Tuning Parameter Estimation in Penalized Least Squares Methodology

The efficiency of the penalized methods (Fan and Li, 2001 Fan , J. , Li , R. ( 2001 ). Variable selection via nonconcave penalized likelihood and its oracle properties . Journal of the American Statistical Association 96 : 1348 – 1360 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) depends strongly on a tuning parameter due to the fact that it controls the extent of penalization. Therefore, it is important to select it appropriately. In general, tuning parameters are chosen by data-driven approaches, such as the commonly used generalized cross validation. In this article, we propose an alternative method for the derivation of the tuning parameter selector in penalized least squares framework, which can lead to an ameliorated estimate. Simulation studies are presented to support theoretical findings and a comparison of the Type I and Type II error rates, considering the L ₁, the hard thresholding and the Smoothly Clipped Absolute Deviation penalty functions, is performed. The results are given in tables and discussion follows.     

Evaluating the Effective Degrees of Freedom of the Delete-a-Group Jackknife

The delete-a-group jackknife is sometimes used when estimating the variances of statistics based on a large sample. We investigate heavily poststratified estimators for a population mean and a simple regression coefficient, where both full-sample and domain estimates are of interest. The delete-a-group (DAG) jackknife employing 30, 60, and 100 replicates is found to be highly unstable, even for large sample sizes. The empirical degrees of freedom of these DAG jackknives are usually much less than their nominal degrees of freedom. This analysis calls into question whether coverage intervals derived from replication-based variance estimators can be trusted for highly calibrated estimates.     

ASYMPTOTIC AND SMALL SAMPLE STATISTICAL PROPERTIES OF RANDOM FRAILTY VARIANCE ESTIMATES FOR SHARED GAMMA FRAILTY MODELS

This paper concerns maximum likelihood estimation for the semiparametric shared gamma frailty model; that is the Cox proportional hazards model with the hazard function multiplied by a gamma random variable with mean 1 and variance θ. A hybrid ML-EM algorithm is applied to 26 400 simulated samples of 400 to 8000 observations with Weibull hazards. The hybrid algorithm is much faster than the standard EM algorithm, faster than standard direct maximum likelihood (ML, Newton Raphson) for large samples, and gives almost identical results to the penalised likelihood method in S-PLUS 2000. When the true value θ₀ of θ is zero, the estimates of θ are asymptotically distributed as a 50–50 mixture between a point mass at zero and a normal random variable on the positive axis. When θ₀ > 0, the asymptotic distribution is normal. However, for small samples, simulations suggest that the estimates of θ are approximately distributed as an x ? (100 ? x)% mixture, 0 ≤ x ≤ 50, between a point mass at zero and a normal random variable on the positive axis even for θ₀ > 0. In light of this, p-values and confidence intervals need to be adjusted accordingly. We indicate an approximate method for carrying out the adjustment.     

A note on estimation in bernoulli trials with dependence

Finite sample properties of estimators for the parameters of a dependent Bernoulli process are investigated using Monte Carlo techniques. A ratio estimator is proposed for the dependence parameter of the model and is compared to the approximate maximum likelihood estimator given by Klotz. It is shown that both estimators have a downward bias that is extreme in certain cases and that samples well in excess of 200 may be necessary before the asymptotic theory can be applied.     

The MLE of Geometric Parameter for a Geometric Process

In this article, we provide the MLE of the ratio parameter of a geometric process and discuss its consistency and asymptotic normality.     

Gaining insight with recursive partitioning of generalized linear models

Recursive partitioning algorithms separate a feature space into a set of disjoint rectangles. Then, usually, a constant in every partition is fitted. While this is a simple and intuitive approach, it may still lack interpretability as to how a specific relationship between dependent and independent variables may look. Or it may be that a certain model is assumed or of interest and there is a number of candidate variables that may non-linearly give rise to different model parameter values. We present an approach that combines generalized linear models (GLM) with recursive partitioning that offers enhanced interpretability of classical trees as well as providing an explorative way to assess a candidate variable's influence on a parametric model. This method conducts recursive partitioning of a GLM by (1) fitting the model to the data set, (2) testing for parameter instability over a set of partitioning variables, (3) splitting the data set with respect to the variable associated with the highest instability. The outcome is a tree where each terminal node is associated with a GLM. We will show the method's versatility and suitability to gain additional insight into the relationship of dependent and independent variables by two examples, modelling voting behaviour and a failure model for debt amortization, and compare it to alternative approaches.     

A consistent method of estimation for the parameters of the three-parameter inverse Gaussian distribution

In this paper, we propose a consistent method of estimation for the parameters of the three-parameter inverse Gaussian distribution. We then discuss some properties of these estimators and show by means of a Monte Carlo simulation study that the proposed estimators perform better than some other prominent estimators in terms of bias and root mean squared error. Finally, we present two real-life examples to illustrate the method of inference developed here.     

Semiparametric score test for varying copula parameter in Markov time series

This article examines a semiparametric test for checking the constancy of serial dependence via copula models for Markov time series. A semiparametric score test is proposed for testing the constancy of the copula parameter against stochastically varying copula parameter. The asymptotic null distribution of the test is established. A semiparametric bootstrap procedure is employed for the estimation of the variance of the proposed score test. Illustrations are given based on simulated series and historic interest rate data.     

Reducing bias of the maximum-likelihood estimation for the truncated Pareto distribution

The Pareto distribution is an important distribution in statistics, which has been widely used in finance, physics, hydrology, geology, astronomy, and so on. Even though the parameter estimation for the Pareto distribution has been well established in the literature, the estimation problem for the truncated Pareto distribution becomes complex. This article investigates the bias and mean-squared error of the maximum-likelihood estimation for the truncated Pareto distribution, and some useful results are obtained.     

Testing the Equality of the Noncentrality Parameters of Two Noncentral t-Distributions with Identical Degrees of Freedom

The problem of testing the equality of the noncentrality parameters of two noncentral t-distributions with identical degrees of freedom is considered, which arises from the comparison of two signal-to-noise ratios for simple linear regression models. A test procedure is derived that is guaranteed to maintain Type I error while having only minimal amounts of conservativeness, and comparisons are made with several other approaches to this problem based on variance stabilizing transformations. The new procedure derived in this article is shown to have good properties and will be useful for practitioners.