The Joint Distribution of the Sum and the Maximum of IID Exponential Random Variables

The authors establish the joint distribution of the sum X and the maximum Y of IID exponential random variables. They derive exact formuli describing the random vector (X, Y), including its joint PDF, CDF, and other characteristics; marginal and conditional distributions; moments and related parameters; and stochastic representations leading to further properties of infinite divisibility and self-decomposability. The authors also discuss parameter estimation and include an example from climatology that illustrates the modeling potential of this new bivariate model.     

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

We investigate the convergence rates of uniform bias-corrected confidence intervals for a smooth curve using local polynomial regression for both the interior and boundary region. We discuss the cases when the degree of the polynomial is odd and even. The uniform confidence intervals are based on the volume-of-tube formula modified for biased estimators. We empirically show that the proposed uniform confidence intervals attain, at least approximately, nominal coverage. Finally, we investigate the performance of the volume-of-tube based confidence intervals for independent non-Gaussian errors.     

Prediction in the linear model under a linear constraint

In this paper we determine the Gauss–Markov predictor of the nonobservable part of a random vector satisfying the linear model under a linear constraint.     

Parameter estimation in nonlinear regression models 1

Fisher information contained in record values, inter-record times and their concomitants from a sample of fixed size is derived in general and explicit expressions are deduced for some specific known bivariate classes of distributions. A comparison between fixed sampling and inverse sampling schemes with equal number of records and concomitants is also carried out. We also consider parameter estimation based on bivariate records and a small simulation study is done.     

Book review

W.G.Cochran: Sampling Techniques, 3rd. Ed. John Wiley & Sons, New York-Santa Barbara-London-Sydney-Toronto 1977. 428 S., £ 12.50; $21.50.

H.Toutenburg: Vorhersage in linearen Modellen. Akademie-Verlag, Berlin 1975, VIII, 176b S., 3 Tab., 28,– M.

O.Kallenberg: Random Measure. Akademie-Verlag, Berlin 1975; Academic Press, London 1976. 104 pp., 28,– M.     

On second-order improved estimation of a gamma scale parameter

This paper concludes our comprehensive study on point estimation of model parameters of a gamma distribution from a second-order decision theoretic point of view. It should be noted that efficient estimation of gamma model parameters for samples ‘not large’ is a challenging task since the exact sampling distributions of the maximum likelihood estimators and its variants are not known. Estimation of a gamma scale parameter has received less attention from the earlier researchers compared to shape parameter estimation. What we have observed here is that improved estimation of the shape parameter does not necessarily lead to improved scale estimation if a natural moment condition (which is also the maximum likelihood restriction) is satisfied. Therefore, this work deals with the gamma scale parameter estimation as a separate new problem, not as a by-product of the shape parameter estimation, and studies several estimators in terms of second-order risk.     

Predicting infectious disease outbreak risk via migratory waterfowl vectors

The spread of an emerging infectious disease is a major public health threat. Given the uncertainties associated with vector-borne diseases, in terms of vector dynamics and disease transmission, it is critical to develop statistical models that address how and when such an infectious disease could spread throughout a region such as the USA. This paper considers a spatio-temporal statistical model for how an infectious disease could be carried into the USA by migratory waterfowl vectors during their seasonal migration and, ultimately, the risk of transmission of such a disease to domestic fowl. Modeling spatio-temporal data of this type is inherently difficult given the uncertainty associated with observations, complexity of the dynamics, high dimensionality of the underlying process, and the presence of excessive zeros. In particular, the spatio-temporal dynamics of the waterfowl migration are developed by way of a two-tiered functional temporal and spatial dimension reduction procedure that captures spatial and seasonal trends, as well as regional dynamics. Furthermore, the model relates the migration to a population of poultry farms that are known to be susceptible to such diseases, and is one of the possible avenues toward transmission to domestic poultry and humans. The result is a predictive distribution of those counties containing poultry farms that are at the greatest risk of having the infectious disease infiltrate their flocks assuming that the migratory population was infected. The model naturally fits into the hierarchical Bayesian framework.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.