A Note on the Complementary Mixture Pareto II Distribution

We introduce a new survival distribution, of Pareto type, that arises from a cure-mixture frailty model. We describe its properties and demonstrate connections with familiar distributions including the Pareto and exponential. We derive its characteristic function and moments.     

Application of Concomitants of Order Statistics of Independent Non Identically Distributed Random Variables in Estimation

In this article, we obtain expressions for the pdf of a single concomitant of order statistic and the joint pdf of a pair of concomitants of order statistics of independent non identically distributed random variables. Using these expressions, we find the means, variances and covariances of order statistics arising from independent non identically distributed bivariate Pareto distributions. A method of estimation of a common parameter involved in several bivariate Pareto distributions using concomitants of order statistics is also discussed.     

Some Limit Results for Pareto Random Variables

We sample m (m ≥ 1) i.i.d. Pareto random variables with the density function x ^?2 (x ≥ 1) and establish two large deviations for the partial sums. In addition, the maxima of sums of the two-tailed Pareto random variables is discussed and some asymptotical forms are obtained also.     

Bayesian inference for the offered optical network unit load

Abstract

In this article, Bayesian inference for the Offered Optical Network Unit Load (OOL) using non-informative, gamma, power function, and gamma-power function priors is considered. Pareto distributed ON-and OFF-periods generated by the ON/OFF sources at an Optical Network Unit (ONU) in an Ethernet Passive Optical Network (EPON) system are assumed for our implementation in this article. A simulation study and a real-data-based illustrative example are given to demonstrate the advantages of the proposed Bayesian method over the large-sample method.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

A method for estimating lorenz curves

This paper proposes a new method for estimating the parameters of Lorenz Curves (LC’s) and fitting LC’s to observed data. The method is very general. It is applicable to any family of LC’s as long as it is given in closed form which is often the case in practice. The method can also be applied to either the LC or to its associated distribution. The estimators are easy to compute as they are obtained one at a time by solving only one equation in one unknown and in many cases the solutions are given in closed-forms. An additional advantage, that is not shared with the currently used method of estimation, is that the method is invariant as to the specification of which variable is written as a function of the other in the LC form. The method is applied to the most commonly suggested LC’s families. An example of real-life data is used to illustrate the methodology. A simulation study is performed to study the properties of the proposed estimators and to compare them with existing ones. The results seem to indicate that the proposed estimators have good properties and they often perform much better than the existing ones.     

Four general multivariate stationary extremal Markovian processes

Two general multivariate stationary Markovian process with maximization structure (denoted by Max-AR(1) and MaxI-AR(1)) are developed respectively. Max-AR(1) is a subclass of MaxI-AR(1). The characterization of the Max-AR(1) and MaxI-AR(1) to be stationary is studied. Some properties of the two maximization processes are derived. Two more related general multivariate stochastic Markovian process with minification structure are analogously constructed (denoted by Min-AR(1) and MinI-AR(1)). Some well known maximization and minification processes are special cases of these four extermal Markovian processes. Two of them are simulated and some point estimations are provided as an illustration of the wide application of these four processes.     

Small-sample estimators of the quantiles of the normal,log-normal and Pareto distributions

Estimators of the quantiles of the normal and log-normal distributions are derived. They are more efficient than the established estimators by a wide margin for small samples and high quantiles of the log-normal distribution. Although their evaluation is iterative, it requires only moderate amount of computing, which is not related to the sample size. The method is also applied to the quantiles of the Pareto distribution, but the resulting estimator is more efficient only in some settings. An application to financial statistics, estimating the return on a unit investment in equity markets over a long term, is presented.