invariant, and is computationally advantageous in comparison to the recently proposed exact local Whittle estimator. The simulation study shows that our estimator has comparable or better mean squared error in finite samples for a variety of models.