Hellinger distance estimation of general bilinear time series models

In the present paper, minimum Hellinger distance estimates for parameters of a bilinear time series model are presented. The probabilistic properties such as stationarity, existence of moments of the stationary distribution and strong mixing property of the model are well known (see for instance [J. Liu, A note on causality and invertibility of a general bilinear time series model, Adv. Appl. Probab. 22 (1990) 247–250; J. Liu, P.J. Brockwell, On the general bilinear time series model, J. Appl. Probab. 25 (1988) 553–564; D.T. Pham, The mixing property of bilinear and generalised random coefficients autoregressive models, Stoch. Process Appl. 23 (1986) 291–300]). We establish, under some mild conditions, the consistency and the asymptotic normality of the minimum Hellinger distance estimates of the parameters of the model.     

On zero-distorted generalized geometric distribution

ABSTRACT

We propose a new generalized geometric distribution which permits inflation/deflation of the zero count probability and study some of its properties. We also present an actuarial application of this distribution and fit it to three datasets used by other researchers. It is observed that the proposed distribution fits reasonably well to these data. Further, in a regression setup, the performance of this distribution is studied vis–a–vis other competing distributions used for explaining variability in a response variable.     

Compounded inverse Weibull distributions: Properties,inference and applications

ABSTRACT

In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be used to model lifetime of series system where the lifetimes follow inverse Weibull distribution and the subgroup size being random follows either geometric or zero-truncated Poisson distribution. Some of the important statistical and reliability properties of each of the distributions are derived. The distributions are found to exhibit both monotone and non-monotone failure rates. The parameters of the distributions are estimated using the expectation-maximization algorithm and the method of minimum distance estimation. The potentials of the distributions are explored through three real life data sets and are compared with similar compounded distributions, viz. Weibull-geometric, Weibull-Poisson, exponential-geometric and exponential-Poisson distributions.     

Bivariate semi α-Laplace distribution and processes

The Laplace distribution is considered as a better choice for modeling whenever data exhibit high kurtosis and heavier tails than Gaussian tails. Even though this is the case, not much work has been done on bivariate Laplace distribution. In this work, we introduce and study a new class of bivariate distributions called bivariate semi α-Laplace distribution, containing bivariate Laplace distributions. Three characterizations of bivariate semi α-Laplace distribution are obtained. Relation with bivariate semi stable distribution is established. An autoregressive model with bivariate semi α-Laplace marginal distributions is developed.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

Seed distributions for the NCAA men’s basketball tournament

Bracketology, the art of successfully picking all the winners in the National Collegiate Athletic Association’s (NCAA) annual men’s Division I college basketball championship tournament, has become a favorite national activity. In spite of the challenges and uncertainty faced in this endeavor, patterns exist in how the seeds appear in each round, particularly the later rounds. This paper statistically analyzes tournaments from 1985 to 2010, finding that the distribution of seeds that win in the rounds beyond the Sweet Sixteen can be modeled as a truncated geometric random variable. This model allows one to consider any set of seeds in each tournament round and compute the probability that these seeds would win in that round; this methodology can evaluate the likelihood of each seed combination in each tournament round, based on past tournament history. Finally, each tournament from 1985 through 2010 is analyzed using this model to assess its likelihood and measure the probability of its occurrence.     

A study of geometric and statistical curvatures for the skew-normal family of distributions

With special reference to the family of skew-normal distributions, we consider geometric curvature of a probability density function as a means to define and identify rare or catastrophic events—a phenomenon common in studying the financial instruments. Further, we study the statistical curvature properties of this family of distributions and discuss the sample size issue, to assess, to what extent the linear and likelihood-based inference of exponential family of distribution can be applicable for the skew-normal family.     

The gamma generalized linear model,log transformation,and the robust Yuen-Welch test for analyzing group means with skewed and heteroscedastic data

ABSTRACT

Alternatives for positively skewed and heteroscedastic data include the Yuen-Welch (YW) test, data transformations, and the generalized linear model (GzLM). Because the GzLM is rarely considered in psychology compared to the other two, we compared these strategies conceptually and empirically. The YW test generally has satisfactory power, but its trimmed mean can deviate substantially from the arithmetic mean, which is often the desired parameter. The gamma GzLM can be used as a substitute for the log transformation and addresses the limitations in inference for the YW and data transformations.     

Characterizations of Life Distributions Using Conditional Expectations of Doubly (Interval) Truncated Random Variables

In this article, we study reliability measures such as geometric vitality function and conditional Shannon's measures of uncertainty proposed by Ebrahimi (1996 Ebrahimi , N. ( 1996 ). How to measure uncertainty in the residual lifetime distribution . Sankhya A 58 : 48 – 56 . [Google Scholar]) and Sankaran and Gupta (1999 Sankaran , P. G. , Gupta , R. P. ( 1999 ). Characterization of lifetime distributions using measure of uncertainty . Calcutta Statist. Assoc. Bull. 49 : 195 – 196 . [Google Scholar]), respectively, for the doubly (interval) truncated random variables. In survival analysis and reliability engineering, these measures play a significant role in studying the various characteristics of a system/component when it fails between two time points. The interrelationships among these uncertainty measures for various distributions are derived and proved characterization theorems arising out of them.     

Geometric Ergodicity and Moment Conditions for a Seasonal GARCH Model with Periodic Coefficients

A seasonal GARCH process with periodic coefficients is considered and conditions for periodic stationarity, geometric ergodicity, β-mixing property with exponential decay rate, and existence of higher-order moments are obtained.