Minimum Asymptotic Variance Research Articles

Non-Parametric Inference for Bivariate Extreme-Value Copulas

Extreme-value copulas arise as the possible limits of copulas of component-wise maxima of independent, identically distributed samples. The use of bivariate extreme-value copulas is greatly facilitated by their representation in terms of Pickands dependence functions. The two main families of estimators of this dependence function are (variants of) the Pickands estimator and the Caperaa-Fougeres-Genest estimator. In this paper, a unified treatment is given of these two families of estimators, and within these classes those estimators with the minimal asymptotic variance are determined. Main result is the explicit construction of an adaptive, minimum-variance estimator within a class of estimators that encompasses the Caperaa-Fougeres-Genest estimator.

Estimation in the Cox Model with Missing Covariate Data

We consider a Cox model for right-censored survival data when a covariate is missing completely at random and when a continuous surrogate covariate and a complete validation sample are available. The likelihood is approximated using a nonparametric estimation of the missing covariate values and we propose new estimators of the regression parameter and of the cumulative hazard function based on the approximated likelihood. We show that the estimators are n 1/2 -consistent and asymptotically Gaussian and give conditions which ensure that the estimator of the regression parameter has the minimal asymptotic variance in the model with missing covariate data.

Optimum percentile estimating equations for nonlinear random coefficient models

In nonlinear random coefficients models, the means or variances of response variables may not exist. In such cases, commonly used estimation procedures, e.g., (extended) least-squares (LS) and quasi-likelihood methods, are not applicable. This article solves this problem by proposing an estimate based on percentile estimating equations (PEE). This method does not require full distribution assumptions and leads to efficient estimates within the class of unbiased estimating equations. By minimizing the asymptotic variance of the PEE estimates, the optimum percentile estimating equations (OPEE) are derived. Several examples including Weibull regression show the flexibility of the PEE estimates. Under certain regularity conditions, the PEE estimates are shown to be strongly consistent and asymptotic normal, and the OPEE estimates have the minimal asymptotic variance. Compared with the parametric maximum likelihood estimates (MLE), the asymptotic efficiency of the OPEE estimates is more than 98%, while the LS-type of procedures can have infinite variances. When the observations have outliers or do not follow the distributions considered in model assumptions, the article shows that OPEE is more robust than the MLE, and the asymptotic efficiency in the model misspecification cases can be above 150%.

Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

We consider a one‐dimensional diffusion process X, with ergodic property, with drift b(x, θ) and diffusion coefficient a(x, θ) depending on an unknown parameter θ that may be multidimensional. We are interested in the estimation of θ and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times ti = iΔ, i = 0, ..., n. We study a particular class of estimating functions of the form ∑f(θ, Xti−1) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.

Information bounds for Gibbs samplers

If we wish to estimate efficiently the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estimators that use no information about the underlying distribution. The empirical estimator has noticeably smaller variance for deterministic sweep. The variance bound for random sweep is in general smaller than for deterministic sweep, but the two are equal if the target distribution is continuous. If the components of the target distribution are not strongly dependent, the empirical estimator is close to efficient under deterministic sweep, and its asymptotic variance approximately doubles under random sweep.

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution functionF that are based on a certain numberkn of largest order statistics can be represented as a statistical tail function al, that is a functionalT applied to the empirical tail quantile functionQn. We study the asymptotic behaviour of such estimators with a scale and location invariant functionalT under weak second order conditions onF. For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality ofT(Qn) ifT is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution andkpn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionalsT re characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made‐to‐order asymptotic behaviour

Improving a linear regression through joint estimation with a probit model

This paper considers joint estimation of a linear equation and a probit model when the explanatory variables are common, but extra observations are available for the binary variable. The practical significance of the problem is outlined and it is shown that the standard single-equation estimator of the linear model can be improved on. An efficient estimator, in the usual sense of minimum asymptotic variance, is derived and discussed. Finite sample variance properties are investigated via Monte Carlo simulation.

SAMPLING PROPERTIES OF AN ESTIMATOR OF A NEW PROCESS CAPABILITY INDEX FOR WEIBULL DISTRIBUTED QUALITY CHARACTERISTICS

Sampling variance of an estimator of the process capability index applicable to a Weibull distributed characteristic has been derived in two ways. The optimal choice of order of the lower quantile corresponding to minimum asymptotic variance of the esti..

On a stochastic approximation procedure based on averaging

Based on the idea of averaging a new stochastic approximation algorithm has been proposed by Bather (1989), which shows a preferable performance for small to moderate sample sizes. In the present paper an almost sure representation is established for this procedure, which gives the optimal rate of convergence with minimal asymptotic variance.

Quasi-Likelihood for Median Regression Models

Abstract This article proposes quasi-likelihood equations for median regression models. The quasi-likelihood can be used for dependent observations such as repeated measurements or time series data. To construct a quasi-likelihood equation, we need to specify the relation between the median and the dispersion and also specify the dependency of the observations. If a monotone transformation of the original observation has a Laplace distribution, then the quasi-likelihood is the exact likelihood. Under moderate assumptions, the quasi-likelihood estimates are consistent and have asymptotically normal distributions. The estimates are also shown to have minimal asymptotic variance within a certain class of consistent estimates. The proposed method is illustrated using data from a clinical trial.

Approximate maximum likelihood frequency estimation

The high-resolution frequency estimators most commonly used, such as MUSIC, ESPRIT and Yule-Walker, determine estimates of the sinusoidal frequencies from the sample covariances of noise-corrupted data. In this paper, a frequency estimation method termed Approximate Maximum Likelihood (AML) is derived from the approximate likelihood function of sample covariances. The statistical performance of AML is studied, both analytically and numerically, and compared with the Cramér-Rao bound as well as the statistical performance corresponding to the aforementioned methods of frequency estimation. AML is shown to provide the minimum asymptotic error variance in the class of all estimators based on a given set of covariances. The implementation of the AML frequency estimator is discussed in detail. The paper also introduces an AML-based procedure for estimating the number of sinusoidal signals in the measured data, which is shown to possess high detection performance.

Asymptotic importance sampling

An importance sampling technique is described which is based on theoretical considerations about the structure of multivariate integrands in domains having small probability content. The method is formulated in the original variable space. Sampling densities are derived for a variety of practical conditions: a single point of maximum loglikelihood; several points; points located at the intersect of several failure surfaces; and, bounded variables. Sampling in the safe domain is avoided and extensive use is made of noncartesian as well as surface coordinates. The parameters of the importance sampling densities are taylored in such a way as to yield asymptotic minimum variance unbiased estimators. The quality and the efficiency of the method improves as the failure probability decreases. Parameter sensitivies are easily computed owing to the use of local surface coordinates. Several examples are provided.

Determination of Estimators with Minimum Asymptotic Covariance Matrices

We give a straightforward condition sufficient for determining the minimum asymptotic variance estimator in certain classes of estimators relevant to econometrics. These classes are relatively broad, as they include extremum estimation with smooth or nonsmooth objective functions; also, the rate of convergence to the asymptotic distribution is not required to be n−½. We present examples illustrating the content of our result. In particular, we apply our result to a class of weighted Huber estimators, and obtain, among other things, analogs of the generalized least-squares estimator for least Lp-estimation, 1 ≤ p < ∞.

Efficiency rate and local deficiency of Huber’s location estimators and of the α-estimators

The paper studies the problem of selecting an estimator with (approximately) minimal asymptotic variance. For every fixed contamination level there is usually just one such estimator in the considered family. Using the first and the second derivative of the asymptotic variance with respect to the parameter which parametrizes the family of estimators the paper gives two examples of how to select the estimator and gives an approximation to a loss which we suffer when we use the estimator with approximately minimal asymptotic variance instead of the estimator with the precisely minimal

Computational aspects of adaptive combination of least squares and least absolute deviations estimators

Dodge and Jurečková (1987) showed that the estimation of linear regression parameter vector by a convex combination of least squares and least absolute deviation estimators could be adapted so that the resulting estimator achieves the minimum asymptotic variance in the model under consideration. The present paper considers the computational aspects of this adaptive estimator; an algorithm based on the iteratively reweighted least squares method is recommended and formally described. Technical details and an effect of the choice of a normalizing constant, appearing in the definition of the estimator, are also discussed. The behavior of the procedure is demonstrated on example.

Estimates of relative risk

For the two-sample problem with proportional hazard functions, we consider estimation of the constant of proportionality, known as relative risk, using complete uncensored data. For this very special case of Cox’s (1972) regression model for survival data, we find a two-step estimate which is asymptotically equivalent to Cox’s partial likelihood estimate, and we show that both estimates are asymptotically optimal (in the sense of minimum asymptotic variance) among all regular rank estimates of relative risk.

Closed Form Estimators of Latent Cell Frequencies

AbstractOne‐stage and two‐stage closed form estimators of latent cell frequencies in multidimensional contingency tables are derived from the weighted least squares criterion. The first stage estimator is asymptotically equivalent to the conditional maximum likelihood estimator and does not necessarily have minimum asymptotic variance. The second stage estimator does have minimum asymptotic variance relative to any other existing estimator. The closed form estimators are defined for any number of latent cells in contingency tables of any order under exact general linear constraints on the logarithms of the nonlatent and latent cell frequencies.

The asymptotic minimum variance estimate of stationary linear single output processes

The problem of minimum error variance estimation of single output linear stationary processes in the presence of weak measurement noise is considered. By applying s domain analysis to the case of single input systems and white observation noise, explicit and simple expressions are obtained for the error covariance matrix of estimate and the optimal Kalman gains both for minimum- and nonminimum-phase systems. It is found that as the noise intensity approaches zero, the error covariance matrix of estimating the output and its derivatives becomes insensitive to uncertainty, in the system parameters. This matrix depends only on the shape of the high frequency tail of the power-density spectrum of the observation, and thus it can be easily determined from the system transfer function. The theory developed is extended to deal with white measurement noise in multiinput systems where an analog- to the single input nonminimum-phase case is established. The results are also applied to colored observation noise problems and a simple method to derive the minimum error covariance matrices and the optimal filter transfer functions is introduced.

On an optimal asymptotic property of the maximum likelihood estimator of a parameter from a stochastic process

This paper is concerned with the estimation of a parameter of a stochastic process on the basis of a single realization. It is shown, under suitable regularity conditions, that the maximum likelihood estimator is the best consistent asymptotically normal estimator in the sense of having minimum asymptotic variance. It also produces the best limiting probability of concentration in symmetric intervals. An application is given for the problem of estimating the mean of the offspring distribution in a Galton-Watson branching process.

A stochastic Newton-Raphson method

Abstract A stochastic approximation procedure of the Robbins-Monro type is considered. The original idea behind the Newton-Raphson method is used as follows. Given n approximations X 1 ,…, X n with observations Y 1 ,…, Y n , a least squares line is fitted to the points ( X m , Y m ),…, ( X n , Y n ) where m n may depend on n . The ( n +1)st approximation is taken to be the intersection of the least squares line with y =0. A variation of the resulting process is studied. It is shown that this process yields a strongly consistent sequence of estimates which is asymptotically normal with minimal asymptotic variance.