Showing papers in "Annals of the Institute of Statistical Mathematics in 1969"

Fitting autoregressive models for prediction

Abstract: This is a preliminary report on a newly developed simple and practical procedure of statistical identification of predictors by using autoregressive models. The use of autoregressive representation of a stationary time series (or the innovations approach) in the analysis of time series has recently been attracting attentions of many research workers and it is expected that this time domain approach will give answers to many problems, such as the identification of noisy feedback systems, which could not be solved by the direct application of frequency domain approach [1], [2], [3], [9].

Nonparametric estimation in markov processes.

Abstract: The problem of statistical inferences in Markov processes has received considerable a t tent ion during the last fifteen years Much of the work consists in carrying over to the Markov case the maximum likelihood and chi-square methods from processes with independent identically dist r ibuted random variables (See, for example, [1] and other references cited there) Alternat ive approaches have also been adopted [11], some of which [7] refer to statistical inferences in more general processes I t is not long ago that presumably the first paper [10] appeared on nonparametr ic estimation of the density in the case of independent identically distr ibuted random variables Soon a number of others ([14], [8], [13], [3], [6]) followed, which by using either similar or different methods obtained fur ther results The purpose of the present paper is to consider the nonparametric estimation of densities in the case of Markov processes The methods being used and results being obtained here are similar to those in [9] Wha t we do specifically here is th is : We first construct asymptotically unbiased est imates for the initial and (two-dimensional) joint densities This is done in section 2 In section 3 these est imates are shown to be consistent in quadrat ic mean, and fur thermore a consistent, in the probability sense, est imate for the transit ion density is obtained Finally, it is proved in section 4 that , under suitable conditions, all three est imators mentioned, properly normalized, are asymptotically normal The appropriate versions of the Central Limit Theorem which are used for this purpose are s ta ted and proved in an appendix, so tha t the continuity of the paper will not be interrupted

The calculation of cumulants via conditioning

Abstract: The cumulants of random variables are important in deriving, for statistics of interest, exact sampling distributions, approximate sampling distributions (as via Cornish-Fisher expansions) and asymptotic sampling distributions (such as asymptotic normality). This note presents a means of calculating cumulants when two or more stages of sampling may be recognized. Given the k-variate random variable (x:, . . . , x~), let A denote an event in the associated probability field. The following properties of first and second order cumulants are well known (see Hansen, Hurwitz and Madow ([4], pp. 61-66) or Feller ([3], p. 164)).

Non-central distributions ofith largest characteristic roots of three matrices concerning complex multivariate normal populations

Abstract: In the case of complex mult ivariate normal distributions (see Goodman [2]), the classical problems concerning MANOVA model, canonical correlation coefficients and covariance model were studied by Khatr i [5] and James [3]. The non-central distributions of the characterist ic (ch.) roots concerning the various test procedures were explicitly given by James [3] in terms of zonal polynomials for complex hermit ian matrices, while Khatr i [5] gave them in integral forms. Here, we shall use the results of James [3] and obtain the distributions of the i th largest ch. roots of three matrices for the three situations mentioned above and for bivariate case, the covariance model is studied in some detail. Incidently, it may be noted that A1-Ani, Sabri [1] obtained the distribution of i th largest ch. root in the null hypothesis on the lines similar to Khatr i [4] where the distribution of the largest ch. root is given under null hypotheses.

A representation of bayes invariant procedures in terms of haar measure

Abstract: It is shown that under fairly general conditions the Bayes procedure among the class of procedures invariant under a group of transformations which leave the statistical problem invariant, is really a Bayes or formal Bayes procedure with respect to a prior measure which is constructed from the right Haar measure on the group and the specified prior. This result is useful in problems where the principle of invariance is applied. Such cases, involving the two action problem or a selection and ranking problem are given as examples.

On the distribution of the latent roots of a positive definite random symmetric matrix I

Abstract: Many problems in multivariate analysis trr~tolve t~e distribution problems of the latent roots of positive definite random symmetric matrices. In particular, the distributions of the latent roots of a Wishart matrix and those of a multivariate quadratic form are very fundamental in the normal multivariate case. In this paper, we shall give the density functions of the following statistics composed of the latent roots of a non-central Wishart matrix and of a non-central multivariate quadratic form. ( i ) The latent roots of the determinantal equations det (~-2XX ~)=0 and det(X-2XAXP)=0 (sections 5.1 and 7.1), (ii) the maximum latent root (sections 5.2 and 7.2), (iii) the traces (sections 5.3 and 7.3). To treat the distribution problems of the latent roots of a non~ central Wishart matrix, we shall introduce a generalized Hermite polynomial with a matrix argument, discuss some properties of it and give its generating function (section 3). We shall also introduce a new function which is appropriate to discuss the distribution of a non-central multivariate quadratic form (section 6).

A note on exponential bounds for binomial probabilities

Abstract: 1 1 one sees immediately where 0 -0. Putting c=-~, p = 3 that (1)does not hold in the stated generality. An inspection of the proof of that inequality shows that the argument of symmetry does not go through. On the other hand, for most of the values (p, c) for which (1) holds, the inequality can be sharpened considerably. It may be remarked that values c>=l-p are of no interest for the considered problem and that the right-hand side of (1) should be independent of p. THEOREM. Let 0__ = 2d + 4c4 I

Power comparisons of tests of two multivariate hypotheses based on individual characteristic roots

Abstract: In this paper, power comparisons are made for tests of each of the following two hypotheses based on individual characteristic roots of a matrix arising in each case: (i) independence between ap-set and aq-set of variates in a (p+q)-variate normal population withp≦q and (ii) equality ofp-dimensional mean vectors ofl p-variate normal populations having a common covariance matrix. At first, a few lemmas are given which help to reduce the central distributions of the largest, smallest, second largest, and the second smallest roots in terms of incomplete beta functions or functions of them. Since the central distribution of the largest root has been discussed by Pillai earlier in several papers ([6], [8], [9], [11], [12], [13]) cdf’s of the three others in the central case are given. Further, the non-central distributions of the individual roots forp-3 are considered for the two hypotheses and that of the smaller root forp=2; that of the largest root forp=2 has been obtained by Pillai earlier, (Pillai [11], Pillai and Jayachandran [14]).

Testing for equality of means, equality of variances, and equality of covariances under restrictions upon the parameter space

Abstract: Abstract : Suppose that the p-dimensional random row vector x has a multivariate normal distribution with mean mu and covariance matrix Sigma. Wilks (Ann. Math. Statist. 17 (1946)) has derived likelihood-ratio tests of the hypothesis H sub mvc and H sub vc against general alternatives H. Here H sub mvc is the hypothesis that the components of the mean vector mu are equal (mu = eta(1, 1, ..., 1) is identical with eta e, where eta is an unknown constant) and that Sigma has the intraclass correlational form Sigma = sigma squared ((1-rho)I + rho e e prime). H sub vc is the hypothesis that Sigma has the intraclass correlational form, mu unrestricted; H is the hypothesis that mu is unrestricted and Sigma is unrestricted, positive definite. In the present paper, we consider likelihood ratio tests of the hypotheses Hr sub mvc and Hr sub vc against general alternatives H, where Hr sub mvc and Hr sub vc differ from H sub mvc and H sub vc, respectively, by constricting the common correlation to lie in an interval rho sub 0 = or < rho < 1. Such tests have practical importance in psychological testing theory, in the analysis of growth curves, and in other contexts.

On nonparametric T-method of multiple comparisons for randomized blocks

Abstract: Some nonparametric generalizations of Tukey’s [9]T-method of multiple comparisons are considered for randomized blocks and the allied efficiency results are studied. For this, the distribution theory of aligned rank order statistics developed in [6], [7] is extended for multiple comparisons along the lines of [5] which deals with one-way layouts.

Distribution of product and quotient of Bessel function variates

Abstract: In this paper, the technique of Mellin transforms is employed to obtain the distribution of the product and quotient of two independent Bessel function random variables. Two different types of Bessel function variates are considered. The results are then specialized to yield a wide variety of classical distributions of importance in applications.

A note on weighing designs

Abstract: Banerjee [1], [2] has shown that the arrangements afforded by a Balanced Incomplete Block Design can be used as an efficient spring balance design. Such designs suffer from one drawback viz., there are only a few or no degrees of freedom left for the estimation of error-variance,σ2. To overcome this difficulty, it has been suggested that the whole design may be repeated a certain number of times to get an estimate of the error variance. In the present note an attempt has been made to give an alternative design where there is no necessity of such repetition. It has been also shown that these designs give a lesser variance of the estimated weights than the repeated design.

On the asymptotic theory of rank order tests for experiments involving paired comparisons

Abstract: The problem of testing the hypothesis of no difference among several treatments for the case when the comparisons between the treatments is possible only in pairs has been considered by Durbin [3], Bradley and Terry [1], Elteren and Noether [4], and Mehra and Puri [6], among others. Following the lines of Sen and Puri [9], a new approach to the asymptotic theory of rank order tests for this problem is developed. This avoids the unnecessarily complicated and lengthy conditional approach of Mehra and Puri [6] and also simplifies the proofs considerably.

Procedures for a best population problem when the criterion of bestness involves a fixed tolerance region

Abstract: In this paper we supply tables of constants necessary to use the procedures developed in [4]. We also present a new procedure for one of the cases not discussed in that paper, as well as a proof that it is parameter-free at levelP*.

On the costwise optimality of hierarchical multiresponse randomized block designs under the trace criterion

Abstract: : Consider the class of general incomplete multiresponse (GIM) designs in which the set of units is divided into blocks of equal size, such that in any block the same subset of responses is measured on each unit. It is shown that with respect to the trace criterion and a reasonable cost restriction, the subclass of hierarchical multiresponse (HM) designs is complete in the sense that given any GIM design, there exists a HM design such that the cost involved under the two designs is the same, but the trace of the covariance matrix of the estimates of the parameters under the HM design is less than or equal to the similar quantity under the GIM design. The results also establish the important fact that there is a large class of situations where the standard multiresponse model (under which all responses are measured on each unit) should not be used. The nonlinear programming problem associated with obtaining the optimum HM design is stated and solved. (Author)