Calcutta Statistical Association Bulletin

A brief review is given of five different interpretations of Bayesian statistical inference.

AMS {2000} Subject Classification: 62A01.

Although measures of diversity and inequality have been extensively proposed in socio-economic and health perspectives, violation of a subtle monotonicity criterion (under stochastic ordering) diminishes their rationality and utility in the context of poverty and other ecomomic indexes. A modification is proposed here to develop certain stochastic ordering monotonicity preserving diversity measures that overcome this drawback. Such measures are shown to be invariant under increasing transformation, and thereby appropriate for (partially) ordered categorical response data models.

AMS{2000) Subject Classification: 62H17, 62-07.

Multivariate density estimation is well known to be a tremendously difficult problem due to the occurrence of phenomena commonly known as the corner effect and the curse of dimensionality. Specifically, histogram density estimation in high dimensions is plagued by the consequence that sampled observations tend to reside with high probability in low density regions of the sample space. In this article we attempt to quantify two central things: in how many dimensions, one starts to really feel the curse of dimensionality, and what sort of sample sizes are needed to do any kind of a reasonable inference in various dimensions. These questions cannot be formulated in a unique way. So the attempt is to derive a broad spectrum of results, which are then illustrated by extensive computation. A number of results may be of independent interest in combinatorics and applied probability. Our subjective conclusion after these extensive computations is that in 3 dimensions one often sees the most drastic effect relative to just one less dimension; in 5 dimensions one feels the curse of high dimensions rather strongly; in 10 dimensions, the feasibility of inference with realistic sample sizes basically vanishes. We also give a subjective minimum sample size recommendation based on the number of dimensions. These calculations are different in character from Epanechnikov(1969).

AMS (2000} Subject Classification: Primary 62H05; Secondary 62G07.

The paper has three components. First, for a realvalued parameter of interest orthogonal (Cox and Reid, 1987) to the nuisance parameter vector, we find a necessary and sufficient condition for the equivalence of second order quantile matching priors and highest posterior density regions matching priors within the class of first order quantile matching priors. Examples are presented to illustrate the result. Second, we develop a quantile matching prior in a normal hierarchical Bayesian model. This prior turns out to be different from the one proposed earlier by Morris (1983). Third, we obtain an exact matching result when the objective is prediction of a real-valued random variable from a location family of distributions.

AMS (2000) Subject Classification: 62F15, 62F25, 62E20

Stochastic partial differential equations (SPDE) are used for stochastic modelling, for instance, in the study of neuronal behaviour in neurophysiology and in building stochastic models for turbulence. Huebner, Khasminskii and Rozovskii (1993) started the investigation of the maximum likelihood estimation of the parameters involved in two types of SPDE's and extended their results for a class of parabolic SPDE's in Huebner and Rozovskii (1995). Prakasa Rao (1998,2000) obtained Bernstein - von Mises type theorems for a class of parabolic SPDE's and investigated the properties of Bayes estimators of parameters involved in such SPDE's. In all the papers cited earlier, it was assumed that a continuous observation of a random field u_ε(x, t) satisfying the SPDE over the region [0, 1] × [0, T] is available. It is obvious that this assumption is not tenable in practice and the problem of interest is to develop methods of estimation of parameters from the random field u_ε(x, t) observed at discrete times t and at discrete positions x or from the Fourier coefficients u_iε(t) observed at discrete time instants. We construct consistent and asymptotically normal estimators of the parameter based on the Fourier coefficients u_iε(t) observed at discrete times t j = j Δ ,     0 ≤ j ≤ n     where   Δ > 0     and     n tends to infinity.

AMS (2000) Subject classification: Primary 62 M 40; Secondary 60 H 15.

The identifiability problem arises when a parametric family of probability distributions is not related to the space of parameters by a one-to-one correspondence. We give a brief review of the recent theoretical development relating to the identifiability problems, specifically focussing on the equivalence classes, conditions for identifiability and determination of identifying functions which lead to reparametrization. We present general definitions of identifiable constraints and model-preserving constraints, and indicate how any given parametric constraint can be expressed as an intersection of an identifiable constraint and a model-preserving constraint. We also present a necessary and sufficient condition for a linear constraint in a linear model to be model-preserving.

AMS{2000} Subject Classification: 62J12, 62J99.

Let N(t) be a counting process indicating the failures (and instantaneous repairs) of a repairable system up to time t. We model the probabilistic behaviour of the process by a nonhomogeneous Poisson process (NHHP) with intensity function λ(t). In the absence of any grounds in favour of a specific parametric form for λ(t) we assume that it is constant between failure epochs and at these epochs receives a jump (the damage-atfrulure- model). Again we don't know the heights of these jumps. Therefore we assume a heirarchical Bayes formulation. The heights of the jumps are assumed to have constrained conditional gamma distributions. The hyperparameters of the gamma distribution are assumed to have known uniform distributions. The exact Bayes solutions are analytically intractable. Hence we provide a scheme of simulation via the Metropolis-Hastings algorithm to estimate the features of the posterior distribution. The methodology is illustrated by applying it to a well known data set.

AMS (2000) Subject Classification: Primary 62F15, 62G05; Secondary 65C05.

We consider two sample survey situations. In one, we have a list of ‘selection units’ (su's) out of which one may suitably draw a sample. But the interest is to estimate the total value of a variable defined on an unknown number of individuals called ‘observational units’ (ou's) which are not directly identifiable but may be contacted through the above su's to “one or more” of which each ‘ou’ is ‘linked’ in a ‘well-defined’ manner. Each collection of ou's so linked is called a ‘network’. Each network is ‘disjoint’ from every other and together they exhaust all the ou's of interest. Through a sample of su's one may observe the variate-values for the ou's in the networks linked to the sampled su's. This approach of reaching samples of ou's through such networks is called ‘network’ sampling. Thompson (1990, 1992) and Thompson and Seber (1996) have given theories of estimation of total or mean and variance estimation when the su's are selected by simple random sampling (SRS) without replacement (WOR) or by stratified SRSWOR methods. Here we extend to general sampling schemes with unequal probabilities.

In the second kind of surveys an initial sample is chosen from a list of identifiable units for many of which the value of a variable is ‘zero’ and only for a few ‘unknown and unidentified’ units the values are positive. The purpose is to estimate the population total with a high coverage of the positive-valued units in the sample. A concept of a ‘neighbourhood’ is then well defined such that given a unit with a positive value some of those in its neighbourhood may also hopefully be positive-valued. So, from an initial sample one may go on ‘adaptively’ extending it by adding successively to each positive-valued unit in the sample each unit in its neighbourhood and stopping only on encountering a ‘neighbouring’ unit that is zero-valued. The relevant theory is developed in the above locations cited. But this covers only SRS with and without replacement (WR and WOR) for the initial sample. We extend here to more general sampling schemes.

AMS (2000) Subject Classification: 62 D05.

A-efficient proper block designs for estimating blank and intersection contrasts are obtained for slope ratio assays. It turns out that a non-equireplicate design is more efficient than the best equireplicate design, whenever an equireplicate design is feasible. Some Aefficient designs are tabulated.

AMS(2000) Subject Classification: 62K05, 62K10.

We consider a single-item one-period inventory model where the market demand is assumed to be random and the item under consideration is obtained from two suppliers who also supply random amounts with means equal to the order placed to the supplier concerned. Under new better than used in expectation (NBUE) assumption on the supply distributions, possibly different, a strategy which maximizes a minimum profit has been proposed. An estimate for this maximin order quantity whenever the (customer) demand distribution is unknown has been obtained and strong consistency of the suggested estimator established.

AMS (2000) Subject Classification: Primary 90B05; Secondary 62G99, 60F15.

AMS (2000) Subject classification: 60E15, 62N05, 62D05.

In this paper we provide a proper statistical analysis of Hillsdale lake (in the suburb of Kansas city in USA) data and propose an estimate of good water clarity level with a desired level of confidence.

AMS (2000} Subject Classification: 62-07.

Pap Smear test (Pap test), a cytological screening test of the cervical squamous intraepithelial lesion (SIL) is a widely accepted screening technique for uterine cervical cancer (UCC). UCC is a major canter burden for Indian women. In this article, we present an in-depth exploratory analysis on various statistical relationships between the Pap test and a set of demographic risk factors, such as Age, Age at consummation of marriage [ACM], parity [PAR] etc. The data had been collected from the Calcutta Medical College Hospital (CMCH) on 308 subjects. Major statistical findings in this article include a possible Age dependent association between PAR and the SIL-status. It seems that a high value of parity at a relatively young age is an important risk factor for developing SIL in future. This finding may be useful in relating to overall female helath issues in developing and underdeveloped country-senario.

AMS (2000) Subject Classification: Primary 62-07, 62Pl0, 92C60, Secondary 92C50, 62F03, 62F10.