Statistical Paradigms

The following sections are included:

• Foreword
• Preface
• List of Contributors
• Contents

We consider two important counterexamples, one due to Basu (1988) and the other due to Wasserman (2004). We argue that while they provide insights about different weaknesses in Frequentist and Bayesian paradigms, too much stress on unbiasedness in the former and the full likelihood in the latter, the counterexamples are not as negative as appeared at first sight. It is shown how the paradigms can deal with the counterexamples without any fundamental change.

Over the past five decades, nonparametric methods have emerged as a viable and often more adaptable way of statistical modeling and analysis of more complex and sometimes interdisciplinary studies. The contemporary evolution of information (and bio-)technology has posed some highly nonstandard interdisciplinary research problems, genuinely in need of statistical appraisal. Some nonparametric perspectives and prospectives in some challenging interdisciplinary research, along with a highlight of some developments in the past fifty years, are appraised here.

Elicitation of prior distributions will necessarily be imprecise and frequently will yield incompatible probability assessments. Identification of the class of distributions compatible with the given information can be used to generate a spectrum of possible Bayes estimates from which to choose. In the absence of compatibility, a Bayes estimate based on a minimally incompatible prior is recommended.

Often in practice when a large number of hypotheses are simultaneously tested, one is willing to allow a few false rejections, say at most k−1, for some fixed k>1. In such a case, the ability of a procedure controlling an error rate measuring at least one false rejection can potentially be improved in terms of its ability to detect false null hypotheses by generalizing this error rate to one that measures at least k false rejections and using procedures that control it. The k-FDR which is the expected proportion of k or more false rejections and a natural generalization of the false discovery rate (FDR) is such a generalized notion of error rate that has recently been introduced and procedures controlling it have been proposed. Many of these procedures are stepup procedures. Some stepdown procedures controlling the k-FDR are presented in this article.

The exponential distribution is an important distribution in the analysis of lifetime data. Its one interesting characterization is that interoccurrence times between the successive events of a Poisson process are independent exponentially distributed. In industrial designed-experiments setting, exponentially distributed response variable can be encountered, for example, when the response is inter-occurrence time between successive defective products when the defectives are produced according to a Poisson process. Traditionally in such a situation, the logtransformed response variable is analyzed using ordinary least squares (OLS). However, this approach has its own limitations and the generalized linear models (GLM) appear to be a good alternative. In the present work, a comparative study is attempted for analyzing 2ⁿ factorial experiments for exponentially distributed response variable among GLM, methods based on Edgeworth expansion and a method based on exact distribution of a pivotal quantity. The comparison is based on theoretical considerations and an extensive simulation study related to the coverages and expected length of confidence intervals (LOCI) for the expected response. The method based on exact distribution of a pivotal quantity turns out to be the best among all the approaches. Results of the analysis of a real dataset agree with the theoretical findings.

The choice of explanatory variables in a multiple linear regression model has been studied for Bayesian prediction. In presence of perfect multicollinearity, the problem is studied when the goal is to predict a vector of future responses. To test the significance of influence of variables for Bayesian prediction in the presence of perfect multicollinearity, Kullback-Leibler (K-L) divergence was decomposed into the discrepancies due to the location and scale parameters, respectively. However, one problem in that decomposition is that the derivation of the distributional form of discrepancy due to the location parameter as well as K-L divergence was very difficult. To overcome this problem, in this paper we consider the predictive influence of variables in a normal multiple linear regression model in the presence of perfect multicollinearity when the goal is to predict a single future response. Then we have derived the distributional results of both discrepancies due to the location and scale parameters, respectively and the test procedure is performed.

Dattatreya Rao et al. (2007) have introduced Wrapped versions of four well-known life testing models viz. Lognormal, Logistic, Weibull and Extreme-Value distributions and mentioned basic characteristics along with graphs of the above said distributions. Here an attempt is made on the lines of Dattatreya Rao et al. (2007) to derive some of the new circular models – Wrapped Half Logistic and Wrapped Binormal distributions by reducing a linear variable to its modulo 2π and using trigonometric moments. Goodness of fit for small samples using Kolmogorov-Smirnov (KS) type Kuiper one sample test [Mardia and Jupp, 2000; Fisher, 1993] and for large samples using χ²-test were employed [Banks et al., 2000]. Three data sets each of sample sizes 10 and 100 respectively are taken for employing Kuiper and χ²-tests.

In this chapter, we focus on the building of an invariant distribution function associated with a non-stationary sample. After discussing some specific problems encountered by non-stationarity inside samples like the “spurious” long memory effect, we build a sequence of stationary processes permitting us to define the concept of meta-distribution for a given non-stationary sample. We use this new approach to discuss some interesting econometric issues in a non-stationary setting, namely forecasting and risk management strategy.

For spline smoothing one can rewrite the smooth estimation as a linear mixed model (LMM) where the smoothing parameter appears as the ratio between the variance of the error terms and the variance of random effects. Smoothing methods that use basis functions with penalization can utilize maximum likelihood (ML) theory in the LMM framework. We introduce the minimum description length (MDL) model selection criterion for LMM's and propose an automatic data-based spline smoothing method based on the MDL criterion. Simulation study shows that the performance of MDL in spline smoothing is close to that of the Bayesian information criterion.

The various impacts of modern communication and information technologies on society cannot be overstated and are reflected in worldwide emergence of digital governance. The purpose of digital governance is to empower the public through information access and analysis by enabling transparency, accuracy, and efficiency for societal good at large. In this context, development and applications of methodologies for geoinformatic hotspot analysis of spatial and temporal data are of utmost importance. The widely used SaTScan^TM software package, is applicable for Bernoulli, Poisson, ordinal, exponential, and normal response models. In this paper, we present a model that is suitable to analyze continuous fractional response data using the upper level set (ULS) scan statistic. The ULS scan statistic, an alternative to the circular scan statistic, uses arbitrarily shaped scanning windows. The circular/elliptic scan statistic uses the Euclidean distance as a sole basis to coalesce cells to form clusters, whereas the ULS scan statistic uses adjacency and the response ratio to combine cells to form candidate zones. The continuous fractional response model is presented after a brief review of the ULS scan statistic and its software implementation. The model is used for hotspot detection with forest cover data for Jalgaon district, Maharashtra State, India, as an illustrative example.

Functional data arise in numerous areas nowadays. When the functional responses evolve with respect to time, the subjects may experience events at different paces with the consequence that the sample curves are improperly aligned for inferential purposes. In particular, the sample mean function without alignment will fail to produce a satisfactory estimator of the true process mean function. In this article a new model for curve alignment or registration is developed from a Bayesian perspective. It incorporates nonparametric spline curve fitting methods with continuous Monte Carlo Markov chain (MCMC) techniques. The functional response curves are fit by nonparametric spline methods with their coefficients treated as random parameters. Similarly, the warping functions are modeled as random spline functions and random shift and amplitude coefficients are also included in the model formulation. An MCMC algorithm is created to estimate the parameters in the model. The performance of the proposed method is evaluated in an empirical study.

This paper considers the problem of nonparametric kernel estimation in panel data models. We examine the finite sample performance of several estimators for estimating a one-way random effects error component model. Monte Carlo experiments show that the pooled estimator that ignores the dependence structure in the model performs well in each trial, but it is not the most efficient estimator since it is generally outperformed in the mean squared sense by the nonparametric feasible generalized least squares estimator, the two-step estimator and an iterative estimator. Although the asymptotic bias and variance of many of the estimators converge at the same rate, the iterative and Two-Step estimators, which have asymptotic improvements, outperform the pooled estimator, but they are not always the best estimators in the simulations. Finally, we employ the estimators in an empirical example focusing on the public capital productivity puzzle to showcase their performance in a real data setting.

This paper presents an overview of Graphical Evaluation and Review Technique (GERT) for the reliability analysis of some consecutive-k systems including m-consecutive-k systems, consecutive-k systems with dependence, multi-state consecutive-k systems. Further, distributions of the waiting times of some patterns are also studied. It is demonstrated that GERT, being straightforward and simple, is generally more efficient than other approaches. This is so since GERT besides providing visual picture of the system makes it possible to analyze the given system in a less inductive manner. GERT is easier to use than the minimal cut set method. In GERT, one has to evaluate a W function, the generating function of the waiting time for the occurrence of the system failure, whereas in minimal cut set method one has to enumerate all possible minimal cut sets leading to system failure. Therefore, once the W function is computed, the reliability of the system consisting of any number of components can then be computed directly. Consecutive-k-out-of-n system models have wide applications in the design of integrated circuits, microwave relay stations in telecommunications, oil pipeline systems, vacuum systems in accelerators, computer ring networks (k-loops), space craft relay stations etc.

A sharp moment bound for sample sum of truncated non-stationary strong-mixing random variables with polynomial decay is obtained when some finite order (> 2) moment of individual random variables exist. It is also shown that this technique of estimation originally due to Ibragimov (1962), later extended by Ghosh and Babu (1977), Babu, Ghosh and Singh (1978), Babu and Singh (1978), Babu (1980), and Dasgupta (1997) breaks down for strong-mixing processes with decay lower than polynomial order. The moment bounds have applications to obtain the rates of convergence in Marcinkiewicz-Zygmund strong law of large numbers, central limit theorem, probabilities of deviations and in quantile processes of the random variables.