Improved estimators of regression coefficients in measurement error models

This paper is concerned with methods of reducing variability and computer time in a simulation study. The Monte Carlo swindle, through mathematical manipulations, has been shown to yield more precise estimates than the “naive” approach. In this study computer time is considered in conjunction with the variance estimates. It is shown that by this measure the naive method is often a viable alternative to the swindle. This study concentrates on the problem of estimating the variance of an estimator of location. The advantage of one technique over another depends upon the location estimator, the sample size, and the underlying distribution. For a fixed number of samples, while the naive method gives a less precise estimate than the swindle, it requires fewer computations. In addition, for certain location estimators and distributions, the naive method is able to take advantage of certain shortcuts in the generation of each sample. The small amount of time required by this “enlightened” naive method often more than compensates for its relative lack of precision.     

Variance estimation for sparse ultra-high dimensional varying coefficient models

This paper considers the problem of variance estimation for sparse ultra-high dimensional varying coefficient models. We first use B-spline to approximate the coefficient functions, and discuss the asymptotic behavior of a naive two-stage estimator of error variance. We also reveal that this naive estimator may significantly underestimate the error variance due to the spurious correlations, which are even higher for nonparametric models than linear models. This prompts us to propose an accurate estimator of the error variance by effectively integrating the sure independence screening and the refitted cross-validation techniques. The consistency and the asymptotic normality of the resulting estimator are established under some regularity conditions. The simulation studies are carried out to assess the finite sample performance of the proposed methods.     

Variance estimation in the analysis of microarray data

Summary.  Microarrays are one of the most widely used high throughput technologies. One of the main problems in the area is that conventional estimates of the variances that are required in the t -statistic and other statistics are unreliable owing to the small number of replications. Various methods have been proposed in the literature to overcome this lack of degrees of freedom problem. In this context, it is commonly observed that the variance increases proportionally with the intensity level, which has led many researchers to assume that the variance is a function of the mean. Here we concentrate on estimation of the variance as a function of an unknown mean in two models: the constant coefficient of variation model and the quadratic variance–mean model. Because the means are unknown and estimated with few degrees of freedom, naive methods that use the sample mean in place of the true mean are generally biased because of the errors-in-variables phenomenon. We propose three methods for overcoming this bias. The first two are variations on the theme of the so-called heteroscedastic simulation–extrapolation estimator, modified to estimate the variance function consistently. The third class of estimators is entirely different, being based on semiparametric information calculations. Simulations show the power of our methods and their lack of bias compared with the naive method that ignores the measurement error. The methodology is illustrated by using microarray data from leukaemia patients.     

An Adjustment to Improve the Bivariate Survivor Function Repaired NPMLE

We recently proposed a representation of the bivariate survivor function as a mapping of the hazard function for truncated failure time variates. The representation led to a class of estimators that includes van der Laan’s repaired nonparametric maximum likelihood estimator (NPMLE) as an important special case. We proposed a Greenwood-like variance estimator for the repaired NPMLE but found somewhat poor agreement between the empirical variance estimates and these analytic estimates for the sample sizes and bandwidths considered in our simulation study. The simulation results also confirmed those of others in showing slightly inferior performance for the repaired NPMLE compared to other competing estimators as well as a sensitivity to bandwidth choice in moderate sized samples. Despite its attractive asymptotic properties, the repaired NPMLE has drawbacks that hinder its practical application. This paper presents a modification of the repaired NPMLE that improves its performance in moderate sized samples and renders it less sensitive to the choice of bandwidth. Along with this modified estimator, more extensive simulation studies of the repaired NPMLE and Greenwood-like variance estimates are presented. The methods are then applied to a real data example. This revised version was published online in September 2005 with a correction to the second author's name.     

Johnson's transformation two-sample trimmed t and its bootstrap method for heterogeneity and non-normality

The present study investigates the performance of Johnson's transformation trimmed t statistic, Welch's t test, Yuen's trimmed t , Johnson's transformation untrimmed t test, and the corresponding bootstrap methods for the two-sample case with small/unequal sample sizes when the distribution is non-normal and variances are heterogeneous. The Monte Carlo simulation is conducted in two-sided as well as one-sided tests. When the variance is proportional to the sample size, Yuen's trimmed t is as good as Johnson's transformation trimmed t . However, when the variance is disproportional to the sample size, the bootstrap Yuen's trimmed t and the bootstrap Johnson's transformation trimmed t are recommended in one-sided tests. For two-sided tests, Johnson's transformation trimmed t is not only valid but also powerful in comparison to the bootstrap methods.     

Some asymptotic behaviour of the bootstrap estimates on a finite sample

Bootstrapping the mean, variance, standard error of the mean, regression coefficient and its standard error is considered. It is shown that at a fixed sample size bootstrap estimates converge to classical sample estimates as the number of bootstrap replications tends to infinity. For the mean, variance and regression coefficient, convergence almost everywhere is proven; for the standard error of the mean and standard error of the regression coefficient, weak convergence is proven. The speed of convergence is illustrated by simulation results.     

The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.     

On the Estimation of the Probability Density by Trigonometric Series

We present in this article an estimator based on a new orthogonal trigonometric series. We give its statistical properties (bias, variance, mean square error, and mean integrated square error) and the asymptotic properties (convergence of variance, convergence of the mean square error, convergence of the mean integrated square error, uniform convergence in probability, and the rate of convergence of the mean integrated square error). The comparison by simulation on a test density between the estimator obtained from a new trigonometric series with Fejer estimator also based on orthogonal trigonometric series, shows that our estimator is more performant in the sense of the mean integrated square error.     

A Convolution Estimator for the Density of Nonlinear Regression Observations

Abstract. The problem of estimating an unknown density function has been widely studied. In this article, we present a convolution estimator for the density of the responses in a nonlinear heterogenous regression model. The rate of convergence for the mean square error of the convolution estimator is of order n ^?1 under certain regularity conditions. This is faster than the rate for the kernel density method. We derive explicit expressions for the asymptotic variance and the bias of the new estimator, and further a data‐driven bandwidth selector is proposed. We conduct simulation experiments to check the finite sample properties, and the convolution estimator performs substantially better than the kernel density estimator for well‐behaved noise densities.     

A Bayesian Approach in Estimating Transition Probabilities of a Discrete-time Markov Chain for Ignorable Intermittent Missing Data

This article focuses on data analyses under the scenario of missing at random within discrete-time Markov chain models. The naive method, nonlinear (NL) method, and Expectation-Maximization (EM) algorithm are discussed. We extend the NL method into a Bayesian framework, using an adjusted rejection algorithm to sample the posterior distribution, and estimating the transition probabilities with a Monte Carlo algorithm. We compare the Bayesian nonlinear (BNL) method with the naive method and the EM algorithm with various missing rates, and comprehensively evaluate estimators in terms of biases, variances, mean square errors, and coverage probabilities (CPs). Our simulation results show that the EM algorithm usually offers smallest variances but with poorest CP, while the BNL method has smaller variances and better/similar CP as compared to the naive method. When the missing rate is low (about 9%, MAR), the three methods are comparable. Whereas when the missing rate is high (about 25%, MAR), overall, the BNL method performs slightly but consistently better than the naive method regarding variances and CP. Data from a longitudinal study of stress level among caregivers of individuals with Alzheimer’s disease is used to illustrate these methods.     

Resampling Methods for Testing a Semiparametric Random Censorship Model

This paper presents a goodness-of-fit test for a semiparametric random censorship model proposed by Dikta (1998 ). The test statistic is derived from a model-based process which is asymptotically Gaussian. In addition to test consistency, the proposed test can detect local alternatives distinct n ^-1/2 from the null hypothesis. Due to the intractability of the asymptotic null distribution of the test statistic, we turn to two resampling approximations. We first use the well-known bootstrap method to approximate critical values of the test. We then introduce a so-called random symmetrization method for carrying out the test. Both methods perform very well with a sample of moderate size. A simulation study shows that the latter possesses better empirical powers and sizes for small samples.     

On testing the number of components in finite mixture models with known relevant component distributions

The limiting distribution of the log-likelihood-ratio statistic for testing the number of components in finite mixture models can be very complex. We propose two alternative methods. One method is generalized from a locally most powerful test. The test statistic is asymptotically normal, but its asymptotic variance depends on the true null distribution. Another method is to use a bootstrap log-likelihood-ratio statistic which has a uniform limiting distribution in [0,1]. When tested against local alternatives, both methods have the same power asymptotically. Simulation results indicate that the asymptotic results become applicable when the sample size reaches 200 for the bootstrap log-likelihood-ratio test, but the generalized locally most powerful test needs larger sample sizes. In addition, the asymptotic variance of the locally most powerful test statistic must be estimated from the data. The bootstrap method avoids this problem, but needs more computational effort. The user may choose the bootstrap method and let the computer do the extra work, or choose the locally most powerful test and spend quite some time to derive the asymptotic variance for the given model.     

Distribution of the two‐sample t‐test statistic following blinded sample size re‐estimation

We consider the blinded sample size re‐estimation based on the simple one‐sample variance estimator at an interim analysis. We characterize the exact distribution of the standard two‐sample t‐test statistic at the final analysis. We describe a simulation algorithm for the evaluation of the probability of rejecting the null hypothesis at given treatment effect. We compare the blinded sample size re‐estimation method with two unblinded methods with respect to the empirical type I error, the empirical power, and the empirical distribution of the standard deviation estimator and final sample size. We characterize the type I error inflation across the range of standardized non‐inferiority margin for non‐inferiority trials, and derive the adjusted significance level to ensure type I error control for given sample size of the internal pilot study. We show that the adjusted significance level increases as the sample size of the internal pilot study increases. Copyright © 2016 John Wiley & Sons, Ltd.     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

Variance Estimation and Asymptotic Confidence Bands for the Mean Estimator of Sampled Functional Data with High Entropy Unequal Probability Sampling Designs

For fixed size sampling designs with high entropy, it is well known that the variance of the Horvitz–Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional, and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz–Thompson estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of 1 week.     

A large-scale monte carlo study of the buckley-james estimator with censored data

Estimation in the presence of censoring is an important problem. In the linear model, the Buckley-James method proceeds iteratively by estimating the censored values than re-estimating the regression coeffi- cients. A large-scale Monte Carlo simulation technique has been developed to test the performance of the Buckley-James (denoted B-J) estimator. One hundred and seventy two randomly generated data sets, each with three thousand replications, based on four failure distributions, four censoring patterns, three sample sizes and four censoring rates have been investigated, and the results are presented. It is found that, except for Type I1 censoring, the B-J estimator is essentially unbiased, even when the data sets with small sample sizes are subjected to a high censoring rate. The variance formula suggested by Buckley and James (1979) is shown to be sensitive to the failure distribution. If the censoring rate is kept constant along the covariate line, the sample variance of the estimator appears to be insensitive to the censoring pattern with a selected failure distribution. Oscillation of the convergence values associated with the B-J estimator is illustrated and thoroughly discussed.     

Goodness-of-fit test for exponentiality based on spacings for general progressive Type-II censored data

There have been numerous tests proposed to determine whether or not the exponential model is suitable for a given data set. In this article, we propose a new test statistic based on spacings to test whether the general progressive Type-II censored samples are from exponential distribution. The null distribution of the test statistic is discussed and it could be approximated by the standard normal distribution. Meanwhile, we propose an approximate method for calculating the expectation and variance of samples under null hypothesis and corresponding power function is also given. Then, a simulation study is conducted. We calculate the approximation of the power based on normality and compare the results with those obtained by Monte Carlo simulation under different alternatives with distinct types of hazard function. Results of simulation study disclose that the power properties of this statistic by using Monte Carlo simulation are better for the alternatives with monotone increasing hazard function, and otherwise, normal approximation simulation results are relatively better. Finally, two illustrative examples are presented.     

How to Select Representative Samples

We give a formal definition of a representative sample, but roughly speaking, it is a scaled‐down version of the population, capturing its characteristics. New methods for selecting representative probability samples in the presence of auxiliary variables are introduced. Representative samples are needed for multipurpose surveys, when several target variables are of interest. Such samples also enable estimation of parameters in subspaces and improved estimation of target variable distributions. We describe how two recently proposed sampling designs can be used to produce representative samples. Both designs use distance between population units when producing a sample. We propose a distance function that can calculate distances between units in general auxiliary spaces. We also propose a variance estimator for the commonly used Horvitz–Thompson estimator. Real data as well as illustrative examples show that representative samples are obtained and that the variance of the Horvitz–Thompson estimator is reduced compared with simple random sampling.     

A Simulation Study of White's Test for Heteroskedasticity in Fixed and Stochastic Regression Models

We study the finite-sample properties of White's test for heteroskedasticity in fixed and stochastic regression models. We compare by simulation White and bootstrap methods when the underlying distribution is symmetric as well as asymmetric. The superior performance of the bootstrap method in small samples does not hold when the underlying distribution is asymmetric.