The effects of zeros in the spectrum on the behavior of the variance for an arithmetic-mean estimate of a homogeneous random field

We study the arithmetic-mean estimate of a homogeneous random field observed on a rectangle and, in particular, the asymptotic behavior of the variance of such an estimate when there are zeros in the spectrum of the process.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 137–144, 1990.     

Improvements of the Maximum Pseudo-Likelihood Estimators in Various Spatial Statistical Models

Abstract

Maximum pseudo-likelihood estimation has hitherto been viewed as a practical but flawed alternative to maximum likelihood estimation, necessary because the maximum likelihood estimator is too hard to compute, but flawed because of its inefficiency when the spatial interactions are strong. We demonstrate that a single Newton-Raphson step starting from the maximum pseudo-likelihood estimator produces an estimator which is close to the maximum likelihood estimator in terms of its actual value, attained likelihood, and efficiency, even in the presence of strong interactions. This hybrid technique greatly increases the practical applicability of pseudo-likelihood-based estimation. Additionally, in the case of the spatial point processes, we propose a proper maximum pseudo-likelihood estimator which is different from the conventional one. The proper maximum pseudo-likelihood estimator clearly shows better performance than the conventional one does when the spatial interactions are strong.     

Presmoothing the transition probabilities in the illness-death model

One major goal in clinical applications of multi-state models is the estimation of transition probabilities. In a recent paper, Meira-Machado et al. (2006) introduce a substitute for the Aalen-Johansen estimator in the case of a non-Markov illness-death model. The idea behind their estimator is to weight the data by the Kaplan-Meier weights pertaining to the distribution of the total survival time of the process. In this paper we propose a modification of Meira-Machado et al. (2006) estimator based on presmoothing. Consistency is established. We investigate the finite sample performance of the new estimator through simulations. Data from a study on colon cancer are used for illustration purposes.     

A model for the relative importance of the criteria in the Multiplicative AHP and SMART

We define the relative importance of any pair of criteria as the substitution rate between the relative gains and losses of the alternatives when we move along an indifference curve. Under the geometric-mean aggregation rule in the Multiplicative AHP and under the arithmetic-mean aggregation rule in SMART, the relative (not the marginal) substitution rate depends neither on the performance of the alternatives under the remaining criteria nor on the units of performance measurement. Hence, it provides a sound argument for distributed decision-making processes where those who judge the criteria are not the same actors as those who assess the performance of the alternatives. The definition has a plausible basis in the psycho-physical research on the relationship between physical stimuli and sensory responses, which shows that human beings are sensitive, not to marginal but to relative changes of the stimulus intensities.     

Estimating spot volatility in the presence of infinite variation jumps

We propose a kernel estimator for the spot volatility of a semi-martingale at a given time point by using high frequency data, where the underlying process accommodates a jump part of infinite variation. The estimator is based on the representation of the characteristic function of Lévy processes. The consistency of the proposed estimator is established under some mild assumptions. By assuming that the jump part of the underlying process behaves like a symmetric stable Lévy process around 0, we establish the asymptotic normality of the proposed estimator. In particular, with a specific kernel function, the estimator is variance efficient. We conduct Monte Carlo simulation studies to assess our theoretical results and compare our estimator with existing ones.     

Limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit theorems with explicit asymptotic variance. In particular, our results show that this estimator is consistent for synchronous data, but possibly biased for non-synchronous observations. Moreover, from our general central limit theorem, we deduce that the estimator can be efficient in the case of a synchronous regular sampling. In the non-synchronous sampling case, the expression of the asymptotic variance is in general less tractable. We study this case more precisely through the example of an alternate sampling.     

Estimating the Density of the Residuals in Autoregressive Models

We consider a (nonlinear) autoregressive model with unknown parameters (vector θ). The aim is to estimate the density of the residuals by a kernel estimator. Since the residuals are not observed, the usual procedure for estimating the density of the residuals is the following: first, compute an estimator for θ; second, calculate the residuals by use of the estimated model; and third, calculate the kernel density estimator by use of these residuals. We show that the resulting density estimator is strong consistent at the best possible convergence rate. Moreover, we prove asymptotic normality of the estimator. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bezier curve smoothing of the Kaplan-Meier estimator

Estimation of a survival function from randomly censored data is very important in survival analysis. The Kaplan-Meier estimator is a very popular choice, and kernel smoothing is a simple way of obtaining a smooth estimator. In this paper, we propose a new smooth version of the Kaplan-Meier estimator using a Bezier curve. We show that the proposed estimator is strongly consistent. Numerical results reveal the that proposed estimator outperforms the Kaplan-Meier estimator and its kernel weighted smooth version in the sense of mean integrated square error. This research is supported by the Korea Research Foundation (1998-015-d00047) made in the program year of 1998.     

On-line Tracking of a Smooth Regression Function

We construct an on-line estimator with equidistant design for tracking a smooth function from Stone–Ibragimov–Khasminskii’s class. This estimator has the optimal convergence rate of risk to zero in sample size. The procedure for setting coefficients of the estimator is controlled by a single parameter and has a simple numerical solution. The off-line version of this estimator allows to eliminate a boundary layer. Simulation results are given. This work is partially supported by a fellowship from the Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel Aviv University.     

The multiple imputations based Kaplan–Meier estimator

We derive the asymptotic distribution of the multiple imputations-based Kaplan–Meier estimator from right censored data with missing censoring indicators. We perform theoretical and numerical comparison studies with a competing semiparametric survival function estimator. We also carry out numerical studies to assess the performance of the proposed estimator when there is model misspecification.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

Nonparametric estimation of multivariate density with direct and auxiliary data and application

We consider the problem of multivariate density estimation, using samples from the distribution of interest as well as auxiliary samples from a related distribution. We assume that the data from the target distribution and the related distribution may occur individually as well as in pairs. Using nonparametric maximum likelihood estimator of the joint distribution, we derive a kernel density estimator of the marginal density. We show theoretically, in a simple special case, that the implied estimator of the marginal density has smaller integrated mean squared error than that of a similar estimator obtained by ignoring dependence of the paired observations. We establish consistency of the marginal density estimator under suitable conditions. We demonstrate small sample superiority of the proposed estimator over the estimator that ignores dependence of the samples, through a simulation study with dependent and non-normal populations. The application of the density estimator in nonparametric classification is also discussed. It is shown that the misclassification probability of the resulting classifier is asymptotically equivalent to that of the Bayes classifier. We also include a data analytic illustration.     

Double-thresholded estimator of extreme value index

The purpose of this Note is to propose an estimator of the extreme value index constructed by using only the number of points exceeding random thresholds. We prove the weak consistency and the asymptotic normality of this estimator. We deduce from this last result that the rate of convergence of our estimator is in a power of the sample size. To our knowledge, this rate of convergence is not reached by any other estimate of the extreme value index. Through a simulation, we compare our estimator to the moment estimator (Dekkers et al., Ann. Statist. 17 (1989) 1833–1855). To cite this article: L. Gardes, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Density Estimation with Replicate Heteroscedastic Measurements

We present a deconvolution estimator for the density function of a random variable from a set of independent replicate measurements. We assume that measurements are made with normally distributed errors having unknown and possibly heterogeneous variances. The estimator generalizes well-known deconvoluting kernel density estimators, with error variances estimated from the replicate observations. We derive expressions for the integrated mean squared error and examine its rate of convergence as n → ∞ and the number of replicates is fixed. We investigate the finite-sample performance of the estimator through a simulation study and an application to real data.     

Completeness and unbiased estimation of mean vector in the multivariate group sequential case

We consider estimation after a group sequential test about a multivariate normal mean, such as a χ² test or a sequential version of the Bonferroni procedure. We derive the density function of the sufficient statistics and show that the sample mean remains to be the maximum likelihood estimator but is no longer unbiased. We propose an alternative Rao-Blackwell type unbiased estimator. We show that the family of distributions of the sufficient statistic is not complete, and there exist infinitely many unbiased estimators of the mean vector and none has uniformly minimum variance. However, when restricted to truncation-adaptable statistics, completeness holds and the Rao-Blackwell estimator has uniformly minimum variance.     

Estimation of the extreme value index and extreme quantiles under random censoring

In this paper, we consider the estimation of the extreme value index and extreme quantiles in the presence of random right censoring. The generalization of the peaks over threshold method is discussed and an adaptation of the moment estimator is proposed. The corresponding extreme quantile estimators are also introduced. We make a start with the analysis of the asymptotic properties of the moment estimator and the corresponding extreme quantile estimator. The finite sample behaviour is illustrated with a small simulation study and through practical examples from survival data analysis.      

The efficiency of the second-order nonlinear least squares estimator and its extension

We revisit the second-order nonlinear least square estimator proposed in Wang and Leblanc (Anne Inst Stat Math 60:883–900, 2008) and show that the estimator reaches the asymptotic optimality concerning the estimation variability. Using a fully semiparametric approach, we further modify and extend the method to the heteroscedastic error models and propose a semiparametric efficient estimator in this more general setting. Numerical results are provided to support the results and illustrate the finite sample performance of the proposed estimator.     

Unbiased estimators in the sense of Lehmann and their discrimination rates

Summary In this paper, we define the index of performance of unbiased estimators in the sense of Lehmann (L-unbiased), which evaluates the power for the estimators to discriminate any wrong values of a parametric function from a correct one. We shall call the indexdiscrimination rate of the estimator. The larger discrimination rate the estimator has, the more desirable it is. An upper bound of discrimination rates is obtained, which is given by thesensitivity of the probability family under consideration. The discrimination rates of several L-unbiased estimators are investigated. Moreover we discuss the conditions under which the L-unbiased estimator is improved in the sense of discrimination rate by the L-unbiased estimator depending only on a sufficient statistic. This research was supported in part by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Science and Culture. The Institute of Statistical Mathematics     

On Kernel-Based Intensity Estimation of Spatial Point Patterns on Linear Networks

We propose an extension of Diggle’s nonparametric edge-corrected kernel-based intensity estimator to the case of events coming from an inhomogenous point pattern on a linear network. We analyze its statistical properties, showing that it is an unbiased estimator of the first-order intensity; we also provide an expression for the variance, and comment on the appropriate bandwidth selection. Our estimator is compared with the current existing equal-split discontinuous kernel density estimator in terms of the mean integrated squared error (MISE). We then use our estimator on two real datasets. We first revisit street crimes in an area of Chicago, obtaining similar results to previously published ones based on a parametric intensity function. Then, we study network-based spatial events consisting of calls to the Police department reporting anti-social behavior in the city of Castellon (Spain).     

Non-parametric estimation of the density of a point process

We consider the Parzen—Rosenblatt type estimator of the density of a point process. Sufficient conditions for convergence of this estimator are established. These results extend those obtained for a random sample.