Accurately and adequately modelling and analyzing relationships in real random phenomena involving several variables are prominent areas in statistical data analysis. Applications of such models are crucial and lead to severe economic and financial implications in human society. Since the beginning of developments in Statistical methodology as the formal scientific discipline, correlation based regression methods have played a central role in understanding and analyzing multivariate relationships primarily in the context of the normal distribution world and under the assumption of linear association. In this paper, we aim to focus on presenting notion of dependence of random variables in statistical sense and mathematical requirements of dependence measures. We consider copula functions and mutual information which are employed to characterize dependence. Some results on copulas and mutual information as measure of dependence are presented and illustrated using real examples. We conclude by discussing some possible research questions and by listing the important contributions in this area.

Models, mathematical or stochastic, which move from one functional form to another through pathway parameters, so that in between stages can be captured, are examined in this article. Models which move from generalized type-1 beta family to type-2 beta family, to generalized gamma family to generalized Mittag-Lef?er family to Levy distributions are examined here. It is known ´ that one can likely ?nd an approximate model for the data at hand whether the data are coming from biological, physical, engineering, social sciences or other areas. Different families of functions are connected through the pathway parameters and hence one will ?nd a suitable member from within one of the families or in between stages of two families. Graphs are provided to show the movement of the different models showing thicker tails, thinner tails, right tail cut off etc.

Fractional designs involve selection from a given set of experimental treatments as subset of treatments to makeup a specified design measure that has such statistical properties as balance, high relative efficiency, D-optimality etc. For decades statisticians have relied on the use Defining Contracts (DC), and Latin Squares (LS) to construct fractional factorial designs. But these methods are shown to have very limited range of applications and sometimes produce designs that are singular. This paper introduces the method of Concentric Balls (CB) for constructing non-singular fractional designs. Each ball consists of treatments that are of equal distance from the center and using a set of rules for selecting treatments from a ball the CB method yields a small set of admissible designs. The best member of this admissible set is the desired design:{Best in the sense of maximizing the determinant of the normalized information matrix or maximizing the relative efficiency of the factorial effects.}Numerical examples show that the CB method covers every range of experimental design conditions and can produce fractional designs that are D-optimal.

Reliable prediction of traffic loads is essential for bridge planning and design. In the study, an improved unbiased Grey Markov forecasting model is applied to predict traffic loads on highway bridges. The comparison between the predicted results and existing practical measurements indicates that high accuracy can be achieved by using the developed model for bridge load prediction. The developed model shows great promise in structural design and durability assessment of highway bridges.

In the Bayesian analysis with a statistical model, it is inevitable to determine a prior distribution of the unknown parameter. Since we encounter more and more complicated models in practical use, we need simple criteria by which we know whether there exists a certain class of prior on the statistical model. Recently, Takeuchi and Amari obtained the geometrical condition that a statistical model admits an alpha parallel prior, one generalization of well-known Jeffreys prior. Matsuzoe, Takeuchi and Amari studied extensively the geometric condition in a curved exponential family. We formulate their result in terms of differential two form called curvature form on statistical model manifolds, which seems more suitable to evaluation of global properties of statistical model. While the trace of two form vanishes in general class of statistical model including exponential family, it does not vanish in the autoregressive moving average model, which is very fundamental and practically important in time series analysis.

A new lifetime class with decreasing failure rate which is obtained by compounding truncated Poisson distribution and a lifetime distribution, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas(1998). A general form of probability, distribution, survival and hazard rate functions as well as its properties will be presented for such a class. This new class of distributions generalizes several distributions which have been introduced and studied in the literature.

Price-Weighted Apportionment Index measures the ?t between log demand distribution and log output distribution. We present the asymptotic sampling distribution of Price-Weighted Apportionment Index by assuming a multinomial distribution for the outcome variables. Our results are based mainly on large-sample normal approximation.