Inference procedures based on the Hellinger distance provide attractive alternatives to likelihood based methods for the statistician. The minimum Hellinger distance estimator has full asymptotic efficiency under the model together with strong robustness properties under model misspecification. However, the Hellinger distance puts too large a weight on the inliers which appears to be the main reason for the poor efficiency of the method in small samples. Here some modifications to the inlier part of the Hellinger distance are provided which lead to substantial improvements in the small sample properties of the estimators. The modified divergences are members of the general class of disparities and satisfy the necessary regularity conditions so that the asymptotic properties of the resulting estimators follow from standard theory. In limited simulations the proposed estimators exhibit better small sample performance at the model and competitive robustness properties in relation to the ordinary minimum Hellinger distance estimator. As the asymptotic efficiencies of the modified estimators are the same as that of the ordinary estimator, the new procedures are expected to be useful tools for applied statisticians and data analysts. AMS (2000) subject classification. Primary 62F10, 62F35; Secondary: 62F12.
Allele sharing method or trait-model-free approach is a robust technique for linkage detection. In allele sharing method, the association between the joint inheritance of the trait and a set of markers in pedigrees is studied without modelling the trait. This paper aims at an efficient implementation of allele-sharing method for binary trait in multi-generation pedigrees that includes construction of a robust identity by descent measure, and the use of a computationally efficient testing procedure to test for linkage. Simulation studies demonstrate that the proposed method performs well on extended pedigrees and appears to outperform the existing allele-sharing approaches on moderate size pedigrees.
In this paper, we provide Poincare-type upper and lower variance bounds for a function g(X) of a discrete integer-valued random variable (r.v.) X, in terms of the (forward) differences of g up to some order. To this end, we investigate a discrete analogue of the Mohr and Noll inequality (1952, Math. Nachr., vol. 7, pp. 55-59), which may be of some independent inter- est in itself. It has been shown by Johnson (1993, Statist. Decisions, vol. 11, pp. 273-278) that for the commonly used absolutely continuous distribu- tions that belong to the Pearson family, the somewhat complicated variance bounds take a very pleasant and simple form. We show here that this is also true for the commonly used discrete distributions. As an application of the proposed inequalities, we study the variance behaviour of the UMVU estimator of log p in Geometric distributions.
Consider the problem of testing the homogeneity of several p-variate normal mean vectors under an order restriction. This is a multivariate extension of Bartholomew’s (Biometrika, 1959) problem. When the covariance matrices are known, this problem has been studied to some extent, for example, by Sasabuchi, Inutsuka and Kulatunga (Biometrika, 1983), Sasabuchi, Miura and Oda (JSCS, 2003) and some others. We are interested in the case when the covariance matrices are common but unknown. In this case, Sasabuchi, Tanaka and Tsukamoto (Ann. Statist., 2003) proposed a test statistic and studied its upper tail probability under the null hypothesis. In the present paper, we provide some tests, which are more powerful than the above test. We derive some theorems about their null distributions and powers. AMS (2000) subject classiflcation. Primary 62F30; secondary 62F03, 62H15.
In this paper we present asymptotic estimates of level crossing probabilities from a Bayesian point of view, based on large deviations. For the Bayesian analysis we choose a finite mixture of conjugate prior distributions to model the uncertainty on the unknown parameters of the two classes of stochastic processes considered: the Brownian motion and the compound Poisson process with upward jumps and negative drift. The estimates of level crossing probabilities are derived as a consequence of large deviation principles for posterior distributions.
Recently, many researchers have devoted themselves to the investigation on the number of replicates needed for experiments in blocks of size two. In practice, experiments in blocks of size four might be more useful than those in blocks of size two. To estimate the main effects and two-factor interactions from a two-level factorial experiment in blocks, we might need many replicates. This article investigates designs with the least number of replicates for factorial experiments in blocks of size four. The methods to obtain such designs are presented.
Consider an invariant prediction problem where the group is transitive on the parameter space. The Haar predictive distribution (Haar inference) is obtained as the formal predictive distribution using the right Haar measure as a prior. This Haar inference is discussed in three different contexts: model matching, Fisherian pivoting, and de Finetti's coherence. It, is shown that the Haar inference enjoys exact probability matching for highest probability density (HPD) regions. Fisherian pivotal distributions agree with Haar inference when pivoting is possible. Further any invariant inference that is essentially different from the Haar inference is incoherent.
Although it is common practice to fit a complex Bayesian model using Markov chain Monte Carlo (MCMC) methods, we provide an alternative sampling-based method to fit a two-stage hierarchical model in which there is conjugacy conditional on the parameters in the second stage. Using the sampling importance resampling (SIR) algorithm, our method subsamples independent samples from an approximate joint posterior density. This is an alternative to a Metropolis-Hastings (MH) algorithm normally used to draw samples from the joint posterior density. We also provide comparison with a Metropolis (MET) algorithm. We illustrate our method using a Poisson regression model which has much interest for the analysis of rare events from small areas. We also illustrate our method using a relatively new logistic regression model. We use four examples, three on Poisson regression and one on logistic regression, and a simulation study on the Poisson regression model to assess the performance of our method relative to the MH and the MET algorithms. AMS (2000) subject classification. 62F15, 62J12. Keywords and phrases. Collapsing, conditional conjugate, hierarchical Bayesian model, MCMC monitoring, Metropolis-Hastings algorithm, Rao-Blackwellized estimator, small areas.
SUMMARY. In a hypothesis testing problem, the classical p-values are often perceived as measurements of the degree of surprise in the data, relative to a hypothesized model. The classical p-values commonly provide a basis for rejection of a hypothesis or a model. In this paper, we develop prior predictive and posterior predictive p-values for one sided hypothesis testing for location parameter problems. We show that for many classes of prior distributions, the infimum of the prior predictive and posterior predictive p-values are equal to the classical p-value, for very general classes of distributions. The results are in spirit similar to that in Casella and Berger (1987) in terms of reconciliation of Bayesian and frequentist evidence. The results are used through many examples relating to the one sided testing problem for location parameter.
SUMMARY. Recent studies demonstrate that volatility exhibits long-range dependence. This article investigates a long memory stochastic volatility model in which the stochastic process governing the volatility is an Autoregressive Fractionally Integrated Moving Average process. Bayesian estimation via Monte Carlo Markov Chain sampling methods is proposed. Besides, Bayesian prediction and smoothing are introduced in the article. The methodologies are applied to daily returns data for illustration.
In standard wavelet methods, the empirical wavelet coe cients are thresholded term by term, on the basis of their individual magnitudes. Information on other coe cients has no in uence on the treatment of particular coe cients. We propose a wavelet shrinkage method that incorporates information on neighboring coe cients into the decision making. The coe cients are considered in overlapping blocks; the treatment of coe cients in the middle of each block depends on the data in the whole block. The asymptotic and numerical performances of two particular versions of the estimator are investigated. We show that, asymptotically, one version of the estimator achieves the exact optimal rates of convergence over a range of Besov classes for global estimation, and attains adaptive minimax rate for estimating functions at a point. In numerical comparisons with various methods, both versions of the estimator perform excellently.
We prove results that relate random correspondences with their measurable selections, thus providing a foundation for viewing random correspondences as "bundles" of random variables.
SUMMARY. Stability of Bayes decision problems under uniform convergence of losses is revisited and sufficient conditions for stability are obtained. The results generalize and com plement the earlier works of Kadane and Chuang, Chuang, and Salinetti. General conditions are also given for the equivalence of two definitions of stability.
The Lorenz curve is a powerful tool for the measurement of inequality in a population of income receivers. Its importance in econometrics is emphasized by the fundamental works of Atkinson (1970), Rothschild and Stiglitz (1973), Dasgupta et al. (1973) or Kakwani (1984) on social welfare preference. Non intersecting Lorenz curves of two alternative income (probability) distributions allow for an unambiguous ranking of these distributions with respect to the concept of concave utility (welfare) functions. Apart from that, the usefulness of nested Lorenz curves in the context of reliability theory for the ranking of life distributions is described by Chandra and Singpurwala (1981) and Kochar (1989). The Lorenz curve plots the proportion of total income earned by various portions of the population when the population is ordered by the size of their incomes. For any distribution supported on [0, oo) with distribution function Fx(x) = Pr(X < x), inverse distribution function
"In this paper an attempt is made to estimate the truncation bias in the mean length of closed birth interval caused by the termination of observations after [a] certain time, through the application of a stochastic model based on certain simplified assumptions. The approach is further evolved to determine the relative contributions of...chance and systematic components to the variance of closed birth interval when the fecundability parameter follows a priori distribution. Some numerical results have also been presented for the purpose of illustration."
"An attempt has been made to reconstruct life tables for India from 1901-11 to 1971-81 and to project for the decades 1981-91 and 1991-2001 by adopting a Brass relational model. The earlier actuarial life tables seem to have been based on a British model of sex differentials in mortality leading to higher life expectancies for females--not in tune with Indian experience. Consistency has been attained in this study by taking recourse to mortality patterns obtained from the Sample Registration System."
A probability distribution for describing the time of first live birth is developed which is more suitable for traditional societies where the age at marriage is low. The model takes account of temporary separation between husband and wife just after marriage and indirectly incorporates adolescent sterility and the restriction on sexual union imposed on younger couples. The model is applied to the data collected in the large scale sample survey entitled “Rural Development and Population Growth—A Sample Survey 1978” conducted by Centre of Population Studies, Banaras Hindu University, India.
"In this paper an open problem by Kendall (1949, 1977) in his population model has been solved. The problem has also been studied in a model which has been modified by the author and made more general." The author's modifications involve the use of alternative marriage rates per unit of time.
AbstractAttempt has been made in this paper to estimate certain parameters (data pertaining to which are either not available or easily reportable) of the human reproductive process as the period of postpartum ammenorrhoea (P.P.A.), number of foetal wastages in between live births etc., using a truncated negative binomial probability model. In view of the hypothesis that the probability of foetal wastages varies from mother to mother, the truncated negative binomial distribution has been compounded by weighing with the best prior Beta distribution of the parameter. Estimation has been made by successive approximation using the method of moments.
