Showing papers on "Asymptotic distribution published in 1994"

Chapter 36 Large sample estimation and hypothesis testing

Abstract: Asymptotic distribution theory is the primary method used to examine the properties of econometric estimators and tests We present conditions for obtaining cosistency and asymptotic normality of a very general class of estimators (extremum estimators) Consistent asymptotic variance estimators are given to enable approximation of the asymptotic distribution Asymptotic efficiency is another desirable property then considered Throughout the chapter, the general results are also specialized to common econometric estimators (eg MLE and GMM), and in specific examples we work through the conditions for the various results in detail The results are also extended to two-step estimators (with finite-dimensional parameter estimation in the first step), estimators derived from nonsmooth objective functions, and semiparametric two-step estimators (with nonparametric estimation of an infinite-dimensional parameter in the first step) Finally, the trinity of test statistics is considered within the quite general setting of GMM estimation, and numerous examples are given

Statistical tests for presence of cyclostationarity

Abstract: The presence of kth-order cyclostationarity is defined in terms of nonvanishing cyclic-cumulants or polyspectra. Relying upon the asymptotic normality and consistency of kth-order cyclic statistics, asymptotically optimal /spl chi//sup 2/ tests are developed to detect the presence of cycles in the kth-order cyclic cumulants or polyspectra, without assuming any specific distribution on the data. Constant false alarm rate tests are derived in both time- and frequency-domain and yield consistent estimates of possible cycles present in the kth-order cyclic statistics. Explicit algorithms for k/spl les/4 are discussed. Existing approaches are rather empirical and deal only with k/spl les/2 case. Simulation results are presented to confirm the performance of the given tests. >

Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions

Abstract: In this article, the construction of confidence regions by approximating the sampling distribution of some statistic is studied. The true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data. In the i.i.d. context, the method has been studied by Wu in regular situations where the statistic is asymptotically normal. The goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions. Essentially, all that is required is that the statistic, suitably normalized, possesses a limit distribution under the true model. Unlike the bootstrap, the convergence to the limit distribution need not be uniform in any sense. The method is readily adapted to parameters of stationary time series or, more generally, homogeneous random fields. For example, an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field.

The asymptotic variance of semiparametric estimators

Abstract: This paper derives a general formula for the asymptotic variance of semiparametric estimators that accounts for the presence of nonparametric estimators of functions. The general formula is specialized to show invariance of the asymptotic variance to the type of nonparametric estimator and to obtain correction terms for estimation of densities and mean-square projections (including conditional expectations). Regularity conditions for the validity of the formula are also given, including primitive conditions for asymptotic normality when series estimators are present. New examples considered include a semiparametric panel probit estimator and a series estimator of the average derivative. Copyright 1994 by The Econometric Society.

Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator

Abstract: This paper investigates the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH(1, 1) model. The rescaled variable (the ratio of the disturbance to the conditional standard deviation) is not required to be Gaussian nor independent over time, in contrast to the current literature. The GARCH process may be integrated (a + a = 1), or even mildly explosive (a + f > 1). A bounded conditional fourth moment of the rescaled variable is sufficient for the results. Consistent estimation and asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix.

A Residual-Based Test of the Null of Cointegration Against the Alternative of No Cointegration

Abstract: This paper proposes a residual-based test of the null of cointegration using a structural single equation model. It is shown that the limiting distribution of the test statistic for cointegration can be made free of nuisance parameters when the cointegrating relation is efficiently estimated. The limiting distributions are given in terms of a mixture of a Brownian bridge and vector Brownian motion. It is also shown that this test is consistent. Critical values are given for standard, demeaned, and detrended cases. Combining results from our test for cointegration with results from the Phillips-Ouliaris test for no cointegration, we find that there is evidence of cointegration between real consumption and real disposable income over the postwar period.

Least squares estimation of a shift in linear processes

Abstract: This paper considers a mean shift with an unknown shift point in a linear process and estimates the unknown shift point (change point) by the method of least squares. Pre-shift and post-shift means are estimated concurrently with the change point. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained when the magnitude of shift is small. It is shown that serial correlation affects the variance of the change point estimator via the sum of the coefficients (impulses) of the linear process. When the underlying process is an ARMA, a mean shift causes overestimation of its order. A simple procedure is suggested to mitigate the bias in order estimation.

Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests

Abstract: This article uses Monte Carlo experiments and response surface regressions in a novel way to calculate approximate asymptotic distribution functions for several well-known unit-root and cointegration test statistics. These allow empirical workers to calculate approximate P values for these tests. The results of the article are based on an extensive set of Monte Carlo experiments, which yield finite-sample quantiles for several sample sizes. Based on these, response surface regressions are used to obtain asymptotic quantiles for many different test sizes. Then approximate distribution functions with simple functional forms are estimated from these asymptotic quantiles.

Kernel Estimation of Partial Means and a General Variance Estimator

Abstract: Econometric applications of kernel estimators are proliferating, suggesting the need for convenient variance estimates and conditions for asymptotic normality. This paper develops a general “delta-method” variance estimator for functionals of kernel estimators. Also, regularity conditions for asymptotic normality are given, along with a guide to verify them for particular estimators. The general results are applied to partial means, which are averages of kernel estimators over some of their arguments with other arguments held fixed. Partial means have econometric applications, such as consumer surplus estimation, and are useful for estimation of additive nonparametric models.

Logarithmic asymptotics for steady-state tail probabilities in a single-server queue

Abstract: We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x-1 log P(W > x) -+ -0* as x - o0 for 0* > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Girtner-Ellis condition for the cumulant generating function of the associated partial sums, i.e. n-1 log Eexp (OSn) --+ (0) as n - oo, plus regularity conditions on the decay rate function 0. The asymptotic decay rate 0* is the root of the equation 0(0) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.

Asymptotics for semiparametric econometric models via stochastic equicontinuity

Abstract: This paper provides a general framework for proving the "square root of" T-consistency and asymptotic normality of a wide variety of semiparametric estimators. The class of estimators considered consists of estimators that can be defined as the solution to a minimization problem based on a criterion function that may depend on a preliminary infinite dimensional nuisance parameter estimator. The method of proof exploits results concerning the stochastic equicontinuity of stochastic processes. The results are applied to the problem of semiparametric weighted least squares estimation of partially parametric regression models. Primitive conditions are given for "square root of" T-consistency and asymptotic normality of this estimator. Copyright 1994 by The Econometric Society.

On the Asymptotics of Constrained $M$-Estimation

Abstract: Limit theorems for an $M$-estimate constrained to lie in a closed subset of $\mathbb{R}^d$ are given under two different sets of regularity conditions. A consistent sequence of global optimizers converges under Chernoff regularity of the parameter set. A $\sqrt n$-consistent sequence of local optimizers converges under Clarke regularity of the parameter set. In either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value. Limit theorems for the optimal value are also obtained, agreeing with Chernoff's result in the case of maximum likelihood with global optimizers.

A Consistent Test for a Unit Root

Abstract: This article investigates several U.S. macroeconomic time series for the presence of a unit root using a newly developed test. This test has stationarity as its null hypothesis, and the alternative is a unit-root process. The test is shown to be consistent, and its asymptotic null distribution is determined. Our findings contrast sharply with those obtained via the standard unit-root tests.

Estimating the Extremal Index

Abstract: SUMMARY The extremal index is an important parameter measuring the degree of clustering of process. The extremal index, a parameter in the interval [0, 1], is the reciprocal of the mean cluster size. Apart from being of interest in its own right, it is a crucial parameter for determining the limiting distribution of extreme values from the process. In this paper we review current work on statistical estimation of the extremal index and consider an optimality criterion based on a bias-variance trade-off. Theoretical results are developed for a simple doubly stochastic process, and it is argued that the main formula obtained is valid for a much wider class of processes. The practical implications are examined through simulations and a real data example.

On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity

Abstract: . Time series with a changing conditional variance have been found useful in many applications. Residual autocorrelations from traditional autoregressive moving-average models have been found useful in model diagnostic checking. By analogy, squared residual autocorrelations from fitted conditional heteroskedastic time series models would be useful in checking the adequacy of such models. In this paper, a general class of squared residual autocorrelations is defined and their asymptotic distribution is obtained. The result leads to some useful diagnostic tools for statisticians using conditional heteroskedastic time series models. Some simulation results and an illustrative example are also reported.

Chapter 45 Estimation and inference for dependent processes

Abstract: This chapter provides an overview of asymptotic results available for parametric estimators in dynamic models. Three cases are treated: stationary (or essentially stationary) weakly dependent data, weakly dependent data containing deterministic trends, and nonergodic data (or data with stochastic trends). Estimation of asymptotic covariance matrices and computation of the major test statistics are covered. Examples include multivariate least squares estimation of a dynamic conditional mean, quasi-maximum likelihood estimation of a jointly parameterized conditional mean and conditional variance, and generalized method of moments estimation of orthogonality conditions. Some results for linear models with integrated variables are provided, as are some abstract limiting distribution results for nonlinear models with trending data.

The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family

Abstract: We here consider testing the hypothesis ofhomogeneity against the alternative of a two-component mixture of densities. The paper focuses on the asymptotic null distribution of 2 log λ n , where λ n is the likelihood ratio statistic. The main result, obtained by simulation, is that its limiting distribution appears pivotal (in the sense of constant percentiles over the unknown parameter), but model specific (differs if the model is changed from Poisson to normal, say), and is not at all well approximated by the conventional χ (2) 2 -distribution obtained by counting parameters. In Section 3, the binomial with sample size parameter 2 is considered. Via a simple geometric characterization the case for which the likelihood ratio is 1 can easily be identified and the corresponding probability is found. Closed form expressions for the likelihood ratio λ n are possible and the asymptotic distribution of 2 log λ n is shown to be the mixture giving equal weights to the one point distribution with all its mass equal to zero and the χ2-distribution with 1 degree of freedom. A similar result is reached in Section 4 for the Poisson with a small parameter value (θ≤0.1), although the geometric characterization is different. In Section 5 we consider the Poisson case in full generality. There is still a positive asymptotic probability that the likelihood ratio is 1. The upper precentiles of the null distribution of 2 log λ n are found by simulation for various populations and shown to be nearly independent of the population parameter, and approximately equal to the (1–2α)100 percentiles of χ (1) 2 . In Sections 6 and 7, we close with a study of two continuous densities, theexponential and thenormal with known variance. In these models the asymptotic distribution of 2 log λ n is pivotal. Selected (1−α) 100 percentiles are presented and shown to differ between the two models.

Nonparametric Estimation of Mean Functionals with Data Missing at Random

Abstract: This article considers a distribution-free estimation procedure for a basic pattern of missing data that often arises from the wellknown double sampling in survey methodology. Without parametric modeling of the missing mechanism or the joint distribution, kernel regression estimators are used to estimate mean functionals through empirical estimation of the missing pattern. A generalization of the method of Cheng and Wei is verified under the assumption of missing at random. Asymptotic distributions are derived for estimating the mean of the incomplete data and for estimating the mean treatment difference in a nonrandomized observational study. The nonparametric method is compared with a naive pairwise deletion method and a linear regression method via the asymptotic relative efficiencies and a simulation study. The comparison shows that the proposed nonparametric estimators attain reliable performances in general.

The Complex Bingham Distribution and Shape Analysis

Abstract: SUMMARY A complex version of the Bingham distribution is defined on the unit complex sphere in Ck. Various statistical properties, including asymptotic normality under high concentration, are derived. Symmetries in the distribution make it a natural tool for the analysis of the shape of landmark data in two dimensions. Strengths and weaknesses of this approach are investigated. Links and contrasts with Bookstein co-ordinates are discussed. A hybrid approach to principal component analysis based on complex and real co-ordinates is suggested.

Maximal Inequalities for Degenerate $U$-Processes with Applications to Optimization Estimators

Abstract: Maximal inequalities for degenerate $U$-processes of order $k, k \geq 1$, are established. The results rest on a moment inequality (due to Bonami) for $k$th-order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes. Rates of uniform convergence are obtained. The maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having $U$-process structure. As an application, a semiparametric regression estimator that maximizes a $U$-process of order 3 is shown to be $\sqrt n$-consistent and asymptotically normally distributed.

Robust Non-parametric Function Estimation

Abstract: The bias of kernel methods based on local constant fits can have an adverse effect when the derivative of the marginal density or that of the regression function is large. The drawback can be repaired by considering a class of kernel estimators based on local linear fits. These estimators have the desired asymptotic properties and can be used to estimate conditional quantiles and to robustify the usual mean regression. The conditional asymptotic normality of these estimators at both boundary and interior points is established. An important consequence of the study is that the proposed method has the desired sampling properties at both boundary and interior points of the support of the design density. Therefore, our procedure does not require boundary modifications. Applications of such a local linear approximation method are discussed.

Nearest Neighbor Estimation of a Bivariate Distribution Under Random Censoring

Abstract: We consider the problem of estimating the bivariate distribution of the random vector $(X, Y)$ when $Y$ may be subject to random censoring. The censoring variable $C$ is allowed to depend on $X$ but it is assumed that $Y$ and $C$ are conditionally independent given $X = x$. The estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of $Y$ given $X = x$ over a range of values of $x$. The weak convergence of the centered estimator multiplied by $n^{1/2}$ is obtained, and a closed-form expression for the covariance function of the limiting process is given. It is shown that the proposed estimator is optimal in the Beran sense. This is similar to an optimality property the product-limit estimator enjoys. Using the proposed estimator of the bivariate distribution, an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established.

Asymptotic filtering theory for univariate ARCH models

Abstract: This paper builds on this earlier work by deriving the asymptotic distribution of the measurement error. This allows us to approximate the measurement accuracy of ARCH conditional variance estimates and compare the efficiency achieved by different ARCH models. We are also able to characterize the relative importance of different kinds of misspecification; for example, we show that misspecifying conditional means adds only trivially (at least asymptotically) to measurement error, while other factors (for example, capturing the "leverage effect," accommodating thick tailed residuals, and correctly modelling the variability of the conditional variance process) are potentially much more important. Third, we are able to characterize a class of asymptotically optimal ARCH conditional variance estimates.

Consistent and Asymptotically Normal Parameter Estimates for Hidden Markov Models

Abstract: Hidden Markov models are today widespread for modeling of various phenomena. It has recently been shown by Leroux that the maximum-likelihood estimate (MLE) of the parameters of a such a model is consistent, and local asymptotic normality has been proved by Bickel and Ritov. In this paper we propose a new clans of estimates which are consistent, asymptoti-cally normal and almost as good an the MLE

Asymptotic Properties of Estimators for the Parameters of Spatial Inhomogeneous Poisson Point Processes

Abstract: Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramer–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes

Abstract: We consider point processes defined through a pairwise interaction potential and admitting a two-dimensional sufficient statistic It is shown that the pseudo maximum likelihood estimate can be stochastically normed so that the limiting distribution is a standard normal distribution This result is true irrespectively of the possible existence of phase transitions The work here is an extension of the work Guyon and Kunsch (1992,Lecture Notes in Statist,74, Springer, New York) and is based on viewing a point process interchangeably as a lattice field

U-Processes in the Analysis of a Generalized Semiparametric Regression Estimator

Abstract: We prove -consistency and asymptotic normality of a generalized semiparametric regression estimator that includes as special cases Ichimura's semiparametric least-squares estimator for single index models, and the estimator of Klein and Spady for the binary choice regression model. Two function expansions reveal a type of U-process structure in the criterion function; then new U-process maximal inequalities are applied to establish the requisite stochastic equicontinuity condition. This method of proof avoids much of the technical detail required by more traditional methods of analysis. The general framework suggests other -consistent and asymptotically normal estimators.