Time series models associated with Mittag-Leffler type distributions and its properties

ABSTRACT

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order α(0 < α ? 1). A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors, and sample path properties are studied. Finally we introduce the q-Mittag-Leffler process associated with the q-Mittag-Leffler distribution.     

The sample monomode and an associated test for discrete monomodality

Abstract

The notions of (sample) mean, median and mode are common tools for describing the central tendency of a given probability distribution. In this article, we propose a new measure of central tendency, the sample monomode, which is related to the notion of sample mode. We also illustrate the computation of the sample monomode and propose a statistical test for discrete monomodality based on the likelihood ratio statistic.     

Invariance of linearity of regression and related characterizations of some classical discrete distributions

Rao (1963) introduced what we call an additive damage model. In this model, original observation is subjected to damage according to a specified probability law by the survival distribution. In this paper, we consider a bivariate observation with second component subjected to damage. Using the invariance of linearity of regression of the first component on the second under the transition of the second component from the original to the damaged state, we obtain the characterizations of the Poisson, binomial and negative binomial distributions within the framework of the additive damage model.     

Multiple nonlinear regression of the Markovian arrival process for estimating the daily global solar radiation

Abstract

Solar radiation is a global ecological phenomenon that affects life everywhere. In this study, a new statistical method, called the Quartiles-Moment's method, is proposed to estimate the scale and shape parameters of the exponentiated Gumbel maximum distribution (EGMD). The Kolomogorov–Smirnov test and the percentiles of the dataset are thus used to fit the dataset of the daily global solar radiation and the corresponding daily maximum temperature with EGMD. Thence, multiple nonlinear regression of the daily global solar radiation and the corresponding daily maximum temperature are produced and compared with the real dataset accordingly.     

A note on the estimation of pr{y<x} for truncation parameter distributions

In this note a relationship in the treatment of the lower and upper truncations considered in Beg (1980) is pointed out and the minimum variance unbiased estimator of P = Pr{Y<X) for the (upper) truncated exponential distribution is obtained.     

Gammaization and wishartness of dependent quadratic forms

In this paper two equivalent sets of necessary and sufficient conditions are derived for dependent quadratic forms to be distributed as multivariate gamma distribution. The procedure also gives a set of necessary and sufficient conditions for principal minors of generalized quadratic forms to be jointly distributed as the joint distribution of principal minors of a Kishart matrix.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

Approximations of the critical region of the fbietkan statistic

The Friedman (1937) test for the randomized complete block design is used to test the hypothesis of no treatment effect among k treatments with b blocks. Difficulty in determination of the size of the critical region for this hypothesis is com¬pounded by the facts that (1) the most recent extension of exact tables for the distribution of the test statistic by Odeh (1977) go up only to the case with k6 and b6, and (2) the usual chi-square approximation is grossly inaccurate for most commonly used combinations of (k,b). The purpose of this paper 2 is to compare two new approximations with the usual x² and F large sample approximations. This work represents an extension to the two-way layout of work done earlier by the authors for the one-way Kruskal-Wallis test statistic.