Largest Characteristic Root Research Articles

On Fitting a Surface

This article deals with the problem of fitting the surface f=g (x) h(y) to the set of points (x/sub i/,y/sub j/,f/j). Functions g(x) and h(y) are supposed to be expressible in terms of orthonormal sets of functions. The desired coefficients of these functions are determined as characteristic vectors corresponding to the largest characteristic root of two materials having common characteristic roots.

Multi-step Sequential and Accelerated Sequential Methodologies for a Replicable Linear Model

We consider a sequence of observable p-dimensional independent and identically distributed N p (A θ, Σ) observations X l, X 2, … where A p × q is a known matrix with rank q(1<q<p), θ q × 1 consists of unknown regression parameters, and Σ is an unknown positive definite matrix. Chatterjee (Chatterjee, S.K. Multi-step sequential procedures for a replicable linear model with correlated variables. In Probability Statistics and Design of Experiments (Proceedings of the R. C. Bose Symposium 1988); Bahadur R.R., Ed.; Wiley Eastern: New Delhi, 1990; 217–226) gave a multi-step sequential methodology to construct a fixed-size confidence region for θ. This methodology used the largest characteristic root of an appropriate conditional dispersion matrix. We propose new multi-step sequential and accelerated methodologies, which are more efficient in lowering the oversampling rate substantially. Accelerated versions of these estimation techniques further reduce sampling operations and these preserve crucial properties associated with the original multi-step methodology of Chatterjee (1988). A substantial part of this investigation includes thorough comparisons of all available methodologies for moderate sample sizes. Some limited robustness issues are addressed. An illustration with realistic data on scores of college students on the college level examination program (CLEP) is also included.

Selective computation—V: The largest characteristic root of a matrix

IN MANY investigations, it is not necessary to find all the characteristic roots of a matrix. All that is required is the largest one. This is the problem we shall consider here. First, we shall consider symmetric matrices. In Section 2, we consider positive definite matrices. In Section 3, we show how to get better bounds with only a very slight increase in work. In Section 4, we consider the accuracy of the estimates. In Section 5, we consider the problem of finding the associated characteristic vector. In Section 6, we consider the Rayleigh quotient. In Section 7, we discuss the case where some of the characteristic roots may be negative. In Section 8, we show how these results may be applied to Sturm-Liouville problems. In Section 9, we give an application to integral equations. Next, we turn to non-negative matrices. By a non-negative matrix we mean one whose elements are non-negative. Matrices of this type arise in many investigations. Finally, we discuss the case of a general matrix. Analogous results for the Sturm-Liouville equation were given in [l]. The simple facts concerning matrix theory that we use may be found in [2].

Spectral Utility Functions and the Design of a Stationary System

abstract: the conventional approach to the design of stochastic systems operates, either directly or indirectly, by minimising the variance of the model. this procedure can be regarded as a natural extension of the stability analysis of a deterministic system, according to which the degree of stability is inversely related to the absolute value of its largest characteristic root. very often, however, the policymaker is not indifferent to the frequency composition of economic fluctuations. he may, for example, have a marked dislike for short-term fluctuations. we formalize this notion by setting up a spectral utility function as a criterion for steady-state optimisation, and show that the results from such an optimisation may conflict with those yielded by the conventional approach. large characteristic roots may not necessarily be bad! the scheme of the paper is as follows. because the ideas involved may be unfamiliar we shall spend some time on a rather intuitive motivation for what follows. this is done in section i. in section ii the notion of a spectral utility function is introduced and its evaluation discussed. we then return to the example of section i to give it a more precise treatment. section iii contains extensions, principally to the multivariate case.;

Asymptotic joint distribution of the largest roots of several multivariate F matrices I. Quasi-independent case

An asymptotic expansion of the joint distribution of k largest characteristic roots C M (i) ( S i S 0 −1), i = 1,…, k, is given, where S' i s and S 0 are independent Wishart matrices with common covariance matrix Σ. The modified second-approximation procedure to the upper percentage points of the maximum of C M (i) ( S i S 0 −1), i = 1,…, k, is also considered. The evaluation of the expansion is based on the idea for studentization due to Welch and James with the use of differential operators and of the perturbation procedure.

The Algebraic Equivalence of the Oberhofer-Kmenta and Theil-Boot Formulae for the Asymptotic Variance of a Characteristic Root of a Dynamic Econometric Model

IN AN important paper, Theil and Boot [4] developed a test for the stability of a dynamic econometric model. The test statistic is the (estimated) absolute value of the largest root of the model's characteristic equation. Theil and Boot made the test operational by providing a formula for the asymptotic variance of the test statistic. This formula involved the asymptotic covariance matrix of the estimated reduced form parameters.1 More recently, Oberhofer and Kmenta [3] provided a formula for the asymptotic variance of the same test statistic, which involved the asymptotic covariance matrix of the estimated structural parameters. This is, of course, more readily available than the covariance matrix of the reduced form estimates. This note shows that the two approaches are algebraically equivalent. That is, starting from a set of structural estimates, it makes no difference whether one applies the Oberhofer-Kmenta formula, or whether one calculates the derived reduced form and then uses the Theil-Boot formula; the results are exactly the same either way. 2. THE OBERHOFER-KMENTA FORMULA Let p represent the absolute value of the largest characteristic root, and p its estimate. Let 0 be the vector of structural coefficients, say of dimension N. Suppose that one has a consistent estimator 0 (say, the three-stage least squares estimate) and its (estimated) asymptotic covariance matrix. Then the Oberhofer-Kmenta result is just that (1) i=a.v. (( a.c. i=, 0j), where a.v. (p) represents the (estimated) asymptotic variance of p and a.c. (0i, 0) represents the

An application of the Rao-Blackwell theorem in preliminary test estimators

Preliminary test estimators are defined for estimating vector parameters. This note illustrates the use of the Rao-Blackwell theorem for uniformly reducing the risk of these estimators when one is based upon the sufficient statistic. The results given are valid for a class of risk functions, including the generalized mean squared error, the trace, and the largest characteristic root.

Sensitivity Comparisons among Tests of the General Linear Hypothesis

Abstract The expected significance level (ESL) of a test, proposed by Dempster and Schatzoff (4) as a criterion for comparing sensitivities of competing test statistics, is defined as the expected value of the observed significance level under a simple alternative hypothesis. For example, the ESL of a lower tailed test of the hypothesis H 0: X∼F (x) against H 1: X∼G (x) is given by ñF(x)dG(x). In this paper, the sensitivities of several proposed tests of the general linear hypothesis are compared by simulating their ESL's under a wide variety of parameterizations. The results indicate that Wilks's likelihood ratio criterion A and Hotelling's T 0 2 provide about the same levels of protection over a wide spectrum of alternatives, whereas Roy's largest characteristic root is shown to be rather insensitive to alternatives involving more than one non-zero root. Other proposed test statistics are shown to be less sensitive as judged by the ESL criterion.

A Mendelian Markov process with binomial transition probabilities

In any finite genetic population, the effect of chance involved in the sampling by which one generation replaces another causes gene frequency change as long as fixation or extinction is not achieved. In genetic terminology this fluctuation in gene frequency due to the finiteness of a population is called 'random genetic drift'. Fisher (1922), using an equation of the diffusion type, was the first to make an effort to describe quantitatively the phenomenon of random genetic drift. In the corrected version of this work (1930), he found the general result that the rate of decrease of the genetic variance in the population due to drift was 1/(2N) per generation, where N is the number of individuals of mating potential. This result was confirmed by various authors using different methods (e.g. Wright, 1931, 1937; Malecot, 1944; Feller, 1950). The particular form of the diffusion model which has had the widest use was introduced into genetical literature by Wright (1945). This equation which plays a fundamental role in the theory of gene frequencies is the Kolmogorov forward equation or-as it is commonly known among physicists-the Fokker-Planck equation. Earlier Kolmogorov (1935) had introduced the steady-state form of the forward equation in genetics. The complete solutions to the problem, however, were obtained much later and are due to Kimura (1955, 1957). The problem of random genetic drift as a problem in finite Markov chains was first considered by Malecot (1944). Based on the largest non-unit characteristic root of the transition probability matrix Malecot obtained the asymptotic rate of decrease of heterozygosity. This is essentially the same as the previously known results due to Fisher and Wright for steady decay. Feller (1951) succeeded in finding a general expression for all the characteristic roots of the transition probability matrix. The use of diffusion approximations requires the assumption of an infinite population: while in fact the state space is discrete, it is treated as though it were continuous. The tacit assumption in such an approach is that for large population size the approximation of the discrete time, discrete space process by the continuous diffusion process does not result in serious error. It was only recently that a rigorous justification of the diffusion approach was given by Watterson (1962) with respect to the processes considered by Wright (1945) and Kimura (1954, 1955). The validity of the diffusion approximation as providing adequate numerical solution to the problem concerning the probability that a particular gene is lost or fixed by a certain time was one of the questions considered by Ewens (1963). Since exact expressions were not available his method of getting exact results consisted in simply powering the transition matrix using a high-speed computer and collecting the appropriate terms.

On the Distribution of the Largest Characteristic Root of a Matrix in Multivariate Analysis

On the Distribution of the Largest Characteristic Root of a Matrix in Multivariate Analysis

On the Monotonic Character of the Power Functions of Two Multivariate Tests

The largest characteristic root has been proposed in [2] as a test statistic in (i) the multivariate analysis of variance test, and (ii) testing that two sets of variates are independent. In this paper it is shown that, in each case, the power function is a monotonically increasing function of each non-centrality parameter, separately. This property was stated in [2] without proof. This provides a stronger result than would be obtained by any direct use of Anderson's Theorem [1] which implies that the power function increases when all the roots are simultaneously increased in the same ratio. The proof of the monotonocity property for the multivariate analysis of variance is given in Section 3, and in Section 4 it is shown how the proof is modified for testing independence between two sets of variates.

Minimax Estimation for Linear Regressions

1. Introduction and Summary. When estimating the coefficients in a linear regression it is usually assumed that the covariances of the observations on the dependent variable are known up to multiplication by some common positive number, say c, which is unknown. If this number c is known to be less than some number k, and if the set of possible distributions of the dependent variable includes "enough" normal distributions (in a sense to be specified later) then the minimum variance linear unbiased estimators of the regression coefficients (see [1]) are minimax among the set of all estimators; furthermore these minimax estimators are independent of the value of k. (The risk for any estimator is here taken to be the expected square of the error.) This fact is closely related to a theorem of Hodges and Lehmann ([3], Theorem 6.5), stating that if the observations on the dependent variable are assumed to be independent, with variances not greater than k, then the minimum variance linear estimators corresponding to the assumption of equal variances are minimax. For example, if a number of observations are assumed to be independent, with common (unknown) mean, and common (unknown) variance that is less than k; and if, for every possible value of the mean, the set of possible distributions of the observations includes the normal distribution with that mean and with variance equal to k; then the sample mean is the minimax estimator of the mean of the distribution. The assumption of independence with common unknown variance is, of course, essentially no less general than the assumption that the covariances are known up to multiplication by some common positive number, since the latter situation can be reduced to the former by a suitable rotation of the coordinate axes (provided that the original matrix of covariances is non-singular). This note consideres the problem of minimax estimation, in the general "linear regression" framework, when less is known about the covariances of the observations on the "dependent variable" than in the traditional situation just described. For example, one might not be sure that these observations are independent, nor feel justified in assuming any other specific covariance structure. It is immediately clear that, from a minimax point of view, one cannot get along without any prior information at all about the covariances, for in that case the risk of every estimator is unbounded. In practice, however, one is typically willing to grant that the covarainces are bounded somehow, but one may not