Journal of Iranian Statistical Society
Volume:11 Issue: 1, 2012

We consider n observations from the GARCH-type model: S = σ2Z, where σ2 and Z are independent random variables. We develop a new wavelet linear estimator of the unknown density of σ2 under four different dependence structures: the strong mixing case, the β- mixing case, the pairwise positive quadrant case and the ρ-mixing case. Its asymptotic mean integrated squared error properties are explored. In each case, we prove that it attains a fast rate of convergence.

We consider a representation of the probability density function of a weighted convolution of the gamma distribution, where a confluent hypergeometric function describes how the differences between the parameters of the components of scale lead to departures from a density range. It is shown that the distributions can be characterized as the product between a gamma density and a confluent hypergeometric function. We give closed-form expressions for the cumulative, survival and hazard rate function. The corresponding moment generating function(m.g.f) and cumulant generating function(c.g.f) have been calculated and their properties have bean discussed.

In this paper, assuming that (X, Y1, Y2)T has a trivariate normal distribution, we derive the exact joint distribution of (X, Y(1), Y(2))^T, where Y(1) and Y(2) are order statistics arising from (Y1, Y2)T. We show that this joint distribution is a mixture of truncated trivariate normal distributions and then use this mixture representation to derive the best (nonlinear) predictiors of X in terms of (Y(1), Y(2))^T. We also predict Y(1) in terms of (X, Y(2))^T, and Y(2) in terms of (X, Y(1))^T. Finally illustrate the usefulness of these results by using real-life data.

In this work, we calculate the limit distribution of the total cost incurred by splitting a tree selected at random from the set of all finite free trees. This total cost is considered to be an additive functional induced by a toll equal to the square of the size of tree. The main tools used are the recent results connecting the asymptotics of generating functions with the asymptotics of their Hadamard product, and the method of moments.

In this paper an approximation for entropy rate of an ergodic Markov chain via sample path simulation is calculated. Although there is an explicit form of the entropy rate here, the exact computational method is laborious to apply. It is demonstrated that the estimated entropy rate of Markov chain via sample path not only converges to the correct entropy rate but also does it exponentially fast.

In this work we give an alterative proof of one of basic properties of zonal polynomials and generalised it for Jack polynomials.