Arnab Kumar Laha

A Bayesian analysis of the change-point problem for directional data

Abstract In this paper, we discuss a simple fully Bayesian analysis of the change-point problem for the directional data in the parametric framework with von Mises or circular normal distribution as the underlying distribution. We first discuss the problem of detecting change in the mean direction of the circular normal distribution using a latent variable approach when the concentration parameter is unknown. Then, a simpler approach, beginning with proper priors for all the unknown parameters – the sampling importance resampling technique – is used to obtain the posterior marginal distribution of the change-point. The method is illustrated using the wind data [E.P. Weijers, A. Van Delden, H.F. Vugts and A.G.C.A. Meesters, The composite horizontal wind field within convective structures of the atmospheric surface layer, J. Atmos. Sci. 52 (1995. 3866–3878]. The method can be adapted for a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts.In this paper we discuss a simple fully Bayesian analysis of the change point problem for the directional data in the parametric framework with circular normal distribution as the underlying distribution. We discuss the problem of detecting change in the mean direction of the circular normal distribution when the concentration parameter is unknown. Beginning with proper priors for all the unknown parameters, the sampling-importance-resampling (SIR) technique is used to obtain the posterior marginal distribution of the change point. The method is illustrated using the wind data (Weijer‘s et. al.(1995)). The method can be adapted to a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts. Note: For softcopy of this paper, please contact the authors - email: arnab@iima.ac.in

SB-Robust Estimators of the Parameters of the Wrapped Normal Distribution

In this article, we study the SB-robustness of various estimators of the mean direction (μ) and the concentration parameter (ρ) of the wrapped normal distribution. The functional corresponding to the sample mean direction is seen to be not SB-robust as an estimator of μ at the family of wrapped normal distributions with varying ρ, whereas the γ-trimmed mean direction is SB-robust at the same family of distributions for the different dispersion measures considered in this article. We also study the SB-robustness of the moment estimator of ρ and also that for a newly introduced trimmed estimator of ρ.

Comparison of treatments in a cataract surgery with circular response.

Circular data are a natural outcome in many biomedical studies, e.g. some measurements in ophthalmologic studies, degrees of rotation of hand or waist, etc. With reference to a real data set on astigmatism induced in two types of cataract surgeries we carry out some two-sample testing problems with the possibility of common or different concentration parameters in the circular set up. Detailed simulation study and the analysis of the data set, including redesigning the cataract surgery data, are carried out.

Portfolio allocation with heavy-tailed returns

In this article we propose two new methods of portfolio allocation which are applicable for all return distributions. The properties of these new methods are compared with that of Markowitzs mean-variance method using extensive simulation. It is found that the new methods perform appreciably in terms of growth of wealth as well as protecting against the downside risk, in situations where the return distributions of one or more of the stocks is heavy-tailed. These methods can be effective substitutes for the mean-variance method which is not applicable for return distributions with heavy-tails having infinite expectation or variance.

A novel sandwich algorithm for empirical Bayes analysis of rank data

Rank data arises frequently in marketing, finance, organizational behavior, and psychology. Most analysis of rank data reported in the literature assumes the presence of one or more variables (sometimes latent) based on whose values the items are ranked. In this paper we analyze rank data using a purely probabilistic model where the observed ranks are assumed to be perturbed versions of the true rank and each perturbation has a specific probability of occurring. We consider the general case when covariate information is present and has an impact on the rankings. An empirical Bayes approach is taken for estimating the model parameters. The Gibbs sampler is shown to converge very slowly to the target posterior distribution and we show that some of the widely used empirical convergence diagnostic tools may fail to detect this lack of convergence. We propose a novel, fast mixing sandwich algorithm for exploring the posterior distribution. An EM algorithm based on Markov chain Monte Carlo (MCMC) sampling is developed for estimating prior hyperparameters. A real life rank data set is analyzed using the methods developed in the paper. The results obtained indicate the usefulness of these methods in analyzing rank data with covariate information.

Modeling Commodity Market Returns: The Challenge of Leptokurtic Distributions

In this chapter, we consider modeling leptokurtic daily log-return distributions of three commodities: gold, silver, and crude oil. Three modeling approaches are tried out namely (a) a two-component mixture of normal distributions model, (b) Variance Gamma (VG) distribution model, and (c) Generalized Secant Hyperbolic (GSH) distribution model. The two-component mixture of normal distributions model is found to be a reasonable model for log-returns on gold and crude oil. The VG distribution model and the GSH distribution model are not found to be suitable for modeling log-returns for any of the three commodities considered in this chapter.

Statistical Challenges with Big Data in Management Science

In the past few years, there has been an increasing awareness that the enormous amount of data being captured by both public and private organisations can be profitably used for decision making. Aided by low-cost computer hardware, fast processing speeds and advancements in data storage technologies, Big Data Analytics has emerged as a fast growing field. However, the statistical challenges that are faced by statisticians and data scientists, while doing analytics with Big Data has not been adequately discussed. In this paper, we discuss the several statistical challenges that are encountered while analyzing Big data for management decision making. These challenges give statisticians significant opportunities for developing new statistical methods. Two methods—Symbolic Data Analysis and Approximate Stream Regression—which holds promise in addressing some of the challenges with Big Data are discussed briefly with real life examples. Two case studies of applications of analytics in management—one in marketing management and the other in human resource management—are discussed.

SB-Robustness of Estimators

Examining the SB-robustness of estimators becomes important in situations where the underlying family of distributions has bounded support or bounded parameter space. Such situations occur routinely when dealing with circular data and statistical quality control. In this paper, we first discuss SB-robustness of estimators in the circular data set-up and review some of the recently obtained results in this regard. Later in the paper, we examine the SB-robustness of some of the commonly used performance measures of control charts which are widely used for control of manufacturing processes. It is shown that for a mean control chart, the False Alarm Probability (FAP), the Average Sample Number when the process is in-control (\(ASN_0\)), the No-Signal Probability (NSP), and the Average Sample Number when the process is out-of-control (\(ASN_1\)) are all SB-robust at the family of all normal distributions with bounded mean and standard deviation. We also show that the above-mentioned performance measures are not SB-robust at the larger family of normal distributions with unbounded mean and standard deviation.

Robustness of tests for directional mean

In this paper we study the robustness of the likelihood ratio, circular mean and circular trimmed mean test functionals in the context of tests of hypotheses regarding the mean direction of circular normal and wrapped normal distributions. We compute the level and power breakdown properties of the three test functionals and compare them. We find that the circular trimmed mean test functional has the best robustness properties for both the above-mentioned distributions. The level and power properties of the test statistics corresponding to these functionals are also studied. Two examples with real data are given for illustration. We also consider the problem of testing the mean direction of the von-Mises–Fisher distribution on the unit sphere and explore the robustness properties of the spherical mean direction and likelihood ratio test functionals.

Bi-objective portfolio optimization using Archive Multi-objective Simulated Annealing

In the current paper, Bi-objective portfolio optimization problem has been tackled using multiobjective optimization framework. Three popular multiobjective optimization algorithms are used for solving this problem. These are: Archive Multi-objective Simulated Annealing (AMOSA) algorithm, Non-dominated Sorting Genetic algorithm II (NSGA-II) and Multi-objective Particle Swarm Optimization using Crowding distance (MOPSOCD). For each algorithm we trace the Pareto optimal front and compare the results by using four comparisons metrics, Spread, Spacing, Set Coverage and Maximum Spread. Comparative results show that NSGA-II performs the best as compared to the other two algorithms.