Asymptotic Estimation Theory Research Articles

AN ADAPTIVE TRIMMED LIKELIHOOD ALGORITHM FOR IDENTIFICATION OF MULTIVARIATE OUTLIERS

This article describes an algorithm for the identification of outliers in multivariate data based on the asymptotic theory for location estimation as described typically for the trimmed likelihood estimator and in particular for the minimum covariance determinant estimator. The strategy is to choose a subset of the data which minimizes an appropriate measure of the asymptotic variance of the multivariate location estimator. Observations not belonging to this subset are considered potential outliers which should be trimmed. For α less than about 0.5, the correct trimming proportion is taken to be that α > 0 for which the minimum of any minima of this measure of the asymptotic variance occurs. If no minima occur for an α > 0 then the data set will be considered outlier free.

A Representation Theorem for the Error of Recursive Estimators

The ultimate objective of this paper is to develop new techniques that can be used for the analysis of performance degradation due to statistical uncertainty for a wide class of linear stochastic systems. For this we need new technical tools similar to those used in [L. Gerencser, Statist. Plann. Inference, 41 (1994), pp. 303-325]. The immediate technical objective is to extend the previous technical results to the Djereveckii--Fradkov--Ljung scheme with enforced boundedness. Our starting point is a standard approximation of the estimation error used in the asymptotic theory of recursive estimation. Tight control of the difference between the estimation error and its standard approximation, referred to as residuals, is a crucial point in our applications. The main technical advance of the present paper is a set of strong approximation theorems for three closely related recursive estimation algorithms in which, for any $q \ge 1$, the $L_q$-norms of the residual terms are shown to tend to zero with rate $N^{-1/2-\varepsilon}$ with some $\varepsilon > 0$. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L. Gerencser, Systems Control Lett., 21 (1993), pp. 347-351]. Two useful corollaries will be derived. In the first a standard transform of the estimation-error process for the basic recursive estimation method, Algorithm CR\@, will be shown to be $L$-mixing, while in the second the asymptotic covariance matrix of the estimator for the same method will be given. Applications to multivariable adaptive prediction and the minimum-variance self-tuning regulator for ARMAX systems will be described.

The sensitivity and specificity of markers for event times

The statistical literature on assessing the accuracy of risk factors or disease markers as diagnostic tests deals almost exclusively with settings where the test, Y, is measured concurrently with disease status D. In practice, however, disease status may vary over time and there is often a time lag between when the marker is measured and the occurrence of disease. One example concerns the Framingham risk score (FR-score) as a marker for the future risk of cardiovascular events, events that occur after the score is ascertained. To evaluate such a marker, one needs to take the time lag into account since the predictive accuracy may be higher when the marker is measured closer to the time of disease occurrence. We therefore consider inference for sensitivity and specificity functions that are defined as functions of time. Semiparametric regression models are proposed. Data from a cohort study are used to estimate model parameters. One issue that arises in practice is that event times may be censored. In this research, we extend in several respects the work by Leisenring et al. (1997) that dealt only with parametric models for binary tests and uncensored data. We propose semiparametric models that accommodate continuous tests and censoring. Asymptotic distribution theory for parameter estimates is developed and procedures for making statistical inference are evaluated with simulation studies. We illustrate our methods with data from the Cardiovascular Health Study, relating the FR-score measured at enrollment to subsequent risk of cardiovascular events.

Risk sensitive identification of linear stochastic systems

Risk-sensitive identification of AR-processes was first considered in [12]. The purpose of this paper is to extend this original approach to ARMA-processes and even to multi-variable linear stochastic systems. We provide a new definition of a risk-sensitive identification criterion. For this we first consider a recursive identification procedure which is parameterized by a weight-matrix K acting on the stochastic gradient. Using the asymptotic theory of recursive estimation a suitably scaled version of the error process will be approximated by a stationary Gaussian process, see Chapter 4.5, Part II of [1]. The new risk sensitive criterion will be defined in terms of this associated stationary Gaussian process in a familiar manner via an exponential-quadratic cost. The main result of the paper is the minimization of the proposed new criterion with respect to the weight-matrix K over a feasible set EK, see (22), where the cost function is known to be finite, Theorem 6.1. This results will then be extended to the case when minimization over a feasible set E°K is considered, see (26), on the complement of which the cost function is known to be infinite, Theorem 6.1. The starting point of our analysis is an expression of the cost function given in LEQG-theory, in particular a result of [10]. A new expression for the cost function will be also given, using stochastic realization theory, as the mutual information rate between two stochastic processes.

Asymptotic estimation theory of multipoint linkage analysis under perfect marker information

We consider estimation of a disease susceptibility locus $\tau$ at a chromosome. With perfect marker data available, the estimator $\htau_N$ of $\tau$ based on $N$-pedigrees has a rate of convergence $N^{-1}$ under mild regularity conditions. The limiting distribution is the arg max of a certain compound Poisson process. Our approach is conditional on observed phenotypes, and therefore treats parametric and nonparametric linkage, as well as quantitative trait loci methods within a unified framework. A constant appearing in the asymptotics, the so-called asymptotic slope-to-noise ratio, is introduced as a performance measure for a given genetic model, score function and weighting scheme. This enables us to define asymptotically optimal score functions and weighting schemes. Interestingly, traditional $N^{-1/2}$ theory breaks down, in that, for instance, the ML-estimator is not asymptotically optimal. Further, the asymptotic estimation theory automatically takes uncertainty of $\tau$ into account, which is otherwise handled by means of multiple testing and Bonferroni-type corrections. Other potential applications of our approach that we discuss are general sampling criteria for planning of linkage studies, appropriate grid size of marker maps, robustness w.r.t. choice of map function (dropping assumption of no interference) and quantification of information loss due to heterogeneity (with linked or unlinked trait loci). We also discuss relations to pointwise performance criteria and pay special attention to weak genetic models, so-called local specificity models.

Simultaneous estimation of coefficients of variation

An improved asymptotic estimation theory for the coefficient of variation γ is developed under the homogeneity hypothesis that several coefficients of variation are the same. Assuming that homogeneity holds, it is advantageous to combine the data to estimate the common coefficient of variation. However, the combined estimator becomes inconsistent when the equality of the hypothesis does not hold. In this situation, estimators based on pretest and ( James and Stein, 1961. Estimation with quadratic loss. Proceeding of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 361–379) principles are proposed. Asymptotic properties of the shrinkage estimator, positive-part and pretest estimators are discussed and compared with the standard and combined estimators. It is demonstrated that the positive part estimator utilizes the sample and nonsample information in a superior way relative to the ordinary shrinkage estimator.

On the Geometry of Censored Models

This paper looks at the process of censoring via the differential geometric theory of Amari (1990) . This theory gives both a conceptual framework and a set of useful tools which help in mastering the asymptotic theory of estimation and testing. After briefly reviewing results from the theory, noting the natural geometry enjoyed by exponential and curved exponential families, the paper examines how censoring can be viewed in a geometric way. In particular the issue of information loss and the sampling properties of estimators are examined. Throughout a visual approach is taken in order to aid intuition and to give insight into the geometric theory

VECTOR AUTOREGRESSIVE PROCESSES WITH NONLINEAR TIME TRENDS IN COINTEGRATING RELATIONS

We extend the conventional cointegrated VAR model to allow for general nonlinear deterministic trends. These nonlinear trends can be used to model gradual structural changes in the intercept term of the cointegrating relations. A general asymptotic theory of estimation and statistical inference is reviewed and a diagnostic test for the correct specification of an employed nonlinear trend is developed. The methods are applied to Finnish interest-rate data. A smooth level shift of the logistic form between the own-yield of broad money and the short-term money market rate is found appropriate for these data. The level shift is motivated by the deregulation of issuing certificates of deposit and its inclusion in the model solves the puzzle of the “missing cointegration vector” found in a previous study.

Gaussian estimation of parametric spectral density with unknown pole

We consider a parametric spectral density with power-law behavior about a fractional pole at the unknown frequency $\omega$. The case of known $\omega$, especially $\omega =0$, is standard in the long memory literature. When $omega$ is unknown, asymptotic distribution theory for estimates of parameters, including the (long) memory parameter, is significantly harder. We study a form of Gaussian estimate. We establish $n$-consistency of the estimate of $\omega$, and discuss its (non-standard) limiting distributional behavior. For the remaining parameter estimates,we establish $\sqrt{n}$-consistency and asymptotic normality.

Information geometry and the quantum estimation problem

For the quantum estimation problem only a few of the advanced techniques of classical information geometry have been developed and applied. An asymptotic theory for parameter estimation needs geometric structures which are difficult to construct in the framework of quantum information geometry. It is proposed how a phase-space information geometry can be constructed, which would interpolate between the classical and quantum ones. To motivate this proposal a short review is presented of some current results on estimation and reconstruction of quantum states.

GMM with Weak Identification

This paper develops asymptotic distribution theory for GMM estimators and test statistics when some or all of the parameters are weakly identified. General results are obtained and are specialized to two important cases: linear instrumental variables regression and Euler equations estimation of the CCAPM. Numerical results for the CCAPM demonstrate that weak-identification asymptotics explains the breakdown of conventional GMM procedures documented in previous Monte Carlo studies. Confidence sets immune to weak identification are proposed. We use these results to inform an empirical investigation of various CCAPM specifications; the substantive conclusions reached differ from those obtained using conventional methods.

Stein–type shrinkage quantile estimation

The simultaneous asymptotic estimation theory of quantiles is considered for an arbitrary population. The Stein–type estimator and its positive version are considered. The relative merits of the proposed estimators are compared with those of the usual estimator using asymptotic quadratic distributional risk those of the usual estimator using asymptotic quadratic distributional risk under local alternatives. It is shown that both proposed estimators are asymptotically superior to the classical estimator. Further, it is demonstrated that the Stein-type estimator is dominated by its positive part

Non-linear regression with multidimensional indices

We consider a non-linear regression model when the index variable is multidimensional. Sufficient conditions on the non-linear function are given under which the least-squares estimators are strongly consistent and asymptotically normally distributed. These sufficient conditions are satisfied by harmonic type functions, which are also of interest in the one dimensional index case where Wu's (Asymptotic theory of non-linear least-squares estimation, Ann. Statist. 9 (1981) 501–513) and Jennrich's (Asymptotic properties of non-linear least-squares estimators, Ann. Math. Statist. 40 (1969) 633–643) sufficient conditions are not applicable.

Small Sample Properties of Certain Cointegration test Statistics: A Monte Carlo Study

This paper reports on the results of a Monte Carlo study. The latter investigates the performance of various versions of the Conformity test CCT$ for the existence and rank of cointegration, as given in Dhrymes (1996b), the likelihood ratio test LRT as given in Johansen (J) (1988), (1991), and the stochastic trends test (SW), as given in Stock and Watson (1988). The design of the experiments allows for small, medium and large stationary roots, and one, two, and three unit roots. The largest system investigated is a quadrivariate VAR(4). Results based on the underlying normal theory indicate that the performance of the CCT is extremely good when the null hypothesis involves the sum of, or individual, (characteristic) roots, some of which are not zero; it does not perform reliably when the sum of the roots under the null involves, in truth,all zero roots. Results based on non-standard asymptotic theory for estimators of zero roots indicate that the CCT has very good power characteristics in detecting the rank of cointegration, but it exhibits some size distortions that can potentially lead to overestimation of the true cointegrating rank. On the other hand, both versions are robust to non normal and dependent error structures. Such results generally hold for sample sizes 100 and 500. For samples of size 100, the LR test performs quite well, in terms of size, when the error process is Gaussian and when small and medium stationary roots are employed in the experimental design, but performs rather poorly in terms of power. The problem is magnified with large stationary roots, and/or non-normal errors. The results improve, as expected, for sample size 500. The SW test performs rather poorly overall, and cannot be recommended for use in empirical applications.

Asymptotic estimation theory for a finite-dimensional pure state model

The optimization of measurement for n samples of pure states are studied. The error of the optimal measurement for n samples is asymptotically compared with the one of the maximum likelihood estimators from n data given by the optimal measurement for one sample.

Asymptotic estimation theory for a finite-dimensional pure state model

The optimization of measurement for n samples of pure states are studied. The error of the optimal measurement for n samples is asymptotically compared with the one of the maximum likelihood estimators from n data given by the optimal measurement for one sample.

Asymptotic parameter estimation for multivariate point processes

The general asymptotic theory of parameter estimation for statistical experiments of any nature developed by A. Wald, L. Le Cam, J. Wolfowitz, J. H~ijek, and others is exposed in the monograph of I. A. Ibragimov and R. Z. Hasminskii [1]. In particular, for a family of regular experiments, they establish general conditions assuring the asymptotic efficiency of the maximum likelihood and Bayesian estimators. For concrete statistical experiments, it is important to obtain conditions that can be easily checked. Such conditions are obtained by Yu. A. Kutoyants for Poisson-type processes [4] and for the inmogeneous Poisson process [2, 3], and by Yu. N. Linkov for counting processes, renewal processes, and for semimartingales [5, 6]. In this paper we establish asymptotic properties of the maximum likelihood estimators and Bayesian estimators based on observations of a multivariate point process. In Sect. 1 we introduce the necessary notation from the theory of random processes and the statistical estimation theory. In this section, we also state the main result and prove it in Sect. 2. In Sect. 3 we indicate the changes in the conditions established in Sect. 2 for inhomogeneous Poisson processes and renewal processes.

A limit theory for long-range dependence and statistical inference on related models

This paper provides limit theorems for multivariate, possibly non-Gaussian stationary processes whose spectral density matrices may have singularities not restricted at the origin, applying those limiting results to the asymptotic theory of parameter estimation and testing for statistical models of long-range dependent processes. The central limit theorems are proved based on the assumption that the innovations of the stationary processes satisfy certain mixing conditions for their conditional moments, and the usual assumptions of exact martingale difference or the (transformed) Gaussianity for the innovation process are dispensed with. For the proofs of convergence of the covariances of quadratic forms, the concept of the multiple Fejér kernel is introduced. For the derivation of the asymptotic properties of the quasi-likelihood estimate and the quasi-likelihood ratio, the bracketing function approach is used instead of conventional regularity conditions on the model spectral density.

Parametric Methods for Bivariate Quantile‐Partitioned Tables and the Efficiency of Corresponding Nonparametric Methods

AbstractIn order to study the agreement between two continuous measurements on a sample of individuals, epidemiologists sometimes partition the original bivariate data into categories defined by the empirical quantiles of the two marginal distributions. The counts in the resulting two‐way contingency table have the empirical bivariate quantile‐partitioned (EBQP) distribution rather than the conventional multinomial distribution. BORKOWF et al. (1997) developed the asymptotic theory and inferential procedures for estimates of measures of agreement calculated from EBQP tables. In this paper, we develop parametric methods for bivariate quantile‐partitioned (BQP) tables. We use these methods to study the efficiency of nonparametric (EBQP) methods and to improve the precision of estimates of measures of agreement. We present computational methods for estimating certain measures of agreement (kappa, weighted kappa, and row proportions) together with their variances. Numerical studies and an example show that nonparametric (EBQP) estimates of kappa and certain row proportions tend to be quite inefficient compared to parametric estimates, whereas nonparametric estimates of weighted kappa can be relatively more efficient for some underlying distributions. Thus, investigators should weigh the efficiency advantages of parametric estimates against possible biases that can result from choosing an inappropriate parametric model when they employ BQP tables to study agreement.

Shrinkage estimation of the proportion in randomized response

This article considers the asymptotic estimation theory for the proportion in randomized response survey usinguncertain prior information (UPI) about the true proportion parameter which is assumed to be available on the basis of some sort of realistic conjecture. Three estimators, namely, the unrestricted estimator, the shrinkage restricted estimator and an estimator based on a preliminary test, are proposed. Their asymptotic mean squared errors are derived and compared. The relative dominance picture of the estimators is presented.