Showing papers on "Asymptotic distribution published in 1996"
TL;DR: In this paper, Monte Carlo simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA): Normal theory maximum likelihood )~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the Satorra-Bentler rescaled X 2(SB) under varying conditions of sample size, model specification, and multivariate distribution.
Abstract: Monte Carlo computer simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood )~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the Satorra-Bentler rescaled X 2 (SB) were examined under varying conditions of sample size, model specification, and multivariate distribution. For properly specified models, ML and SB showed no evidence of bias under normal distributions across all sample sizes, whereas ADF was biased at all but the largest sample sizes. ML was increasingly overestimated with increasing nonnormality, but both SB (at all sample sizes) and ADF (only at large sample sizes) showed no evidence of bias. For misspecified models, ML was again inflated with increasing nonnormality, but both SB and ADF were underestimated with increasing nonnormality. It appears that the power of the SB and ADF test statistics to detect a model misspecification is attenuated given nonnormally distributed data.
4,168 citations
TL;DR: In this paper, the asymptotic distribution of standard test statistics is described as functionals of chi-square processes, and a transformation based upon a conditional probability measure yields an asymptic distribution free of nuisance parameters, which can be easily approximated via simulation.
Abstract: Many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses. This paper studies the asymptotic distribution theory for such tests. The asymptotic distributions of standard test statistics are described as functionals of chi-square processes. In general, the distributions depend upon a large number of unknown parameters. We show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters, and we show that this transformation can be easily approximated via simulation. The theory is applied to threshold models, with special attention given to the so-called self-exciting threshold autoregressive model. Monte Carlo methods are used to assess the finite sample distributions. The tests are applied to U.S. GNP growth rates, and we find that Potter's (1995) threshold effect in this series can be possibly explained by sampling variation.
2,327 citations
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TL;DR: The authors employed response surface regressions based on simulation experments to calculate asymptotic distribution functions for the likelihood ratio tests for cointegration proposed by Johansen and provided tables of critical values that are very much more accurate than those available previously.
Abstract: This paper employs response surface regressions based on simulation experments to calculate asymptotic distribution functions for the likelihood ratio tests for cointegration proposed by Johansen The paper provides tables of critical values that are very much more accurate than those available previously However the principal contributions of the paper are a set of data les that contain estimated asymptotic quantiles obtained from response surface estimation and a computer program for utilizing them This program which is freely available via the Internet can easily be used to calculate asymptotic critical values and P values Graphs of some of the tabulated distribution functions are also provided An empirical example motivated by the European Economic and Monetary Union proposed in the Maastricht Treaty suggests that not all the countries of the European Union may qualify initially for participation in the EMU.
1,841 citations
TL;DR: In this article, the authors provide a framework for understanding the cross-section and time series approaches which have been used to test the convergence hypothesis and show how these alternative approaches make different assumptions on whether the data are well characterized by a limiting distribution, and how the choice of an appropriate testing framework is depend on both the specific null and alternative hypotheses under consideration as well as on the initial conditions characterizing the data being studied.
856 citations
TL;DR: In this paper, the authors provide a proof of the consistency and asymptotic normality of the quasi-maximum likelihood estimator in GARCH(1,1) and IGARCH (1, 1) models, showing that the presence of a unit root in the conditional variance does not affect the limiting distribution of the estimators.
Abstract: This paper provides a proof of the consistency and asymptotic normality of the quasi-maximum likelihood estimator in GARCH(1,1) and IGARCH(1,1) models. In contrast to the case of a unit root in the conditional mean, the presence of a unit root in the conditional variance does not affect the limiting distribution of the estimators ; in both models, estimators are normally distributed. In addition, a consistent estimator of the covariance matrix is available, enabling the use of standard test statistics for inference.
488 citations
TL;DR: In this paper, the authors present a study of extreme stock market price movements, using data for an index of the most traded stocks on the New York Stock Exchange for the period 1885-1990, showing empirically that the extreme returns obey a Frechet distribution.
Abstract: This article presents a study of extreme stock market price movements. According to extreme value theory, the form of the distribution of extreme returns is precisely known and independent of the process generating returns. Using data for an index of the most traded stocks on the New York Stock Exchange for the period 1885-1990, the author shows empirically that the extreme returns obey a Frechet distribution. Copyright 1996 by University of Chicago Press.
446 citations
TL;DR: Based on the kernel integrated square difference and applying a central limit theorem for degenerate V-statistic proposed by Hall (1984), the authors proposed a consistent nonparametric test of closeness between two unknown density functions under quite mild conditions.
Abstract: Based on the kernel integrated square difference and applying a central limit theorem for degenerate V-statistic proposed by Hall (1984), this paper proposes a consistent nonparametric test of closeness between two unknown density functions under quite mild conditions. We only require the unknown density functions to be bounded and continuous. Monte Carlo simulations show that the proposed tests perform well for moderate sample sizes.
390 citations
01 Jan 1996
TL;DR: The method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates, whereas previous stochastic analyses typically consider mean and mean square behavior.
Abstract: This paper presents an analysis of stochastic gradient-based adaptive algorithms with general cost functions. The analysis holds under mild assumptions on the inputs and the cost function. The method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates, whereas previous stochastic analyses typically consider mean and mean square behavior. The parameter estimates are shown to enter a small neighborhood about the optimum value and remain there for a finite length of time. Furthermore, almost sure exponential bounds are given for the rate of convergence of the parameter estimates. The asymptotic distribution of the parameter estimates is shown to be Gaussian with mean equal to the optimum value and covariance matrix that depends on the input statistics. Specific adaptive algorithms that fall under the framework of this paper are signed error least mean squre (LMS), dual sign LMS, quantized state LMS, least mean fourth, dead zone algorithms, momentum algorithms, and leaky LMS.
316 citations
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TL;DR: Two measures of sensitivity to initial conditions in nonlinear stochastic dynamic systems are proposed, one of which relates Fisher information with initial-value sensitivity in dynamical systems and a simple method for choosing the bandwidth is proposed.
Abstract: Using locally polynomial regression, we develop nonparametric estimators for the conditional density function and its square root, and their partial derivatives. Two measures of sensitivity to initial conditions in nonlinear stochastic dynamic systems are proposed, one of which relates Fisher information with initial-value sensitivity in dynamical systems. We propose estimators for these, and show asymptotic normality for one of them. We further propose a simple method for choosing the bandwidth. The methods are illustrated by simulation of two well-known models in dynamical systems.
277 citations
TL;DR: In this paper, the authors developed nonparametric estimators for the conditional density function and its square root, and their partial derivatives, and showed asymptotic normality for one of them.
Abstract: Using locally polynomial regression, we develop nonparametric estimators for the conditional density function and its square root, and their partial derivatives. Two measures of sensitivity to initial conditions in nonlinear stochastic dynamic systems are proposed, one of which relates Fisher information with initial-value sensitivity in dynamical systems. We propose estimators for these, and show asymptotic normality for one of them. We further propose a simple method for choosing the bandwidth. The methods are illustrated by simulation of two well-known models in dynamical systems.
274 citations
TL;DR: In this paper, the authors consider the non-Gaussian stable distributions as a model of financial returns and show that returns are often much more leptokurtic than is consistent with normality.
Abstract: Publisher Summary Financial asset returns are the cumulative outcome of a vast number of pieces of information and individual decisions arriving continuously in time. According to the Central Limit Theorem, if the sum of a large number of iid random variates has a limiting distribution after appropriate shifting and scaling, the limiting distribution must be a member of the stable class. It is therefore natural to assume that asset returns are at least approximately governed by a stable distribution if the accumulation is additive, or by a log-stable distribution if the accumulation is multiplicative. The Gaussian is the most familiar and tractable stable distribution, and therefore either it or the log-normal has routinely been postulated to govern asset returns. However, returns are often much more leptokurtic than is consistent with normality. This naturally leads one to consider also the non-Gaussian stable distributions as a model of financial returns.
TL;DR: In this article, the authors developed a noniterative, easily computed estimator of β for models in which some components of X are discrete, which is n ½ consistent and asymptotically normal.
Abstract: Others have developed average derivative estimators of the parameter β in the model E(Y|X = x) = G(xβ), where G is an unknown function and X is a random vector. These estimators are noniterative and easy to compute but require that X be continuously distributed. This article develops a noniterative, easily computed estimator of β for models in which some components of X are discrete. The estimator is n ½ consistent and asymptotically normal. An application to data on product innovation by German manufacturers illustrates the estimator's usefulness.
TL;DR: An algorithm is presented for estimating the distribution of a fit measure by drawing bootstrap samples from the model-expected proportions, the so-called nonnaive bootstrap method, from five different data sets, and results show that the asymptotic chi-square distribution is not at all valid for sparse data.
Abstract: When sparse data have to be fitted to a log-linear or latent class model, one cannot use the theoretical chi-square distribution to evaluate model fit, because with sparse data the observed cross-table has too many cells in relation to the number of observations to use a distribution that only holds asymptotically. The choice of a theoretical distribution is also difficult when model-expected frequencies are 0 or when model probabilities are estimated 0 or 1. The authors propose to solve these problems by estimating the distribution of a fit measure, using bootstrap methods. An algorithm is presented for estimating this distribution by drawing bootstrap samples from the model-expected proportions, the so-called nonnaive bootstrap method. For the first time the method is applied to empirical data of varying sparseness, from five different data sets. Results show that the asymptotic chi-square distribution is not at all valid for sparse data.
TL;DR: In this article, the asymptotic properties of the kernel estimate of sliced inverse regression are investigated, and it turns out that regardless of kernel function, the distribution remains the same for a wide range of smoothing parameters.
Abstract: To explore nonlinear structures hidden in high-dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression, Li and Duan proposed the sliced inverse regression (SIR) method which is simple to use. In this paper, the asymptotic properties of the kernel estimate of sliced inverse regression are investigated. It turns out that regardless of the kernel function, the asymptotic distribution remains the same for a wide range of smoothing parameters.
TL;DR: This paper obtained strong Bahadur representations for a general class of M-estimators that satisfy the condition that the random variables are independent but not necessarily identically distributed random variables.
Abstract: We obtain strong Bahadur representations for a general class of M-estimators that satisfies $\Sigma_i \psi (x_i, \theta) = o(\delta_n)$, where the $x_i$'s are independent but not necessarily identically distributed random variables The results apply readily to M-estimators of regression with nonstochastic designs More specifically, we consider the minimum $L_p$ distance estimators, bounded influence GM-estimators and regression quantiles Under appropriate design conditions, the error ratesobtained for the first-order approximations are sharp in these cases We also provide weaker and more easily verifiable conditions that suffice for an error rate that is suboptimal but strong enough for deriving the asymptotic distribution of M-estimators in a wide variety of problems
TL;DR: In this paper, the authors show that the restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations, and give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates in non-normal cases.
Abstract: The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. The result does not require normality or boundedness of the rank p of design matrix of fixed effects. Moreover, we give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates (MLE) in non-normal cases. As an application, we show for all unconfounded balanced mixed models of the analysis of variance the REML (ANOVA) estimates are consistent; and are also A.N. provided the models are nondegenerate; the MLE are consistent (A.N.) if and only if certain constraints on p are satisfied.
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TL;DR: Bootstrap as mentioned in this paper is a method for estimating the distribution of an estimator or test statistic by resampling one's data, which can be used to substitute computation for mathematical analysis if calculating the asymptotic distribution of a statistic or estimator is difficult.
Abstract: The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one's data. It amounts to treating the data as if they were the population for the purpose of evaluating the distribution of interest. Under mild regularity conditions, the bootstrap yields an approximation to the distribution of an estimator or test statistic that is at least as accurate as the approximation obtained from first-order asymptotic theory. Thus, the bootstrap provides a way to substitute computation for mathematical analysis if calculating the asymptotic distribution of an estimator or statistic is difficult. The maximum score estimator Manski (1975, 1985), the statistic developed by Ha..rdle et al. (1991) for testing positive- definiteness of income-effect matrices, and certain functions of time- series data (Blanchard and Quah 1989, Runkle 1987, West 1990) are examples in which evaluating the asymptotic distribution is difficult and bootstrapping has been used as an alternative.1 In fact, the bootstrap is often more accurate in finite samples than first-order asymptotic approximations but does not entail the algebraic complexity of higher-order expansions. Thus, it can provide a practical method for improving upon first-order approximations. First-order asymptotic theory often gives a poor approximation to the distributions of test statistics with the sample sizes available in applications. As a result, the nominal levels of tests based on asymptotic critical values can be very different from the true levels. The information matrix test of White(1982) is a well-known example of a test in which large finite- sample distortions of level can occur when asymptotic critical values are used (Horowitz 1994, Kennan and Neumann 1988, Orme 1990, Taylor 1987). Other illustrations are given later in this chapter. The bootstrap often provides a tractable way to reduce or eliminate finite- sample distortions of the levels of statistical tests.
TL;DR: In this article, the authors derive the asymptotic distribution of their estimators and show how to compute estimated standard errors, which can either be used alone or as prepivoting devices in bootstrap analysis.
Abstract: Cook and Stefanski have described a computer-intensive method, the SIMEX method, for approximately consistent estimation in regression problems with additive measurement error. In this article we derive the asymptotic distribution of their estimators and show how to compute estimated standard errors. These standard error estimators can either be used alone or as prepivoting devices in a bootstrap analysis. We also give theoretical justification to some of the phenomena observed by Cook and Stefanski in their simulations.
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01 Dec 1996TL;DR: In this article, the authors introduce a general definition of marginal log-linear parameters and describe conditions for a marginal loglinear parameter to be a smooth parameterization of the distribution and to be variation independent.
Abstract: Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. This paper introduces a general definition of marginal log-linear parameters and describes conditions for a marginal log-linear parameter to be a smooth parameterization of the distribution and to be variation independent. Statistical models defined by imposing affine restrictions on the marginal log-linear parameters are investigated. These models generalize ordinary log-linear and multivariate logistic models. Sufficient conditions for a log-affine marginal model to be nonempty and to be a curved exponential family are given. Standard large-sample theory is shown to apply to maximum likelihood estimation of log-affine marginal models for a variety of sampling procedures.
TL;DR: In this article, the authors consider a regression model in which the mean function may have a discontinuity at an unknown point and propose an estimate of the location of the discontinuity based on one-side nonparametric regression estimates of the mean functions.
Abstract: We consider a regression model in which the mean function may have a discontinuity at an unknown point. We propose an estimate of the location of the discontinuity based on one-side nonparametric regression estimates of the mean function. The change point estimate is shown to converge in probability at rate 0(n-1) and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models. Confidence regions for the location and size of the change are also discussed.
TL;DR: In this paper, an additive nonparametric regression model with known link function is proposed and the asymptotic distribution of this regression estimate is given, and the practical performance is investigated via an application to the study of migration between East and West Germany.
Abstract: SUMMARY We consider estimation of an additive nonparametric regression model with known link function. The asymptotic distribution of this regression estimate is given. The practical performance is investigated via an application to the study of migration between East and West Germany.
TL;DR: In this article, the estimation of the multivariate regression function m(x1, …, xd) = E[ψ(Yd)|X1 = x1, Xd = xd], and its partial derivatives, for stationary random processes Yi, Xi using local higher-order polynomial fitting was considered.
TL;DR: It is shown that the largest slope in a fixed size sample of slopes has an approximate Reverse Weibull distribution fitted to the largest slopes and the location parameter used as an estimator of the Lipschitz constant.
Abstract: A number of global optimisation algorithms rely on the value of the Lipschitz constant of the objective function. In this paper we present a stochastic method for estimating the Lipschitz constant. We show that the largest slope in a fixed size sample of slopes has an approximate Reverse Weibull distribution. Such a distribution is fitted to the largest slopes and the location parameter used as an estimator of the Lipschitz constant. Numerical results are presented.
TL;DR: In this article, a two-parameter long memory process (the Gegenbauer process) is considered and the estimator of one of the parameters converges faster than the usual O p (T 1 2 ) to the unit-root limiting distribution, i.e., a functional of Brownian motions.
TL;DR: In this paper, the authors prove the consistency of an estimator of the unknown parameters based on the periodogram and derive its asymptotic distribution for fractional ARIMA time series with innovations that have infinite variance.
Abstract: Consider the fractional ARIMA time series with innovations that have infinite variance. This is a finite parameter model which exhibits both long-range dependence (long memory) and high variability. We prove the consistency of an estimator of the unknown parameters which is based on the periodogram and derive its asymptotic distribution. This shows that the results of Mikosch, Gadrich, Kluppelberg and Adler for ARMA time series remain valid for fractional ARIMA with long-range dependence. We also extend the limit theorem for sample autocovariances of infinite variance moving averages developed in Davis and Resnick to moving averages whose coefficients are not absolutely summable.
TL;DR: This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η), H(0) : γ = γ(0), based on the pseudolikelihood L, where ϕ̂ is a consistent estimator of ϕ, the nuisance parameter.
Abstract: This paper concerns the asymptotic distribution of the likehood ratio statistic T for testing H 0 : θ = θ 0 based on the pseudolikelihood L(θ, φ), where φ is a simple estimator of φ. We show that the asymptotic distribution of T under H 0 is a weighted sum of indepepdent x 1 2 -variables where the weights involve the asymptotic joint covariance matrix of φ and the score function for θ. Some sufficient conditions are provided for the limiting distribution to be x 2 . The result is extended to allow θ 0 to be a boundary value of the θ parameter space, and φ to be misspecified in L(θ, φ). We also examine the issue of power loss when φ is misspecified in L(θ, φ). Several examples including variance component models, multivariate survival models, genetic linkage analysis and the Behrens-Fisher problem are presented to demonstrate the scope of the problems considered and to illustrate the results.
TL;DR: It is shown that an asymPTotically precise one-term correction to the asymptotic distribution function of the classical Cramйr-von Mises statistic approximates the exact distribution function remarkably closely for sample sizes as small as 7 or even smaller.
Abstract: It is shown that an asymptotically precise one-term correction to the asymptotic distribution function of the classical Cramйr-von Mises statistic approximates the exact distribution function remarkably closely for sample sizes as small as 7 or even smaller. This correction can be quickly evaluated, and hence it is suitable for the computation of practically exact p-values when testing simple goodness of fit. Abstract2 It is shown that an asymptotically precise one-term correction to the asymptotic distribution function of the classical Cramer-von Mises statistic approximates the exact distribution function remarkably closely for sample sizes as small as 7 or even smaller. This correction can be quickly evaluated, and hence it is suitable for the computation of practically exact $p$-values when testing simple goodness of fit. Similar findings hold for Watson's rotationally invariant modification, where a sample size of 4 appears to suffice.
TL;DR: Gill and Lewbel recently introduced a test for the rank of a matrix based on the LDU decomposition as mentioned in this paper, which is incorrect except in a very limited problem, and the test can be used to produce a valid test asymptotically equivalent to the minimum-X 2 test.
Abstract: Gill and Lewbel recently introduced a test for the rank of a matrix based on the LDU decomposition. Unfortunately, the asymptotic distribution suggested by them is incorrect except in a very limited problem. In general, the asymptotic distribution is that of a highly complicated nonlinear function of a normally distributed random vector that appears to defy useful characterization. The LDU decomposition can be used to produce a valid test asymptotically equivalent to the minimum-X 2 test.
TL;DR: In this article, asymptotic normality of certain classes of M-and R-estimators of the slope parameter vector in linear regression models with long memory moving average errors was discussed.
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TL;DR: In this paper, a vector conditional heteroscedastic autoregressive nonlinear (CHARN) model is proposed to estimate the conditional mean and the conditional variance (volatility) matrix of the past.
Abstract: We consider a vector conditional heteroscedastic autoregressive nonlinear (CHARN) model in which both the conditional mean and the conditional variance (volatility) matrix are unknown functions of the past. Nonparametric estimators of these functions are constructed based on local polynomial fitting. We examine the rates of convergence of these estimators and give a result on their asymptotic normality. These results are applied to estimation of volatility matrices in foreign exchange markets. Estimation of the conditional covariance surface for the Deutsche Mark/US Dollar (DEM/USD) and Deutsche Mark/British Pound (DEM/GBP) daily returns show negative correlation when the two series have opposite lagged values and positive correlation elsewhere. The relation of our findings to the capital asset pricing model is discussed.