Extreme Value Analysis for Progressively Type-II Censored Order Statistics

An account to extreme value theory for progressively Type-II censored order statistics is presented which enables us to handle limit laws for upper and lower extreme, intermediate and central progressively Type-II censored order statistics within one framework. We illustrate that the extreme value analysis for progressively Type-II censored order statistics is connected to limit laws for sums of independent but not-identically distributed exponential random variables. Moreover, we show that the limits are transformations of extreme value distributions and illustrate the connection to extreme value analysis for order statistics.     

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Linear estimation of location and scale parameters in the generalized gamma distribution

The expressions for moments of order statistics from the generalized gamma distribution are derived. Coefficients to get the BLUEs of location and scale parameters in the generalized gamma distribution are computed. Some simple alternative linear unbiased estimates of location and scale parameters are also proposed and their relative efficiencies compared to the BLUEs are studied.     

Simultaneous estimation of ordered parameters

The problem of estimating ordered parameters is encountered in biological, agricultural, reliability and various other experiments. Consider two populations with densities f₁(x₁-ω₁) and f₂(x₂-ω₂) where ω₁#ω₂. The estimation of ω₁,ω₂) with the loss function, the sum of squared errors, is studied. when f_i is the f_i(,_i,,_i ²) density with ,_i known, i=1,2; we obtain a class of minimax estimators. When ω₁ #ω₂ we show some of these estimators are improved by the maximum likelihood estimator. For a general f_i we give sufficient conditions for the minimaxity of the analogue of the Pitman estimator.     

Abel series distributions with applications to fluctuations of sample functions of stochastic processes

A two-parameter class of discrete distributions, Abel series distributions, generated by expanding a suitable pa,rametric function into a series of Abel polynomials is discussed. An Abel series distribution occurs in fluctuations of sample functions of stochastic processes and has applications in insurance risk, queueing, dam and storage processes. The probability generating function and the factorial moments of the Abel series distributions are obtained in closed forms. It is pointed out that the name of the generalized Poisson distribution of Consul and Jain is justified by the form of its generating function. Finally it is shown that this generalized Poisson distribution is the only member of the Abel series distributions which is closed under convolution.     

Correlation between the numbers of two types of children in a family with the mpsd for the family size

The family size (sibship size) N is regarded as an integer-valued random variable having a Modified Power Series distribution (MPSD) of Gupta (1974). The family produces two types of children, with probabilities p and q (p+q =1) . It is proved that the correlation between the numbers B and C of these children is positive or negative according as the function log f(θ) is convex or concave with respect to the function g(θ), (see Section 2). This condition is a simple and a natural extension of the one given by Rao et al (1973). Several examples are discussed to illustrate the result.     

Multivariate regression models for discrete and continuous repeated measurements

A general class of multivariate regression models is considered for repeated measurements with discrete and continuous outcome variables. The proposed model is based on the seemingly unrelated regression model (Zellner, 1962) and an extension of the model of Park and Woolson(1992). The regression parameters of the model are consistently estimated using the two-stage least squares method. When the out come variables are multivariate normal, the two-stage estimator reduces to Zellner’s two-stage estimator. As a special case, we consider the marginal distribution described by Liang and Zeger (1986). Under this this distributional assumption, we show that the two-stage estimator has similar asymptotic properties and comparable small sample properties to Liang and Zeger's estimator. Since the proposed approach is based on the least squares method, however, any distributional assumption is not required for variables outcome variables. As a result, the proposed estimator is more robust to the marginal distribution of outcomes.     

ESTIMATING PARAMETERS IN AUTOREGRESSIVE MODELS IN NON-NORMAL SITUATIONS: ASYMMETRIC INNOVATIONS

The estimation of coefficients in a simple autoregressive model is considered in a supposedly difficult situation where the innovations have an asymmetric distribution. Two distributions, gamma and generalized logistic, are considered for illustration. Closed form estimators are obtained and shown to be efficient and robust. Efficiencies of least squares estimators are evaluated and shown to be very low. This work is an extension of that of Tiku, Wong and Bian [1] Tiku, M. L., Wong, W. K. and Bian, G. 1999. Time Series Models with Asymmetric Innovations. Commun. Stat.-Theory Meth., 28: 1131–1160.  [Google Scholar] who give solutions for a simple AR(1) model.

     

THE MEDIAN ESTIMATE OF THE AUTOREGRESSIVE LOCATION PARAMETER

ABSTRACT

The concept of generalized order statistics was introduced by Kamps (1995 Kamps , U. ( 1995 ) A Concept of Generalized Order Statistics . Germany : B. G. Teubner Stuttgart [Crossref] , [Google Scholar]) to unify several concepts that have been used in statistics such as order statistics, record values, and sequential order statistics. Estimation of the parameters of the Burr type XII distribution are obtained based on generalized order statistics. The maximum likelihood and Bayes methods of estimation are used for this purposes. The Bayes estimates are derived by using the approximation form of Lindley (1980 Lindley , D. V. ( 1980 ). Approximate Bayesian methods . J. Trabajos de Estadistica 31 : 223 – 237 .[Crossref] , [Google Scholar]). Estimation based on upper records from the Burr model is obtained and compared by using Monte Carlo simulation study. Our results are specialized to the results of AL-Hussaini and Jaheen (1992 AL-Hussaini , E. K. , Jaheen , Z. F. ( 1992 ). Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model . J. Statist. Comput. Simul. 41 : 31 – 40 .[Taylor &; Francis Online] , [Google Scholar]) which are based on ordinary order statistics.     

MULTI-LEVEL FACTORIAL DESIGNS WITH MINIMUM NUMBERS OF LEVEL CHANGES

The order of experimental runs in a fractional factorial experiment is essential when the cost of level changes in factors is considered. The generalized foldover scheme given by [1] Coster, D. C. and Cheng, C. S. 1988. Minimum cost trend free run orders of fractional factorial designs. The Annals of Statistics, 16: 1188–1205. [Crossref], [Web of Science ®] , [Google Scholar]gives an optimal order to experimental runs in an experiment with specified defining contrasts. An experiment can be specified by a design requirement such as resolution or estimation of some interactions. To meet such a requirement, we can find several sets of defining contrasts. Applying the generalized foldover scheme to these sets of defining contrasts, we obtain designs with different numbers of level changes and then the design with minimum number of level changes. The difficulty is to find all the sets of defining contrasts. An alternative approach is investigated by [2] Cheng, C. S., Martin, R. J. and Tang, B. 1998. Two-level factorial designs with extreme numbers of level changes. The Annals of Statistics, 26: 1522–1539. [Crossref], [Web of Science ®] , [Google Scholar]for two-level fractional factorial experiments. In this paper, we investigate experiments with all factors in slevels.