Showing papers on "Asymptotic distribution published in 1971"

Inference about the change-point from cumulative sum tests

Abstract: SUMMARY The point of change in mean in a sequence of normal random variables can be estimated from a cumulative sum test scheme. The asymptotic distribution of this estimate and associated test statistics are derived and numerical results given. The relation to likelihood inference is emphasized. Asymptotic results are compared with empirical sequential results, and some practical implications are discussed. The cumulative sum scheme for detecting distributional change in a sequence of random variables is a well-known technique in quality control, dating from the paper of Page (1954) to the recent expository account by van Dobben de Bruyn (1968). Throughout the literature on cumulative sum schemes the emphasis is placed on tests of departure from initial conditions. The purpose of this paper is to examine a secondary aspect: estimation of the index T in a sequence {xt}, where the departure from initial conditions has taken place. The work is closely related to an earlier paper by Hinkley (1970), in which maximum likelihood estimation and inference were discussed. We consider specifically sequences of normal random variables x1, ..., xT, say, where initially the mean 00 and the variance o2 are known. A cumulative sum, cusum, scheme is used to detect possible change in mean from 00, and for simplicity suppose that it is a one-sided scheme for detecting decrease in mean. Then the procedure is to compute the cumulative sums t

Nonparametric Estimate of Regression Coefficients

Abstract: 1. Summary and introduction. The present investigation is a follow up of [7] to a class of multiple regression problems, and is devoted to the construction of an estimate of regression parameter vector based on suitable rank statistics. Asymptotic linearity of these rank statistics in the multiple regression set up is established and the asymptotic multi-normality of the derived estimates is deduced. There exists the choice of the score-generating function to every basic distribution so that the asymptotic distribution of the estimates is the same as that of maximallikelihood estimates.

The analysis of several 2× 2 contingency tables

Abstract: SUMMARY Consider data arranged into k 2 x 2 contingency tables. The principal result is the derivation of a statistical test for making an inference on whether each of the k contingency tables has the same relative risk. The test is based on a conditional reference set and can be regarded as an extension of the Fisher-Irwin treatment of a single 2 x 2 contingency table. Both exact and asymptotic procedures are presented. The analysis of k 2 x 2 contingency tables is required in several contexts. The two principal ones are (i) the comparison of binary response random variables, i.e. random variables taking on the values zero or one, for two treatments, over a spectrum of different conditions or populations; and (ii) the comparison of the degree of association among two binary random variables over k different populations. Cochran (1954) has investigated this problem with respect to testing if the success probability for each of two treatments is the same for every contingency table. Cochran's recommendation is that the equality of the two success probabilities should be tested using the total number, summed over all tables, of successes for one of the treatments. Cochran considers the asymptotic distribution of the total number of successes, for one of the treatments, conditional on all marginals being fixed in every table. He recommends this technique whenever the difference between the two populations on a logistic or probit scale is nearly constant for each contingency table. The constant logistic difference is equivalent to the relative risk being equal for each table. Mantel & Haenlszel (1959), in an important paper discussing retrospective studies, have also proposed an asymptotic method for analysing several 2 x 2 contingency tables. Their worlk on this problem was evidently done independently of Cochran, for their method is exactly the same as Cochran's except for a modification dealing with the correction factor associated with a finite population. Birch (1964) and Cox (1966) clarified the problem by showing, that under the assumption of constant logistic differences for each table, same relative risk, the conditional distribution of the total number of successes, for one of the treatments, leads to a uniformly most powerful unbiased test. Birch and Cox also derived the exact probability distribution of this conditional random variable under the given model. In this paper, we investigate the more general situation where the difference between the logits in each table is not necessarily constant. Procedures are derived for making an inference with regard to the hypothesis of constant logistic differences. Both the exact and asymptotic distributions are derived for the null and nonnull cases. This problem has been discussed by several investigators. A constant logistic difference corresponds to no interaction between the treatments and the k populations. The case k = 2 corresponds to one in which Bartlett (1935) has derived both an exact and an asymptotic procedure. Norton (1945)

Asymptotic Properties of Maximum Likelihood Estimators for the Independent Not Identically Distributed Case

Abstract: Conditions are established under which maximum likelihood estimators are consistent and asymptotically normal in the case where the observations are independent but not identically distributed. The key concept employed is uniform integrability; and the required convergence theorems which involve uniform integrability, and are of independent interest, appear in the appendix. A motivational example involving estimation under variable censoring is presented. This example invokes the full generality of the theorems with regard to lack of i.i.d. and lack of densities $\operatorname{wrt}$ Lebesgue or counting measure.

On the estimation of a harmonic component in a time series with stationary independent residuals

Abstract: the et are independently and identically distributed random variables each with mean zero and finite variance, and the gu(O) are specified functions of a vector-valued parameter 0. Whittle ( 1952) proposed an approximate least squares method of simultaneously estimating O and the angular frequencies, sine and cosine coefficients, of each harmonic term from observations (X1, ..., Xn) and derived heuristically the asymptotic distribution of the estimators. This paper presents rigorous proofs of Whittle's statements concerning the asymptotic distribution, formulated precisely as limit theorems, for the special case of independent residuals, where Xt = E(Xt) +et, so that the parameter 0 disappears. The arguments used here suggest how one can deal with the general case, and proofs for this will be given in a subsequent paper.

A chi-square statistic with random cell boundaries'

Abstract: In testing goodness of fit to parametric families with unknown parameters, it is often desirable to allow the cell boundaries for a chi-square statistic to be functions of the estimated parameter values. Suppose $M$ cells are used and $m$ parameters are estimated using BAN estimators based on sample. Then A. R. Roy and G. S. watson showed that in the univariate case the asymptotic null distribution of the chi-square statistic is that of $\sum^{m - m - 1}_1 Z^2_t + \sum^{m - 1}_{M - m} \lambda_t Z^2_t$, where $Z_t$ are independent standarad normal and the constants $\lambda_t$ lie between 0 and 1. They further observed that in the location-scale case the $\lambda_t$ are independent of the parameters if the cell boundaries are chosen in a natural way, and that in any case all $\lambda_t$ approach 0 as $M$ is appropraitely increased. We extend all of these results to the case of rectangular cells in any number of dimensions. Moreover, we give a method for numerical computation of the exact cdf of the asymptotic distribution and provde a short table of crticial points for testing goodness-of-fit to the univariate normal family.

The estimation of coherence and group delay

Abstract: SUMMARY The usual estimation of coherence is known to be badly biased towards zero if over the band of frequencies used for its estimation the phase is changing rapidly, as in the case of large group delay. The usual asymptotic distribution theory for spectral estimates is of no value in this circumstance, since it does not involve the group delay. A new theory is used to obtain estimates of coherence and group delay, which is itself of considerable physical interest, and the asymptotic distribution of the estimates is obtained.

The Bernstein-Von Mises Theorem for Markov Processes

Abstract: Since the appearance of P. Billingsley's monograph [2] on the large sample inference in Markov processes in which the weak consistency and asymptotic normality of the maximum likelihood estimate was investigated, there has been considerable interest in the further development of the theory along other directions. Billingsley's work was mainly concerned with extending the results of H. Cramer ([4] page 500). Among more recent developments one might mention the proof of the almost sure consistency of the maximum likelihood estimator following the ideas of A. Wald by G. Roussas [7], and the results on asymptotic Bayes estimates obtained by Lorraine Schwartz [9]. In the present paper we extend to Markov processes one of the fundamental results in the asymptotic theory of inference, viz., the approach of the posterior density (in a sense to be made precise later) to the normal. When the observed chance variables are independent and identically distributed, this result was obtained by L. LeCam in [5] (page 309). The same author offers another derivation of this result in [6]. Special cases of the theorem were first given by S. Bernstein and R. von Mises (for reference see [5]). The regularity conditions satisfied by the transition probability density of the Markov process are given in Section 1. We prove in Theorem 2.4 of Section 2 those properties of the maximum likelihood estimator that are needed for the proof of the main result of the paper given in Section 3 (Theorem 3.1). Theorem 3.1 is stated in a form which is general enough to include the Bernstein-von Mises theorem as well as the somewhat sharper versions that are available when it is known that the prior probability distribution has a finite absolute moment of order $m$. Theorem 3.2 deduces these results as a consequence of Theorem 3.1. The latter result also enables us to prove a theorem on the asymptotic behavior of regular Bayes estimates. This is done in Theorem 4.1 of Section 4.

Approximate solution of the equations for aerosol agglomeration

Abstract: The integrodifferential equation for the agglomeration and settling of aerosols is reduced to a set of ordinary nonlinear equations for the moments. In general these equations do not form a closed set. By choosing a specific functional form for the distribution of particle volumes, the parameters of which are functions of time, the system is closed since all the moments are expressible directly in terms of the chosen parameters, and an n-parameter function then leads to a set of n ordinary differential equations. The specific case of a log-normal distribution for particle sizes is discussed in detail. For the case of pure Brownian agglomeration analytic solutions are obtained and the asymptotic distribution is explored. When gravitational settling and agglomeration are included, the system may be readily solved numerically.

On Obtaining Large-Sample Tests from Asymptotically Normal Estimators

Abstract: This is an extension of Wald's asymptotic test procedure based on unrestricted maximum-likelihood estimators. Wald showed that under certain regularity conditions the test statistic has a limiting central chi-square distribution under the hypothesis and a limiting noncentral chi-square distribution under a sequence of local alternatives. We extend this procedure, allowing it to be based on a broader class of estimators and to obey simpler and less restrictive conditions. Sufficient conditions for validity of the limiting distributions are local twice-differentiability of the left side of the hypothesis and, under a sequence of local alternatives, asymptotic normality of the estimator of the parameter defining the distribution and stochastic convergence (to the appropriate asymptotic value) of the estimator of the covariance matrix. The required asymptotic behavior is verified for the case of independent sampling from two normal distributions and formulas are presented which aid in computing the test statistic.

Asymptotic Behavior of a Class of Confidence Regions Based on Ranks in Regression

Abstract: Asymptotic behavior of a class of confidence regions, based on rank statistics, for the regression parameter vector is considered. These regions are shown to be asymptotically bounded and ellipsoidic in probability. Asymptotic normality of their center of gravities is also proved. It is noted that the asymptotic efficiencies of these regions when defined in terms of ratio of Lebesgue measures corresponds to that of corresponding test statistics that are used to define these regions. Similar conclusion holds for their center of gravities, where now asymptotic efficiency is defined as inverse ratio of their generalized limiting variances. Also a class of consistent estimators is given for some functionals of the underlying distributions. Finally simultaneous confidence intervals, based on the above center of gravity, for linear parametric functions are shown to have asymptotic coverage probability $1 - \alpha$. Basic to this work are two papers, one by the author [4] and one by Jureckova [3].

Point Estimation of Forces of Transition in Demographic Models

Abstract: SUMMARY Occurrence/exposure rates are developed as estimators for the forces of transition of a homogeneous, time-continuous Markov process with at most countable state space in the case where the number of such forces is finite. Conditions for consistency and asymptotic normality of the estimators are stated.

The Exact Distribution of Wilks' Criterion

Abstract: 1. Summary. In this paper the exact distribution of Wilks' likelihood ratio criterion for testing linear hypothesis about regression coefficients is discussed. The exact distribution, in the most general case, is given in simple algebraic functions which can be computed without much difficulty. Explicit expressions for the density function as well as for the cumulative distribution function are given under the null hypothesis.

Distribution of the Canonical Correlations and Asymptotic Expansions for Distributions of Certain Independence Test Statistics

Abstract: The sample canonical correlations between two sets of variates are given a representation as the roots of a determinantal equation involving independent matrix variates having simple standardized distributions. This result is applied to obtain asymptotic formulas for the non-null distributions of three criteria for testing the hypothesis of independence of two sets of variates.

A problem in accelerated life testing

Abstract: Parameters of the Power Rule Model are estimated using data from censored sample life tests conducted at accelerated environments. By amending the functional form of the model, it is possible to obtain estimators that are asymptotically independent. The asymptotic normality of these estimators is ascertained by examining the shapes of their relative likelihood functions. Approximate confidence intervals, and an exact plausibility interval for the mean life (exponential case) at use conditions environment are obtained.

Linear Estimation of the Parameters of the Extreme-Value Distribution Based on Suitably Chosen Order Statistics

Abstract: Best linear unbiased estimates (BLUEs) based on a few order statistics are found for the location and scale parameters of the extreme-value distribution (Type-I asymptotic distribution of smallest values), when one or both parameters are unknown, such that the estimates have maximum efficiencies among the BLUEs based on the same number of order statistics. These estimates are then compared with the BLUEs and asymptotically best linear estimates (ABLEs) based on a few order statistics whose ranks were determined from the spacings that maximize the asymptotic efficiencies of the ABLEs. An application to the Weibull distribution is given.

A note on the asymptotic normality of the distribution of the number of empty cells in occupancy problems.

Abstract: : Two elementary proofs of the asymptotic normality of the distribution of the number of empty cells in occupancy problems are provided. (Author)

Statistical Inference in Random Coefficient Regression Models Using Panel Data

Abstract: In this chapter we modify Rao’s statistical theory to suit our study of non-experimental situations. We restrict ourselves to situations where we have panel data. We specify a regression equation with coefficients random across units but coming from the same multivariate distribution. We develop an estimation procedure which produces at least asymptotically efficient estimators of the parameters of the model.

Exact Finite Sample Density Functions of GCL Estimators of Structural Coefficients in a Leading Exactly Identifiable Case

Abstract: As is true of any statistical application, testing econometric statistical hypotheses involves constructing critical regions with prescribed probabilities of Type I error and determining probabilities of Type II error under alternative hypotheses. Such constructions and determinations presuppose knowledge of exact statistical distribution functions of econometric estimators and test statistics either directly, or indirectly, as in the case where a statistical distribution function that approximates the exact distribution with known margin of error is used. If this requisite information is not available, econometric statistical inference remains guesswork. Partly because of the complicated nature of systems of simultaneous econometric structural equations and the large numbers of structural constants required to characterize distribution functions of econometric statistics, however, only a few exact marginal distributions of econometric estimators and test statistics have been extracted so far. Th...

A test for the homogeneity of a mixture

Abstract: An aggregate is assumed to contain particles of different types and different sizes. The problem is to determine whether or not the various types of particles are thoroughly mixed. A test for homogeneity is presented and the asymptotic distribution of the test statistic is derived.

The Distribution of Linear Combinations of the Sample Occupancy Numbers.

Abstract: Assume that a random sample of n observations has been taken from the multinomial distribution with N equiprobable cells. Let si, i = 0, 1, 2, …, n, be the number of cells occurring i times in the sample. Let W=∑i=0nwisi be a linear combination of the si's. In this paper we obtain a sufficient condition under which the asymptotic distribution of W is normal as n and N tend to infinity such that n/N → α>0.