Showing papers on "Asymptotic distribution published in 1980"

The Lagrange Multiplier Test and its Applications to Model Specification in Econometrics

Abstract: Many econometric models are susceptible to analysis only by asymptotic techniques and there are three principles, based on asymptotic theory, for the construction of tests of parametric hypotheses. These are: (i) the Wald (W) test which relies on the asymptotic normality of parameter estimators, (ii) the maximum likelihood ratio (LR) procedure and (iii) the Lagrange multiplier (LM) method which tests the effect on the first order conditions for a maximum of the likelihood of imposing the hypothesis. In the econometric literature, most attention seems to have been centred on the first two principles. Familiar " t-tests " usually rely on the W principle for their validity while there have been a number of papers advocating and illustrating the use of the LR procedure. However, all three are equivalent in well-behaved problems in the sense that they give statistics with the same asymptotic distribution when the null hypothesis is true and have the same asymptotic power characteristics. Choice of any one principle must therefore be made by reference to other criteria such as small sample properties or computational convenience. In many situations the W test is attractive for this latter reason because it is constructed from the unrestricted estimates of the parameters and their estimated covariance matrix. The LM test is based on estimation with the hypothesis imposed as parametric restrictions so it seems reasonable that a choice between W or LM be based on the relative ease of estimation under the null and alternative hypotheses. Whenever it is easier to estimate the restricted model, the LM test will generally be more useful. It then provides applied researchers with a simple technique for assessing the adequacy of their particular specification. This paper has two aims. The first is to exposit the various forms of the LM statistic and to collect together some of the relevant research reported in the mathematical statistics literature. The second is to illustrate the construction of LM tests by considering a number of particular econometric specifications as examples. It will be found that in many instances the LM statistic can be computed by a regression using the residuals of the fitted model which, because of its simplicity, is itself estimated by OLS. The paper contains five sections. In Section 2, the LM statistic is outlined and some alternative versions of it are discussed. Section 3 gives the derivation of the statistic for

Logistic-normal distributions:Some properties and uses

Abstract: SUMMARY The logistic transformation applied to a ^-dimensional normal distribution produces a distribution over the d-dimensional simplex which can sensibly be termed a logistic-norma l distribution. Such distributions, implicitly used in a number of recent applications, are here given a formal identity and some useful properties are recorded. A main aim is to extend the area of application from the restricted role as a substitute for the Dirichlet conjugate prior class in the analysis of multinomial and contingency table data to the direct statistical description and analysis of compositional and probabilistic data.

An Empirical Investigation of Goodness-of-Fit Statistics for Sparse Multinomials

Abstract: Traditional discussions of goodness-of-fit tests for multinomial data consider asymptotic chi-squared properties under the assumption that all expected cell frequencies become large. This condition is not always satisfied, however, and another asymptotic theory must be considered. For testing a specified simple hypothesis, Morris (1975) and Hoist (1972) gave conditions for the asymptotic normality of the Pearson and likelihood ratio statistics when both the sample size and number of cells become large (even if the expected cell frequencies remain small). Monte Carlo techniques are used to examine the applicability of the normal approximations for moderate sample sizes with moderate numbers of cells.

A Reliability Distribution With Increasing, Decreasing, Constant and Bathtub-Shaped Failure Rates

Abstract: A distribution with one scale and two shape parameters is studied. The distribution can describe increasing (I), decreasing (D), constant and bathtub-shaped (B) failure rates. This motivates the working name, IDB distribution. The IDB distribution is motivated by mixtures of a set of IFR distributions but can also be given a competing risk interpretation. For mixed distributions a more general result on the initial slope of the failure rate is given. Asymptotic results for the ML estimation of survival probabilities are given, and when possible compared with ML estimation based on the Weibull, Rayleigh and exponential distributions. Also, the asymptotic gain from classifying failures into two categories is illustrated. One application to real data is given.

Stable discrete adaptive control

Abstract: The paper presents a proof of stability of the model reference adaptive control problem for the discrete case.

Interpreting the Likelihood Ratio Statistic in Factor Models When Sample Size is Small

Abstract: The use of the likelihood ratio statistic in testing the goodness of fit of the exploratory factor model has no formal justification when, as is often the case in practice, the usual regularity conditions are not met. In a Monte Carlo experiment it is found that the asymptotic theory seems to be appropriate when the regularity conditions obtain and sample size is at least 30. When the regularity conditions are not satisfied, the asymptotic theory seems to be misleading in all sample sizes considered.

Asymptotic normality of prediction error estimators for approximate system models

Abstract: A general class of parameter estimation methods for stochastic dynamical systems is studied. The class contains the least squares method, output-error methods, the maximum likelihood method and several other techniques. It is shown that the class of estimates so obtained are asymptotically normal and expressions for the resulting asymptotic covariance matrices are given. The regularity conditions that are imposed to obtain these results, are fairly weak. It is, for example, not assumed that the true system can be described within the chosen model set, and, as a consequence, the results in this paper form a part of the so-called approximate modeling approach to system identification. It is also noteworthy that arbitrary feedback from observed system outputs to observed system inputs is allowed and stationarity is not required

Uniform Asymptotic Normality of the Maximum Likelihood Estimator

Abstract: A very general result concerning the weak consistency and uniform asymptotic normality of the maximum likelihood estimator is presented. The result proves to be of particular value in establishing uniform asymptotic normality of randomly normalized maximum likelihood estimators of parameters in stochastic processes. The only conditions imposed are certain regularity conditions on the (random) information function, easily verified in practice. Application of the result is briefly considered.

Robust analysis of variance based upon a likelihood ratio criterion

Abstract: SUMMARY Robust tests of general linear hypotheses in linear models are developed. These are likeli- hood ratio type tests in the same sense that M-estimates are maximum likelihood type estimates. Construction of the tests suggests a decomposition of the data into terms analogous to classical sums of squares, providing a robust analysis of variance. Asymptotic efficiency and robustness properties of the tests are the same as those of the M-estimates upon which they are based. Parameter estimation is usually only a first step in the analysis of data arising fromn a linear model. A classical least squares analysis often focuses upon the analysis of variance, which tests simultaneous hypotheses on large subsets of the parameters. Since the terms in a classical analysis of variance are quadratic forms in least squares estimates, one would expect that the sensitivity of the estimates to departures from normality should be inherited by the tests. In fact, for moderate to heavy tailed error distributions or in the presence of outliers, it appears that the classical F test does lose power. Calculations of relative efficiency for proce- dures proposed in this paper substantiate on theoretical grounds the possible inefficiency and lack of power of classical F tests. In this paper, Huber's M-estimates are adapted to hypothesis tests which can be termed likelihood ratio type tests. These procedures naturally generalize and bear a striking re- semblance to classical F tests. Robustness and efficiency properties of M-estimates apply directly to the proposed tests. Hence the case to be made for using likelihood ratio type tests rather than classical F tests is the same as that for using M-estimates in favour of least squares estimates: possible poor performance of the classical methods may be overcome with methods which perform well both when classical assumptions are met and when they are not. The proposed methods are natural, intuitive and as easily computed as M-estimates.

A general correlation coefficient for directional data and related regression problems

Abstract: SUMMARY A correlation coefficient p2 iS proposed for bivariate angular distributions and for bivariate distributions on general manifolds. In the cylindrical case p2 iS the coefficient of Mardia (1976), and for the bivariate angular case it is a modified version of the correlation coefficient of Mardia & Puri (1978). Some properties of p2 are examined and compared with those of other bidirectional correlation coefficients. In particular, this coefficient is found to be closely connected with important exponential families of distributions. Further, the asymptotic distribution of the sample version of p2 under the hypothesis of independence does not depend on the marginal distributions. Thus it is asymptotically robust against concentration in the bivariate angular case. The regression models arising from complete dependence as measured by p2 are examined. A numerical example is given.

Asymptotic Distribution of Symmetric Statistics

Abstract: Sequences of $m$th order symmetric statistics are examined for convergence in law. Under appropriate conditions, a limiting distribution exists and is equivalent to that of a linear combination of products of Hermite polynomials of independent $N(0, 1)$ random variables. Connections with the work of von Mises, Hoeffding, and Filippova are noted.

A Simple Asymptotic Estimate for the Index of a Stable Distribution

Abstract: SUMMARY A simple asymptotic estimate is constructed for the index of a stable distribution based on order statistics from a distribution in its domain of attraction. The asymptotic distribution of the estimate is then found in case the order statistics are taken from the stable distribution itself.

Properties of Some Estimators for the Errors-in-Variables Model

Abstract: The limiting behavior of estimators for several errors-in-variables models is investigated. It is assumed that an estimator of the covariance matrix of the measurement error is available. Models are delineated on the basis of the prior knowledge of the error structure. In all cases the limiting distribution of the estimators, standardized by $n^{\frac{1}{2}}$, is normal. Modifications of the estimators that guarantee finite moments and improve the small sample behavior of the estimators are presented.

Autoregressive and window estimates of the inverse correlation function

Abstract: SUMMARY The concept of the inverse correlation function of a stationary process xt was first introduced by Cleveland (1972), who also introduced the autoregressive and the window methods for estimating this function. The asymptotic distribution of the estimates provided by these two methods is derived and their asymptotic covariance structure is shown to be in accordance with a remark of Parzen (1974). The results are extended to show that the two procedures suggested by Durbin (1959, 1961) for estimating the parameters of a moving average model are asymptotically efficient, relative to maximum likelihood in the Gaussian case. Some key word8: Akaike's information criterion; Autoregressive spectral estimate; Inverse correlation function; Inverse covariance function; Moving average model; Window spectral estimate.

The asymptotic-distribution of the scan statistic under uniformity

Abstract: The problem of testing uniformity on [0, 1] against a clustering alternative, is considered. Naus has shown that the generalized likelihood ratio test yields the scan statistic $N(d)$. The asymptotic distribution of $N(d)$ under the null hypothesis of uniformity is considered herein, and related to the version of the scan statistic defined for points from a Poisson process. An application of the above yields distributional results for the supremum of a stationary Gaussian process with a correlation function that is tent-like in shape, until it flattens out at a constant negative value.

Asymptotic Validity of F Tests for the Ordinary Linear Model and the Multiple Correlation Model

Abstract: In this article we establish under fairly general conditions that the F tests used in the linear model and the correlation model are asymptotically valid in the presence of nonnormality, in that their sizes are unaffected, asymptotically, by this nonnormality. Similar results could be derived for Scheffe-type simultaneous confidence intervals as well as the one-sided t tests used in these models. Finally, we find the asymptotic distribution of the sample variance and show why the size of a χ2 test about the variance for the linear model is not asymptotically valid in the presence of nonnormal errors.

The General Distribution of the Error Rate of a Classification Procedure With Application to Logistic Regression Discrimination

Abstract: The large-sample distribution of the error rate of an arbitrary estimator of the optimal classification rule is given. The asymptotic distribution of the logistic regression estimator is found. These results are used to show that the efficiency of logistic regression classification in some nonnormal cases is low. This suggests that maximum likelihood discrimination should be used whenever possible.

Testing for New Better than Used in Expectation with Incomplete Data

Abstract: This article proposes a test for the problem of testing that a life distribution is an exponential distribution against the alternative that it is new better than used in expectation, not exponential, on the basis of randomly right censored data. The test statistic is an analog of the “total time on test” test statistic.

Asymptotic normality of MRPP statistics from invariance principles of u-statistics

Abstract: Multi-response permutation procedures (MRPP) were recently introduced to test differences between a priori classified groups of objects ( Mielke, Berry Johnson, 1976; Mielke, 1979 ). The null distributions of the MRPP statistics were initially conjectured to be asymptotically normal for some specified conditions within the setting of a sequence of finite populations due to Madow ( 1948 ). Asymptotic normality of a class of MRPP statistics (under the null hypothesis) is shown in two cases: (i) the setting which considers the populations to be the samples resulting from sequential independent identically distributed (i.i.d.) sampling (sampling from infinite populations) and (ii) the setting of a sequence of increasingly large finite populations (sampling from finite populations). The results are direct applications of the weak convergence of a U-statistic process in the i.i.d. case to a Brownian motion (Bhattacharyya and Sen, 1977) and of the weak convergence of a U-statistic process in the finite populatio...

The Properties of Moment Estimators for the Weibull Distribution Based on the Sample Coeffkient of Variation

Abstract: Moment estimators k* and b* for the shape parameter, k, and scale parameter, b, of the twoparameter form of the Weibull distribution, are given. The estimators are based on the sample coefficient of variation and are shown to be asymptotically efficient with respect to the Cramer-Rao lower bound, and to be asymptotically normally distributed. The asymptotic distribution is determined and a table for calculating the estimators and their approximate variance-covariance matrix is given. Examples of the use of the method and the construction of large sample confidence intervals and confidence regions are given.

Averaging Methods for the Asymptotic Analysis of Learning and Adaptive Systems, with Small Adjustment Rate

Abstract: Recently proven theorems concerning weak convergence of nonMarkovian processes to diffusions, together with an averaging and a stability method, are applied to two (learning or adaptive) processes of current interest: (1) an automata model for route selection in telephone traffic routing; (2) an adaptive quantizer for use in the transmission of random signals in communication theory The models are chosen because they are prototypes of a large class to which the methods can be applied The technique of application of the basic theorems to such processes is developed Suitably interpolated and normalized “learning or adaptive” processes converge weakly to a diffusion, as the “learning or adaptation” rate goes to zero For small learning rates, the qualitative properties (eg, asymptotic (large-time) variances and parametric dependence) of the processes can be determined from the properties of the limit

Quantile Functions, Convergence in Quantile, and Extreme Value Distribution Theory.

Abstract: : The aim of this paper is to summarize the probability theory of quantile functions. The contributions of this paper are: (1) to emphasize the duality of quantile functions with distribution functions (sec. 1); (2) to explicitly define the notions of 'convergence in quantile' and 'convergence in r-mean quantile' (sec 2); (3) provide simple proofs of the distribution theory of extreme values (sec 4); and (4) emphasize the role of tail exponents of quantile functions and density-quantile functions in providing easy to apply criteria for the extreme value distributions corresponding to a specified distribution. (Author)

On the Treatment of Autocorrelated Errors in the Multiperiod Prediction of Dynamic Simultaneous Equation Models

Abstract: Estimation of parametric multiple time series models has been a major topic in recent work in statistics and econometrics (e.g., Hannan [1970], Wilson [1973], Dhrymes and Erlat [1974], and Hatanaka [1976]). However, relatively little has been reported on their prediction property, particularly when the model involves autocorrelated errors. For the model with uncorrelated errors, Goldberger, Nagar and Odeh [1962] and Dhrymes [1973] have obtained the asymptotic distribution of the reduced form coefficient estimates derived from the structural form estimates. It obviously serves as the asymptotic distribution of the one period ahead prediction of the model. Schmidt [1974] recently derived the asymptotic distribution of multiperiod ahead predictions for such a model, i.e., simultaneous equation autoregressive model (or alternatively called "dynamic model" in econometrics) with exogenous variables (ARX) (see also Brissimis and Gill [1978]). When the disturbances of the models are autocorrelated, it is known that the derivation of the simplified prediction scheme, not to mention its asymptotic distribution, becomes complicated even for the single equation model. Bloomfield [1972] and Yamamoto [1978] derived the asymptotic mean square error of one period and multiperiod predictions for the single equation autoregressive moving average (ARMA) models, respectively. In this paper, first we derive the optimal prediction scheme for multiperiod prediction of a simultaneous equation autoregressive model with exogenous variables, whose disturbances obey either autoregressive or moveing average process. The complication due to the error autocorrelation is handled by the introduction of the backward representation of the model, and the optimal predictor is given by a relatively simple formula with matrix notations. Secondly, for the unknown parameter case, we derive the asymptotic distribution of the optimal prediction scheme with the consistent estimates of the parameters. Our results are quite general, and we show, with a few examples, that they are easily modified to various single and simultaneous equation models of simpler specifications. The scope of this paper is as follows. Section 2 presents two types of model representation. The first is the state variable representation suggested by

The Asymptotic Distribution of the Sample Autocorrelations for an Integrated ARMA Process

Abstract: The behavior of the sample autocorrelation function, r(k), for an integrated autoregressive moving average time series is examined. The nonnormal asymptotic distribution of r(k) is characterized as a function of lag k and the parameters of the process. The validity of the approximation in moderate-sized samples is examined.