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This manuscript describes a mathematical epidemiological model of COVID-19 to investigate the dynamics of this pandemic disease and we have fitted this model to the current COVID-19 cases in Italy. We have obtained the basic reproduction number which plays a crucial role on the stability of disease free equilibrium point. Backward bifurcation with respect to the cure rate of treatment occurs conditionally. It is clear from the sensitivity analysis that the developments of self immunities with proper maintaining of social distancing of the exposed and asymptomatic individuals play key role for controlling the disease. We have validated the model by considering the COVID-19 cases of Italy and the future situations of epidemicity in Italy have been predicted from the model. We have estimated the basic reproduction number for the COVID-19 outbreak in Italy and effective reproduction number has also been studied. Finally, an optimal control model has been formulated and solved to realize the positive impacts of adapting lock down by many countries for maintaining social distancing.

Certain discrepancies in data sources for estimating the household consumption expenditure to derive Gross Domestic Product(GDP) of India are discussed in this paper. Simple implementable strategies are suggested to improve the estimation of GDP.

We review a few unusual aspects of Bootstrap and some of the recent theoretical as well as methodological advances. We discuss the handling of non-linearity by Bootstrap through a numerical example in Section 2 . Application to the estimation of high-dimensional inverse covariance matrix is presented in Section 3 with emphasis on the Augmented Bootstrap and a Bayesian version of it. Another high dimensional example, namely, Random Forest and its offshoot random survival forest (Ishwaran et al. (2008) [32] ) are discussed in Section 4 . Bootstrap for massive data, introduced by Kleiner et al. (2011) [35] , is discussed in Section 4 . In Section 5 , we discuss some aspects of Bootstrap in the context of hypothesis testing in high-dimension.

To determine the socio-medical profile of child workers in Indian railways. This was a prospective longitudinal survey over one year. The children up to the age of 14 y entering the reserved second class sleeper compartment of long distance trains in Indian railways were included. The data were collected regarding their profile by the investigators either by interviewing or observation as per situation in the moving train in a predesigned proforma. Data pertaining to the social, demographic and medical aspects were collected. A total of eighty one children were noted in 22500 kilometers of train journey. Sex distribution (8:1) was unequal with males outnumbering females. This study, first of its kind, attempted to delineate a distinctive socio-medical profile in a special group of children. It might, on the behalf of social pediatrics, increase the awareness and help the society to formulate a rehabilitation policy in collaboration with national and international organizations.

Usual derivation of BIC for the marginal likelihood of a model or hypothesis via Laplace approximation does not hold for a change-point which is a discrete parameter. We provide an analogue l BIC, which is a lower bound to the marginal likelihood of a model with change points and has an approximation error up to Op(1) like standard Schwartz BIC. Several applications are provided covering simulated r.v.'s and real financial figures on short-term interest rate.

We survey some recent work in high dimensional multiple testing estimation and other multivariate inference problems depending on random matrices and graphical problems. Different approaches to these problems are explored featuring classical procedures like the Benjamini-Hochberg multiple tests, empirical Bayes and Bayes tests and estimates and use of shrinkage as a major method in estimation. The choice of topics reflects to some extent our taste and interests.

We consider the situation in which a large database needs to be analyzed to identify a few important predictors of a given quantitative response variable. There is a lot of evidence that in this case classical model selection criteria, such as the Akaike information criterion or the Bayesian information criterion (BIC), have a strong tendency to overestimate the number of regressors. In our earlier papers, we developed the modified version of BIC (mBIC), which enables the incorporation of prior knowledge on a number of regressors and prevents overestimation. In this article, we review earlier results on mBIC and discuss the relationship of this criterion to the well-known Bonferroni correction for multiple testing and the Bayes oracle, which minimizes the expected costs of inference. We use computer simulations and a real data analysis to illustrate the performance of the original mBIC and its rank version, which is designed to deal with data that contain some outlying observations. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper, we consider the classification problem when the predictors are multivariate binary random variables. Variables are modeled as independent, but not necessarily identical, Bernoulli. A triangular array for parameters, ( p 11 ( n ) , … , p 1 d ( n ) , p 21 ( n ) , … , p 2 d ( n ) ) , is assumed to allow parameters to change and the number of the variables, d, to increase for adopting more flexible models as the sample size, n, increases. Our results are obtained under moderate assumptions on the triangular array of the probability vectors. We use maximum likelihood estimators for the parameters and plug them into the Bayes classifier. This is a plug-in classifier, a sort of objective Bayes rule. It is shown in Wilbur et al. [2002. Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial DNA fingerprints. Biometrics 58, 378–386] via simulations that the plug-in rule classifies quite well even when the assumption of independence is violated. The main interest in this paper is in the complex case of d / n ν → c for some ν > 0 and c > 0 for which very little is known. Using linearity of the plug-in rule, we show its persistence, a generalization of the notion of consistency, when the variance of the plug-in rule or a quantity measuring signal to noise ratio is divergent; otherwise we show there exists an example of non-persistence of the plug-in rule. In case of non-persistence, we introduce the notion of sparsity and overcome non-persistence by selecting a subset of the variables. This shows why a variable selection procedure may be effective especially for contemporary practical problems with high dimensional data [Wilbur et al., 2002. Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial DNA fingerprints. Biometrics 58, 378–386].

We adopt boosting for classification and selection of high-dimensional binary variables for which classical methods based on normality and non singular sample dispersion are inapplicable. Boosting seems particularly well suited for binary variables. We present three methods of which two combine boosting with the relatively classical variable selection methods developed in Wilbur et al. (2002). Our primary interest is variable selection in classification with small misclassification error being used as validation of proposed method for variable selection. Two of the new methods perform uniformly better than Wilbur et al. (2002) in one set of simulated and three real life examples.

We discuss the Empirical Bayes approach to the problem of multiple testing and compare it with a very popular frequentist method of Benjamini and Hochberg aimed at controlling the false discovery rate. Our main focus is the ‘sparse mixture’ case, when only a small proportion of tested hypotheses is expected to be false. The specific parametric model we consider is motivated by the application to detecting genes responsible for quantitative traits, but it can be used in a variety of other applications. We define some Parametric Empirical Bayes procedures for multiple testing and compare them with the Benjamini and Hochberg method using computer simulations. We explain some similarities between these two approaches by placing them within the same framework of threshold tests with estimated critical values. Copyright © 2007 John Wiley & Sons, Ltd.

We establish weak and strong posterior consistency of Gaussian process priors studied by Lenk [1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 (402), 509–516] for density estimation. Weak consistency is related to the support of a Gaussian process in the sup-norm topology which is explicitly identified for many covariance kernels. In fact we show that this support is the space of all continuous functions when the usual covariance kernels are chosen and an appropriate prior is used on the smoothing parameters of the covariance kernel. We then show that a large class of Gaussian process priors achieve weak as well as strong posterior consistency (under some regularity conditions) at true densities that are either continuous or piecewise continuous. © 2005 Elsevier B.V. All rights reserved.

PARAMETRIC MODELS FOR INCUBATION DISTRIBUTION IN PRESENCE OF LEFT AND RIGHT CENSORING

We discuss the possibility of constructing pseudo panel data from cross-sectional data, sampled at different points in time, by aligning individuals sharing some common characteristics into groups called “cohorts”. Based on a real-life example on income distribution in the USA, we construct and validate a pseudo panel data and compare this with real panel data. The agreement is encouraging.

In a normal example of Stone (1979, J. Roy. Statist. Soc. Ser. B 41, 276–278), Berger et al. (2003, J. Statist. Plann. Inference 112, 241–258) showed BIC may be a poor approximation to the logarithm of Bayes Factor. They proposed a Generalized Bayes Information Criterion (GBIC) and a Laplace approximation to the log Bayes factor in that problem. We consider a fairly general case where one has p groups of observations coming from an arbitrary general exponential family with each group having a different parameter and r observations. We derive a GBIC and a Laplace approximation to the integrated likelihood, under the assumption that p → ∞ and r → ∞ (and some additional restrictions, which vary from example to example). The general derivation clarifies the structure of GBIC. A general theorem is presented to prove the accuracy of approximation, and the worst possible approximation error is derived for several examples. In several numerical examples, the Laplace approximation and GBIC are seen to be quite good. They perform much better than BIC.

There is no easy extension of the Kaplan-Meier and Nelson-Aalen estimators to the bivariate case, and estimating bivariate survival distributions nonparametrically is associated with various nontrivial problems. The Dabrowska estimator will, for example, associate negative mass to some subsets. Bayesian methods hold some promise as they will avoid the negative mass problem, but are also prone to difficulties. We simplify and extend an example by Pruitt to show that the posterior distribution from a Dirichlet process prior is inconsistent. We construct a different nonparametric prior via Beta processes and provide an updating scheme that utilizes only the most relevant parts of the likelihood, and show that this leads to a consistent estimator.

Mahalanobis (1893–1972) became involved with the analysis of anthropologic data in the 1920s and developed his measure of generalized distance in 1930. In the Indian Statistical Institute, he made important theoretical and practical contributions to sample surveys and became Chairman of the United Nations Sub-Commission on Statistical Sampling. Keywords: generalized distance; sample surveys; Indian Statistical Institute; United Nations

A variety of pseudo-Bayes factors have been proposed, based on using part of the data to update an improper prior, and using the remainder of the data to compute the Bayes factor. A number of these approaches are of a bootstrap or cross-validation nature, with some type of average being taken over the data used for updating. Asymptotic characteristics of a number of these pseudo-Bayes factors are discussed, and it is shown how many behave quite differently from ordinary Bayes factors. It is also shown that arguments of predictive optimality, based on simply inserting the empirical distribution in place of the 'true predictive distribution', can be misleading; the particular example of this that is studied is the argument given in Bernardo and Smith [1994. Bayesian Theory. Wiley, Chichester] to the effect that the geometric intrinsic Bayes factor has an optimal predictive property.

Consider the optimal assembly problem of assigning blades of different but known momenta to fixed positions of a rotor of circular cross section used in hydroturbine. The assembly has to be such that the resultant imbalance of the rotor after assembly is a minimum and the momenta are more or less uniformly spread over the circular section. When the number of blades and consequently the number of possible permutations is also large , the optimal assembly is difficult to obtain. A few heuristic approaches for the assembly are suggested and the statistical properties for the assembly arc studied under an appropriate data based model (for the distribution of momenta). We compare the performance of the proposed techniques with the optimal arrangement when the number of blades is small. It is also found that the imbalance arising out of the proposed assembly can be predicted with a high degree of accuracy. based only on first a few large momenta spacings when the blade momenta follow sowe standard distributions.

One way of handling composite hypotheses by SPRT's is to integrate out nuisance parameters under both the null and alternative hypotheses and then form an SPRT. If the prior is improper, Wald inequalities will not hold in general. We provide sufficient conditions and then verify that the usual sequential |t|-test satisfies these conditions but Wald's |t|-test does not satisfy these conditions and in fact cannot satisfy a strengthened form of these inequalities in general. Nonetheless, simulations show Wald's inequalities for A, B, corresponding to usual small α, β, will usually hold.

It is difficult to enumerate the people in India who are engaged in various small-scale industries in the unorganized sector because they are concentrated in small regional pockets. In estimating separately the total numbers of workers earning principally through ten respective single-industries in the unorganized small-scale sector in a specific district in rural India, through numerical illustrations we have two observations to report: (1) A traditional stratified two-stage sampling scheme is ineffective for some of the industries because of failures to capture the earners concentrated in priorly unknown locations. (2) An adaptive sampling scheme extending the initial sample by appropriate 'network' formations based on well-defined 'neighbourhoods' brings about dramatic improvements exploiting clustering tendencies of earners by different industries.

We explore two proposals for finding empirical Bayes prediction intervals under a normal regression model. The coverage probabilities and expected lengths of such intervals are studied and compared via appropriate higher-order asymptotics.

Stone (J. Roy. Statist. Soc. Ser. B 41 (1979) 276) showed that BIC can fail to be asymptotically consistent. Note, however, that BIC was developed as an asymptotic approximation to Bayes factors between models, and that the approximation is valid only under certain conditions. The counterexample of Stone arises in situations in which BIC is not an adequate approximation. We develop some new approximations to Bayes factors, that are valid for the situation considered in Stone (1979) and discuss related issues of consistency.

In Bayesian model selection, or hypothesis testing, difficulties arise when improper noninformative priors are used to calculate the Bayes factors. Several methods have been proposed to remove these difficulties. In this paper we discuss a unified derivation of some of these methods which shows that in some qualitative or conceptual sense, these methods are no more than a fixed number of observations away from a Bayes factor based on noninformative priors, and are close to each other and to certain Bayes factors based on low information proper priors which include priors recommended by Jeffreys (1961).

A geological map is the representation, on a two-dimensional plane, of the disposition of three-dimensional rock bodies exposed on the earth's surface. The problem of mapping is essentially that of dividing an area into “homogeneous” subregions on the basis of the exposed rock types. Automatic Bayesian methods of model selection using default Bayes factors have been employed to solve the problem of choosing a set of boundaries between “homogeneous” subregions, assuming no complication excepting low-angle tilting affected rock bodies. The method is tested on two data sets. A sampling scheme for optimum allocation of observation points is also presented.

The paper presents new measures of divergence between prior and posterior which are maximized by the Jeffreys prior. We provide two methods for proving this, one of which provides an easy to verify sufficient condition. We use such divergences to measure information in a prior and also obtain new objective priors outside the class of Bernardo’s reference priors.

With reference to the quasi-likelihood arising from an unbiased estimating function, we consider a large class of test statistics which includes the likelihood ratio, Rao's score and Wald's statistics in particular. We study Bartlett adjustability and third-order power in a possibly non-iid setting and provide explicit formulae. Since the relevant Bartlett identities may not hold while working with a quasi-likelihood, our results can differ from those based on the usual likelihood. The prospects regarding posterior Bartlett adjustability have been briefly indicated.

We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, log-spline models, Dirichlet processes and interval censoring.

Summary This is a brief history of the evolution of official and academic Statistics in India which focuses mainly on the period 1930 to 1960 but traces its origins in antiquity and recent history. We also comment on how Statistics has continued to evolve since the 1960's. This is a history of both institutions and people, who built and shaped them, and of ideas. Resume Ceci est une coute histoire de l'evolution de la Statistique officielle et acadeemique en Inde. Nous retracons l'hisoire a patir des origines de l'antiquite jusqu'a l'histoire plus recente mais potons plus d'atention sur la peiode des annees 1930 a 1960. Ceci et lhistoire d'institutions, des gens qui les batirent et les formerent eet d'idees.

We consider consistency of the posterior in the context of right censored data. We establish posterior consistency when the distribution of the lifetime has a Dirichlet distribution and also investigate the case when the prior is generated through a prior for the distribution of the observable (Z,Δ). We also show that a naive extension of these methods to interval censored data leads to peculiar estimates.

A Dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of Bayesian density estimation. In the recent years, efficient Markov chain Monte Carlo method for the computation of the posterior distribution has been developed. The method has been applied to data arising from different fields of interest. The important issue of consistency was however left open. In this paper, we settle this issue in affirmative.

The paper deals with the aging property of a one-component system supported by an identical, inactive standby and a perfect repair facility. We assume that all the lifetimes and repair times induced by the operating and under-repair components are mutually independent and repair times are arbitrary. It is shown that the lifetime of a system that begins with one (both) operative component (s) having NWUE (NBUE) lifetimes is NWUE (NBUE).

Aim To assess the clinical profile of 80 chronic hepatitis C patients in a tertiary health care center in Northern India and also to study the efficacy and tolerability of pegylated interferon (Peg-IFN) α 2b and ribavirin therapy in a cohort of chronic hepatitis C patients. Methods Thirty subjects with chronic hepatitis C (CH–C) with genotypes 2 and 3 received Peg-IFN α 2b 1.5 μg/kg subcutaneously weekly plus daily ribavirin 800 mg for 24 weeks .Subjects with genotype 1 infection received therapy for 48 weeks with ribavirin 1000 mg/day and Peg-IFN α 2b dose remained the same. The primary end point was the sustained viral response (SVR). Drug dosage was modified or temporarily discontinued if anemia or bone marrow suppression developed. Results The clinical profile of chronic hepatitis C infected patients showed decompensated cirrhosis in the more elderly patients. Genotype 3 was the commonest genotype and was seen in 21 (70%) patients. The mean baseline HCV RNA was high. SVR was achieved less commonly with genotype 1 than with genotype 2/3. Patients who became negative for HCV RNA at 4-weeks (rapid virological response or RVR) and 12 weeks (early virological response or EVR) of treatment showed significantly higher sustained virological response (SVR) rates. Similarly, patients who showed normalization of ALT level at 4-weeks and 12-weeks of treatment showed significant high rate of SVR. Overall treatment was well tolerated. Conclusion In our region, CHC subjects have high viral load and genotype 3 being the most common. Treatment with Peg-IFN α 2b and ribavirin is effective and well tolerated. Genotype 1 was more resistant to the treatment. Patients who achieved RVR and EVR are more likely to achieve SVR. Although the numbers of patients in this study was small, considering the paucity of data of treatment from India, the data is relevant.

In this paper, we establish some optimality properties of the multiple testing rule induced by the horseshoe estimator due to Carvalho, Polson, and Scott (2010, 2009) from a Bayesian decision theoretic viewpoint. We consider the two-groups model for the data and an additive loss structure such that the total loss is equal to the number of misclassified hypotheses. We use the same asymptotic framework as Bogdan, Chakrabarti, Frommlet, and Ghosh (2011) who introduced the Bayes oracle in the context of multiple testing and provided conditions under which the Benjamini-Hochberg and Bonferroni procedures attain the risk of the Bayes oracle. We prove a similar result for the horseshoe decision rule up to $O(1)$ with the constant in the horseshoe risk close to the constant in the oracle. We use the Full Bayes estimate of the tuning parameter $\tau$ . It is worth noting that the Full Bayes estimate cannot be replaced by the Empirical Bayes estimate, which tends to be too small.

We prove a theorem justifying the regularity conditions which are needed for Path Sampling in Factor Models. We then show that the remaining ingredient, namely, MCMC for calculating the integrand at each point in the path, may be seriously flawed, leading to wrong estimates of Bayes factors. We provide a new method of Path Sampling (with Small Change) that works much better than standard Path Sampling in the sense of estimating the Bayes factor better and choosing the correct model more often. When the more complex factor model is true, PS-SC is substantially more accurate. New MCMC diagnostics is provided for these problems in support of our conclusions and recommendations. Some of our ideas for diagnostics and improvement in computation through small changes should apply to other methods of computation of the Bayes factor for model selection., Published in at http://dx.doi.org/10.1214/12-STS403 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The hazard rate h(t) of a lifetime random variable is assumed to be a constant equal to a up to time r and another constant equal to b thereafter. The parameters T and (a, b) are assumed to be independent apriori with r having a uniform prior on [t1, t2], 0 < t1 < t2 < ∞ while the prior of (a, b) is assumed to be smooth, It is proved that the marginal posterior mode of r is n-consistent; the marginal posterior mass of T is concentrated around an n−1 neighborhood of the unknown parameter value; the posterior distribution of (a, b) can be approximated by a Normal distribution; a, b T are asymptotically independent aposieriori; and one can approximate the posterior mean and variance of (a, b) by easily computable quantities, The accuracies of these approximations are examined by a simulation study.

This relatively brief, yet detailed survey of a collection of beautiful, important results is a fitting tribute both to Wald and the vitality of Sequential Analysis by one of its most creative prac...

This paper considers, in the multiparameter case, perturbed ellipsoidal and highest posterior density regions with both Bayesian and frequentist validity up to o(n -1 ).