On the performance of some non-parametric estimators of the conditional survival function with interval-censored data

Simple nonparametric estimators of the conditional distribution of a response variable given a continuous covariate are often useful in survival analysis. Since a few nonparametric estimation options are available, a comparison of the performance of these options may be of value to determine which approach to use in a given application. In this note, we compare various nonparametric estimators of the conditional survival function when the response is subject to interval- and right-censoring. The estimators considered are a generalization of Turnbull’s estimator proposed by Dehghan and Duchesne (2011) and two nonparametric estimators for complete or right-censored data used in conjunction with imputation methods, namely the Nadaraya-Watson and generalized Kaplan-Meier estimators. We study the finite sample integrated mean squared error properties of all these estimators by simulation and compare them to a semi-parametric estimator. We propose a rule-of-thumb based on simple sample summary statistics to choose the most appropriate among these estimators in practice.     

Estimation of fractional integration in the presence of data noise

A comparative study is presented regarding the performance of commonly used estimators of the fractional order of integration when data is contaminated by noise. In particular, measurement errors, additive outliers, temporary change outliers, and structural change outliers are addressed. It occurs that when the sample size is not too large, as is frequently the case for macroeconomic data, then non-persistent noise will generally bias the estimators of the memory parameter downwards. On the other hand, relatively more persistent noise like temporary change outliers and structural changes can have the opposite effect and thus bias the fractional parameter upwards. Surprisingly, with respect to the relative performance of the various estimators, the parametric conditional maximum likelihood estimator with modelling of the short run dynamics clearly outperforms the semiparametric estimators in the presence of noise that is not too persistent. However, when a non-zero mean is allowed for, it may reverse the conclusion.     

Nonparametric estimation of the conditional tail index and extreme quantiles under random censoring

The estimation of the tail index and extreme quantiles of a heavy-tailed distribution is addressed when some covariate information is available and the data are randomly right-censored. Several estimators are constructed by combining a moving-window technique (for tackling the covariate information) and the inverse probability-of-censoring weighting method. The asymptotic normality of these estimators is established and their finite-sample properties are investigated via simulations. A comparison with alternative estimators is provided. Finally, the proposed methodology is illustrated on a medical dataset.     

Bias from misspecification of the component variances in a normal mixture

Bias in parameter estimates can be substantial when heteroscedastic normal mixtures are misspecified as homoscedastic normal mixtures, and vice versa. We show through simulations that the maximum likelihood estimators under the false assumption of equal variances are inconsistent and bias in parameter estimates is appreciable and even substantial when the mixture components are not well-separated. Finite sample bias in parameter estimates is close to the asymptotic bias even for a sample size of 200 or less. When homoscedastic normal mixtures are misspecified as heteroscedastic normal mixtures, the maximum likelihood estimators are consistent. However, the maximum likelihood estimators under a correctly specified homoscedastic mixture model converge to the true parameter values faster than those under a misspecified heteroscedastic mixture model. The bias of the maximum likelihood estimators is less dependent on the lower bound imposed on the component variances to ensure that the likelihood is bounded under the false assumption of unequal variances when the sample size is 500 or more and the component distributions are well-separated. An example is given to demonstrate the effects of a misspecification of the component variances on estimates of the prevalence of hypertension using normal mixtures.     

Strong consistency of ML estimators using partial fraction and elementary subsystem representation of multivariable systems

The strong consistency of ML estimators for partial fraction matrix (PFM) and elementary subsystem (ESS) representations of vector ARMA processes will be discussed for the case when the structure is overestimated. The main results of the paper are the off-line strong consistency theorems of ML estimators for PFM representation, when the number of poles is overestimated and for the ESS representations when the number of ESS's associated with the poles is overestimated.     

Estimation of the proportion ratio under a simple crossover trial

The proportion ratio (PR) of patient response is one of the most commonly used indices for measuring the relative treatment effect in a randomized clinical trial (RCT). Assuming a random effect multiplicative risk model, we develop two point estimators and three interval estimators in closed forms for the PR under a simple crossover RCT. On the basis of Monte Carlo simulation, we evaluate the performance of these estimators in a variety of situations. We note that the point estimator using a ratio of two arithmetic means of patient response probabilities over the two groups (distinguished by the order of treatment-received sequences) is generally preferable to the corresponding one using a ratio of two geometric means of patient response probabilities. We note that the three interval estimators developed in this paper can actually perform well with respect to the coverage probability when the number of patients per group is moderate or large. We further note that the interval estimator based on the ratio of two arithmetic means of patient response probabilities with the logarithmic transformation is probably the best among the three interval estimators discussed here. We use a simple crossover trial studying the suitability of two new inhalation devices for patients who were using a standard inhaler device delivering Salbutamol published elsewhere to illustrate the use of these estimators.     

Recursive smoothing algorithms for the estimation of signals from uncertain observations via mixture approximations

Considering discrete-time systems with uncertain observations when the signal model is unknown, but only covariance information is available, and the signal and the observation additive noise are correlated and jointly Gaussian, we present recursive algorithms for suboptimal fixed-point and fixed-interval smoothing estimators. To derive the algorithms, we employ a technique consisting in approximating the conditional distributions of the signal given the observations by Gaussian distributions, taking successive approximations of the mixtures of normal distributions. The expectation of these approximations provides us with the suboptimal estimators. In a numerical simulation example, the performance of the proposed estimators is compared with that of linear ones, via the sample mean square values of the corresponding estimation errors.     

On choosing the characterization for smoothed perturbation analysis

Using second derivative estimators of the GI/G/1 queue as an illustrative example, the authors demonstrate that smoothed perturbation analysis estimators are not necessarily as distribution-free as infinitesimal perturbation analysis estimators, in the sense that the appropriate choice of conditioning quantities-the so-called characterization-may depend on the underlying distribution. Through a different choice of characterization, the authors derive an estimator that works for distributions for which a previously derived estimator fails. The importance of finding the appropriate characterization, or set of conditioning quantities, when applying the technique of smoothed perturbation analysis is illustrated by the contrast between two estimators of the same quantity (second derivative of mean steady-state system time) based on different characterizations of the simple path     

A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter

Nine parametric estimators of the location and scale parameters of a two-parameter Weibull distribution are compared in terms of their bias and efficiency in a simulation study. The estimators considered are the maximum likelihood estimators (MLE), moment estimators (ME), generalized spacing estimators (GSE), modified maximum likelihood estimators I (MMLE-I), modified maximum likelihood estimators II (MMLE-II), Tiku's modified maximum likelihood estimators (TMMLE), least-squares estimators (LSE), weighted least-squares estimators (WLSE) and percentile estimators (PCE). The aim of the comparisons is to identify the most efficient estimators among these nine estimators for different shape parameters and sample sizes.     

Accuracy analysis of the sine-wave parameters estimation by means of the windowed three-parameter sine-fit algorithm

This paper investigates the accuracy of the sine-wave parameter estimators provided by the Weighted Three-Parameter Sine-Fit (W3PSF) algorithm when a generic cosine window is adopted. Simple expressions for the estimators are derived, which allows a very simple implementation of that algorithm. Moreover, it is shown that the W3PSF algorithm can be well approximated by the classical weighted Discrete Time Fourier Transform (DTFT) when the number of analyzed waveform cycles is high enough. Under that constraint the asymptotic mean square errors (MSEs) of the estimated parameters and the expected sum-squared fitting error can also be evaluated with good accuracy using very simple expressions. Then, the statistical performances of the W3PSF algorithm and the classical 3PSF algorithm are compared through computer simulations in the case of noisy or noisy and harmonically distorted sine-waves. The performed analysis allows us to identify when the W3PSF algorithm outperforms the classical 3PSF algorithm.     

Closed-form error estimators for the linear strain and quadratic strain tetrahedron finite elements

Closed-form representations of the Zienkiewicz-Zhu error estimators for straight edge linear and quadratic strain tetrahedral elements are presented. The closed-form representation of the element error estimators was developed using symbolic algebra. Dramatic computational time savings result in the element error estimator evaluation phase of the FEM process when the closed-form representation is used as compared with the use of Gaussian cubature.     

Inference for the Generalization Error

In order to compare learning algorithms, experimental results reported in the machine learning literature often use statistical tests of significance to support the claim that a new learning algorithm generalizes better. Such tests should take into account the variability due to the choice of training set and not only that due to the test examples, as is often the case. This could lead to gross underestimation of the variance of the cross-validation estimator, and to the wrong conclusion that the new algorithm is significantly better when it is not. We perform a theoretical investigation of the variance of a variant of the cross-validation estimator of the generalization error that takes into account the variability due to the randomness of the training set as well as test examples. Our analysis shows that all the variance estimators that are based only on the results of the cross-validation experiment must be biased. This analysis allows us to propose new estimators of this variance. We show, via simulations, that tests of hypothesis about the generalization error using those new variance estimators have better properties than tests involving variance estimators currently in use and listed in Dietterich (1998). In particular, the new tests have correct size and good power. That is, the new tests do not reject the null hypothesis too often when the hypothesis is true, but they tend to frequently reject the null hypothesis when the latter is false.     

On the use of the bootstrap for estimating functions with functional data

The bootstrap methodology for functional data and functional estimation target is considered. A Monte Carlo study analyzing the performance of the bootstrap confidence bands (obtained with different resampling methods) of several functional estimators is presented. Some of these estimators (e.g., the trimmed functional mean) rely on the use of depth notions for functional data and do not have received yet much attention in the literature. A real data example in cardiology research is also analyzed. In a more theoretical aspect, a brief discussion is given providing some insights on the asymptotic validity of the bootstrap methodology when functional data, as well as a functional parameter, are involved.     

A smoothing principle for the Huber and other location M-estimators

A smoothing principle for M-estimators is proposed. The smoothing depends on the sample size so that the resulting smoothed M-estimator coincides with the initial M-estimator when n→∞. The smoothing principle is motivated by an analysis of the requirements in the proof of the Cramér-Rao bound. The principle can be applied to every M-estimator. A simulation study is carried out where smoothed Huber, ML-, and Bisquare M-estimators are compared with their non-smoothed counterparts and with Pitman estimators on data generated from several distributions with and without estimated scale. This leads to encouraging results for the smoothed estimators, and particularly the smoothed Huber estimator, as they improve upon the initial M-estimators particularly in the tail areas of the distributions of the estimators. The results are backed up by small sample asymptotics.     

Eliminating bias due to censoring in Kendall’s tau estimators for quasi-independence of truncation and failure

While the currently available estimators for the conditional Kendall’s tau measure of association between truncation and failure are valid for testing the null hypothesis of quasi-independence, they are biased when the null does not hold. This is because they converge to quantities that depend on the censoring distribution. The magnitude of the bias relative to the theoretical Kendall’s tau measure of association between truncation and failure due to censoring has not been studied, and so its importance in real problems is not known. We quantify this bias in order to assess the practical usefulness of the estimators. Furthermore, we propose inverse probability weighted versions of the conditional Kendall’s tau estimators to remove the effects of censoring and provide asymptotic results for the estimators. In simulations, we demonstrate the decrease in bias achieved by these inverse probability weighted estimators. We apply the estimators to the Channing House data set and an AIDS incubation data set.     

On the estimation of the linear relation when the error variances are known

The problem of consistent estimation in measurement error models in a linear relation with not necessarily normally distributed measurement errors is considered. Three possible estimators which are constructed as different combinations of the estimators arising from direct and inverse regression are considered. The efficiency properties of these three estimators are derived and the effect of non-normally distributed measurement errors is analyzed. A Monte-Carlo experiment is conducted to study the performance of these estimators in finite samples.     

Likelihood analysis of the multivariate ordinal probit regression model for repeated ordinal responses

We consider the analysis of longitudinal ordinal data, meaning regression-like analysis when the response variable is categorical with ordered categories, and is measured repeatedly over time (or space) on the experimental or sampling units. Particular attention is given to the multivariate ordinal probit regression model, in which the correlation between ordered categorical responses on the same unit at different times (or locations) is modeled with a latent variable that has a multivariate normal distribution. An algorithm for maximum likelihood analysis of this model is proposed and the analysis is demonstrated on an example. Simulations clarify the extent to which maximum likelihood estimators can be more efficient than generalized estimating equations (GEE) estimators of regression coefficients and the extent to which likelihood ratio tests can be more accurate than tests based on standard errors and approximate normality of GEE estimators.     

Estimating smooth distribution function in the presence of heteroscedastic measurement errors

Measurement error occurs in many biomedical fields. The challenges arise when errors are heteroscedastic since we literally have only one observation for each error distribution. This paper concerns the estimation of smooth distribution function when data are contaminated with heteroscedastic errors. We study two types of methods to recover the unknown distribution function: a Fourier-type deconvolution method and a simulation extrapolation (SIMEX) method. The asymptotics of the two estimators are explored and the asymptotic pointwise confidence bands of the SIMEX estimator are obtained. The finite sample performances of the two estimators are evaluated through a simulation study. Finally, we illustrate the methods with medical rehabilitation data from a neuro-muscular electrical stimulation experiment.     

A comparison of fractal dimension estimators based on multiple surface generation algorithms

Fractal geometry has been actively researched in a variety of disciplines. The essential concept of fractal analysis is fractal dimension. It is easy to compute the fractal dimension of truly self-similar objects. Difficulties arise, however, when we try to compute the fractal dimension of surfaces that are not strictly self-similar. A number of fractal surface dimension estimators have been developed. However, different estimators lead to different results. In this paper, we compared five fractal surface dimension estimators (triangular prism, isarithm, variogram, probability, and variation) using surfaces generated from three surface generation algorithms (shear displacement, Fourier filtering, and midpoint displacement). We found that in terms of the standard deviations and the root mean square errors, the triangular prism and isarithm estimators perform the best among the five methods studied.     

Restricted regression estimation in measurement error models

The problem of consistent estimation of the regression coefficients when some prior information about the regression coefficients is available is considered. Such prior information is expressed in the form of exact linear restrictions. The knowledge of covariance matrix of measurement errors that is associated with explanatory variables is used to construct the consistent estimators. Some consistent estimators are suggested which satisfy the exact linear restrictions also. Their asymptotic properties are derived and analytically analyzed under a multivariate ultrastructural model with not necessarily normally distributed measurement errors. The finite sample properties of the estimators are studied through a Monte-Carlo simulation experiment.