Showing papers in "Calcutta Statistical Association Bulletin in 1970"

Asymptotic Distribution of a Class of Multivariate Rank Order Statistics

Abstract: A VARIETY of rank tests for the multivariate multi-sample location and scale problems are now available in the literature. Chatterjee and Sen (1964, 1966) considered the median and the rank-sum tests for location. Puri and Sen (1966) extended the rank permutation idea of the previous authors to general Chernoff-Savage (1958) statistics and obtained the multivariate generalizations of the latter theorems (see also Tamura (1966)). Finally, Sen (1965) and Sugiura (1965) considered some other tests based on appropriate U-statistics. The rank statistics to be considered here are neither U-statistics nor statistics that satisfy the regularity conditions of the ChernoffSavage theorems. These of course include the median test statistics as a special case, and more generally, are useful for the censored data problem, where the censoring is made by the pooled sample quantiles. In this respcet, the theory developed here generalizes the results of Gastwirth (1965, 1966) and Sen (1967) to the multivariate case, though by an entirely different approach _based on the asymptotic behaviour of the empirical distribution functions. Multivariate generalization of a different type of tests for censored data (cf. Basu (1967)) follows more easily and is just appended as a remark.

An Alternative Derivation of the Distribution of the Direction and Collinearity Statistics in Discriminant Analysis

Abstract: or populations. This hypothesis, H, of goodness of fit of l'x consists of two aspects, (i) collinearity aspect, i.e., whether a single discriminant function could be adequate at all, and (ii) direction aspect, i.e., whether the single function which could discriminate adequately, agrees with the assigned function l'x. The name collinearity aspect follows from the fact that a single discriminant function is adequate only when the group means are collinear. Bartlett (1951) constructed an over-all criterion for testing H and then factorized it into two factors corresponding to the direction and collinearity aspects of H. Wilks' 1\ criterion is 1\ = I Wx I I I Bx+ Wx I and the 'residual' Wilks' 1\, when the hypothetical function is eliminated, is 1\R = 1\/{l'Wxlfl' (Bx+ Wx)l} = 1\//\1 ... (1.2)

Four and Six Level Second Order Rotatable Designs

Abstract: l. INTRODUCTION. Since the introduction of rotatable designs by Box and Hunter (1957) attempt to construct these designs have been made by a large number of researchers. In literature, no rotatable design exists which has four levels for each of the factors. In practice, situations often arise when designs with four levels are required. This calls for the construction of ro!atable designs which have four levels for each of the factors. In the present paper, we have given a method of construction of 4 level second order rotatable designs. A method of obtaining designs with factors at 6 levels each is also described.

On Response Surface Designs with Equi-Spaced Doses*

Abstract: THE second order rotatable designs (SORD) available in literature are not generally available with equi-spaced levels. Designs with equi-spaced levels are sometimes desired by the experimenter. The present paper aims at obtaining some second order response surface designs with equi-spaced levels. It has been shown that though the proposed designs are non-rotatable, they are more efficient at certain specific points than the corresponding rotatable designs. Two series of SORD with equi-spaced levels are also presented. For the exploration of response surfaces, Box and Hunter (1957) introduced second order rotatable designs (SORD). Mainly, four series of SORD are available in literature, viz.,

Invariant Problems of Linear Inference and Related Designs

Abstract: THE principle of in variance in the selection of inference procedures is wellknown [Lehmann, (1959)]. We shall consider here some linear inferential problems and their invariance under the symmetric group of permutations of (all or a subset of) the parameters occurring in such problems. It is first shown that the problem in general is not invariant underthis group. Next we characterise the ·particular form of the problem when it is invariant. Finally, these concepts are utilised to derive optimum designs for such invariant problems wrt various known optimality criteria, viz., A-, Dand £-optimality criteria [Kiefer (1958), (1959)]. More particularly, we shall be concerned with the following two problems: (a) derivation of linear inferential problems which are invariant under the symmetric group of permutations of {all or a subset of) the parameters; (b) derivation of optimum experimental designs for such invariant linear inferential problems wrt the various known optimality criteria listed above. We shall deal with the problem {a) in sections 2 to 4 while problem {b) will be treated in section 5.

On Some Tests in Multivariate Mixed Effects Model with Incomplete Data

Abstract: WHILE testing the hypotheses regarding various components of the multivariate fixed, random or mixed model, we generally assume that each of the observations is made on p characters ( p > 1). Under this situation standard tests are available (Anderson (1958), Roy and Gnanadesikan (1959), Gnanadesikan (1956), Roy and Roy (1958), Chakravorti (1968)). However, if it so happens that all these p characters can be measured at some levels, while at other levels some characters are omitted or the experimental conditions are such that they cannot be measured at these levels, then in this case the data remain incomplete. In this article we shall discuss some of the problems of testing of hypotheses in the later case stated above. To discuss the problems we have considered two-way classified multivariate mixed effects model with one observation per cell.

On the Equivalence of Confidence Sets Based on Bayesian and Non­Bayesian Procedures*

Abstract: SUPPOSE that we have a random samples of observations, s = (xl> . .. , Xn), on a continuous random variable with density function f(x, 8) depending on k unknown parameters 8 = (81 , ... , 8k). Our main object is to use the sample s to make probability statements relevant to the values of some k' ( < k) components of 8 which, with no loss of generality, we take to be (8~> ... , 8k·) = 0 (say). Consider the Bayesian and frequentist solutions to this problem. The Bayesian first specifies a prior distribution 1r(8) for 8 and combining this with the likelihood, arrives at a posterior distribution of 8 and hence to the appropriate marginal distribution of 0 from which he makes a probability statement of the form