Estimation in the Positive Stable Shared Frailty Cox Proportional Hazards Model

Shared frailty models are of interest when one has clustered survival data and when focus is on comparing the lifetimes within clusters and further on estimating the correlation between lifetimes from the same cluster. It is well known that the positive stable model should be preferred to the gamma model in situations where the correlated survival data show a decreasing association with time. In this paper, we devise a likelihood based estimation procedure for the positive stable shared frailty Cox model, which is expected to obtain high efficiency. The proposed estimator is provided with large sample properties and also a consistent estimator of standard errors is given. Simulation studies show that the estimation procedure is appropriate for practical use, and that it is much more efficient than a recently suggested procedure. The suggested methodology is applied to a dataset concerning time to blindness for patients with diabetic retinopathy.     

Use of vector divisions in solving quasi-likelihood equations for a Poisson model

Maximum-likelihood estimation technique is known to provide consistent and most efficient regression estimates but often this technique is tedious to implement, particularly in the modelling of correlated count responses. To overcome this limitation, researchers have developed semi- or quasi-likelihood functions that depend only on the correct specification of the mean and variance of the responses rather than on the distribution function. Moreover, quasi-likelihood estimation provides consistent and equally efficient estimates as the maximum-likelihood approach. Basically, the quasi-likelihood estimating function is a non-linear equation constituting of the gradient, Hessian and basic score matrices. Henceforth, to obtain estimates of the regression parameters, the quasi-likelihood equation is solved iteratively using the Newton–Raphson technique. However, the inverse of the Jacobian matrix involved in the Newton–Raphson method may not be easy to compute since the matrix is very close to singularity. In this paper, we consider the use of vector divisions in solving quasi-likelihood equations. The vector divisions are implemented to form secant method formulas. To assess the performance of the use of vector divisions with the secant method, we generate cross-sectional Poisson counts using different sets of mean parameters. We compute the estimates of the regression parameters using the Newton–Raphson technique and vector divisions and compare the number of non-convergent simulations under both algorithms.     

Scalable statistical inference for averaged implicit stochastic gradient descent

Stochastic gradient descent (SGD) provides a scalable way to compute parameter estimates in applications involving large‐scale data or streaming data. As an alternative version, averaged implicit SGD (AI‐SGD) has been shown to be more stable and more efficient. Although the asymptotic properties of AI‐SGD have been well established, statistical inferences based on it such as interval estimation remain unexplored. The bootstrap method is not computationally feasible because it requires to repeatedly resample from the entire data set. In addition, the plug‐in method is not applicable when there is no explicit covariance matrix formula. In this paper, we propose a scalable statistical inference procedure, which can be used for conducting inferences based on the AI‐SGD estimator. The proposed procedure updates the AI‐SGD estimate as well as many randomly perturbed AI‐SGD estimates, upon the arrival of each observation. We derive some large‐sample theoretical properties of the proposed procedure and examine its performance via simulation studies.     

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.     

The Gaussian rank correlation estimator: robustness properties

The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix obtained from pairwise Gaussian rank correlations is always positive semidefinite, and very easy to compute, also in high dimensions. We compare the properties of the Gaussian rank correlation with the popular Kendall and Spearman correlation measures. A simulation study confirms the good efficiency and robustness properties of the Gaussian rank correlation. In the empirical application, we show how it can be used for multivariate outlier detection based on robust principal component analysis.     

Maximum likelihood estimators in regression models with infinite variance innovations

In this paper we consider the problem of maximum likelihood (ML) estimation in the classical AR(1) model with i.i.d. symmetric stable innovations with known characteristic exponent and unknown scale parameter. We present an approach that allows us to investigate the properties of ML estimators without making use of numerical procedures. Finally, we introduce a generalization to the multivariate case.     

Robust estimation and selection for single-index regression model

In this paper, we consider a single-index regression model for which we propose a robust estimation procedure for the model parameters and an efficient variable selection of relevant predictors. The proposed method is known as the penalized generalized signed-rank procedure. Asymptotic properties of the proposed estimator are established under mild regularity conditions. Extensive Monte Carlo simulation experiments are carried out to study the finite sample performance of the proposed approach. The simulation results demonstrate that the proposed method dominates many of the existing ones in terms of robustness of estimation and efficiency of variable selection. Finally, a real data example is given to illustrate the method.     

Adjusted empirical likelihood models with estimating equations for accelerated life tests

This article proposes an adjusted empirical likelihood estimation (AMELE) method to model and analyze accelerated life testing data. This approach flexibly and rigorously incorporates distribution assumptions and regression structures by estimating equations within a semiparametric estimation framework. An efficient method is provided to compute the empirical likelihood estimates, and asymptotic properties are studied. Real-life examples and numerical studies demonstrate the advantage of the proposed methodology.     

Local asymptotic normality for Student-Lévy processes under high-frequency sampling

There is considerable interest in parameter estimation in Lévy models. The maximum likelihood estimator is widely used because under certain conditions it enjoys asymptotic efficiency properties. The toolkit for Lévy processes is the local asymptotic normality which guarantees these conditions. Although the likelihood function is not known explicitly, we prove local asymptotic normality for the location and scale parameters of the Student-Lévy process assuming high-frequency data. In addition, we propose a numerical method to make maximum likelihood estimates feasible based on the Monte Carlo expectation-maximization algorithm. A simulation study verifies the theoretical results.     

ITERATIVE ANALYSIS OF GENERALIZED LATTICE DESIGNS

Generalized lattice designs are defined. They include as special cases the square and rectangular lattice designs, and the α-designs defined by Patterson and Williams (1976). An iterative procedure is given for the combined estimation of variety effects in generalized lattice designs with optimal or near optimal efficiency factors. This procedure, together with an approximate variance matrix, enables the analysis of efficient generalized lattice designs to be carried out on mini computers.     

Improving the best linear unbiased estimator for the scale parameter of symmetric distributions by using the absolute value of ranked set samples

Ranked set sampling is a cost efficient sampling technique when actually measuring sampling units is difficult but ranking them is relatively easy. For a family of symmetric location-scale distributions with known location parameter, we consider a best linear unbiased estimator for the scale parameter. Instead of using original ranked set samples, we propose to use the absolute deviations of the ranked set samples from the location parameter. We demonstrate that this new estimator has smaller variance than the best linear unbiased estimator using original ranked set samples. Optimal allocation in the absolute value of ranked set samples is also discussed for the estimation of the scale parameter when the location parameter is known. Finally, we perform some sensitivity analyses for this new estimator when the location parameter is unknown but estimated using ranked set samples and when the ranking of sampling units is imperfect.     

Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application

Abstract. We consider a bidimensional Ornstein–Uhlenbeck process to describe the tissue microvascularization in anti‐cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one‐dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein–Uhlenbeck diffusion. We compare this estimator with the one obtained by the well‐known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions.     

Applying Least Absolute Deviation Regression to Regression-type Estimation of the Index of a Stable Distribution Using the Characteristic Function

Least absolute deviation regression is applied using a fixed number of points for all values of the index to estimate the index and scale parameter of the stable distribution using regression methods based on the empirical characteristic function. The recognized fixed number of points estimation procedure uses ten points in the interval zero to one, and least squares estimation. It is shown that using the more robust least absolute regression based on iteratively re-weighted least squares outperforms the least squares procedure with respect to bias and also mean square error in smaller samples.     

Instrumental variable estimation in ordinal probit models with mismeasured predictors

Researchers in the medical, health, and social sciences routinely encounter ordinal variables such as self‐reports of health or happiness. When modelling ordinal outcome variables, it is common to have covariates, for example, attitudes, family income, retrospective variables, measured with error. As is well known, ignoring even random error in covariates can bias coefficients and hence prejudice the estimates of effects. We propose an instrumental variable approach to the estimation of a probit model with an ordinal response and mismeasured predictor variables. We obtain likelihood‐based and method of moments estimators that are consistent and asymptotically normally distributed under general conditions. These estimators are easy to compute, perform well and are robust against the normality assumption for the measurement errors in our simulation studies. The proposed method is applied to both simulated and real data. The Canadian Journal of Statistics 47: 653–667; 2019 © 2019 Statistical Society of Canada     

Minimum distance estimation in linear regression with strong mixing errors

Abstract

Minimum distance estimation on the linear regression model with independent errors is known to yield an efficient and robust estimator. We extend the method to the model with strong mixing errors and obtain an estimator of the vector of the regression parameters. The goal of this article is to demonstrate the proposed estimator still retains efficiency and robustness. To that end, this article investigates asymptotic distributional properties of the proposed estimator and compares it with other estimators. The efficiency and the robustness of the proposed estimator are empirically shown, and its superiority over the other estimators is established.     

An iterative sparse algorithm for the penalized maximum likelihood estimator in mixed effects model

In this paper, we propose a new iterative sparse algorithm (ISA) to compute the maximum likelihood estimator (MLE) or penalized MLE of the mixed effects model. The sparse approximation based on the arrow-head (A-H) matrix is one solution which is popularly used in practice. The A-H method provides an easy computation of the inverse of the Hessian matrix and is computationally efficient. However, it often has non-negligible error in approximating the inverse of the Hessian matrix and in the estimation. Unlike the A-H method, in the ISA, the sparse approximation is applied “iteratively” to reduce the approximation error at each Newton Raphson step. The advantages of the ISA over the exact and A-H method are illustrated using several synthetic and real examples.     

Reliability estimation in a multicomponent stress–strength model under generalized half-normal distribution based on progressive type-II censoring

In this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example.     

αn–Designs

This paper defines a broad class of resolvable incomplete block designs called α_n–designs, of which the original α–designs are a special case with n = 1. The statistical and mathematical properties of α–designs extend naturally to these n –dimensional designs. They are a flexible class of resolvable designs appropriate for use in factorial experiments, in constructing efficient t –latinized resolvable block designs, and for enhancing the existing class of α–designs for a single treatment factor.     

M-Estimation for partially functional linear regression model based on splines

ABSTRACT

M-estimation is a widely used technique for robust statistical inference. In this paper, we study robust partially functional linear regression model in which a scale response variable is explained by a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, we use polynomial splines to approximate the slop parameter. The estimation procedure is easy to implement, and it is resistant to heavy-tailederrors or outliers in the response. The asymptotic properties of the proposed estimators are established. Finally, we assess the finite sample performance of the proposed method by Monte Carlo simulation studies.     

Efficiently weighted estimating equations with application to proportional excess hazards

A general approach to estimation, that can lead to efficient estimation in two stages, is presented. The method will not always be available, but sufficient conditions for efficiency are provided together with four examples of its use: (1) estimation of the odds ratio in 1:M matched case-control studies with a dichotomous exposure variable; (2) estimation of the relative hazard in a two-sample survival setting; (3) estimation of the regression parameters in the proportional excess hazards model; and (4) estimation in a partly linear parametric additive hazards model. The method depends upon finding a family of weighted estimating equations, which includes a simple initial equation yielding a consistent estimate and also an equation that yields an efficient estimate, provided the optiomal weights are used.