For square contingency tables with ordered categories, we decompose the symmetry model into three models; i.e., the palindromic symmetry, the marginal means equality, and the cumulative subsymmetry models. The palindromic symmetry model is also decomposed into the generalized palindromic symmetry and the extended marginal homogeneity models. The decompositions are applied to the unaided vision data.

A fundamental problem encountered in statistics is that of testing the equality of scale parameters against umbrella alternative with at least one strict inequality. In this article, a nonparametric test based on U-statistic by considering the subsample minima and maxima for several sample scale problem against umbrella alternative, when peak of the umbrella is known, is proposed. The proposed test have the advantage of not requiring the several distribution functions to have a common median, but rather any common quantile of order q, 0 ? q ?1, (not necessarily ½) which is assumed to be known. Pitman efficiency indicate that the proposed test is equivalent to the test B proposed by Gaur, Mahajan and Arora (2012).

Distributional properties of two non-adjacent dual generalized order statistics have been used to characterize distributions. Further, one sided contraction and dilation for the dual generalized order statistics are discussed and then the results are deduced for generalized order statistics, order statistics and adjacent dual generalized order statistics and generalized order statistics.

Recently Sarhan and Balakrishnan (Journal of Multivariate Analysis, 98, 1508 - 1527, 2007) introduced a new singular bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Sarhan-Balakrishnan did not discuss any estimation procedure of the unknown parameters. In Sarhan-Balakrishnan model, there is no scale parameter. Unfortunately without the presence of any scale parameter, it is dif?cult to use it for any data analysis purposes. We introduce a scale parameter in the model and it becomes a four-parameter bivariate model. The usual maximum likelihood calculation involves a four dimensional optimization problem. We discuss the maximum likelihood estimation of the unknown parameters using EM algorithm, and it involves only a one-dimensional optimization calculation at each M-step of the EM algorithm. One data analysis has been performed for illustrative purposes. The performance of the EM algorithm is very satisfactory.

An integral part of successful risk management in modern financial markets is the accurate calculation of the price sensitivities of the underlying asset. There are a number of recent research papers that have focused on this important issue. A strand of literature has applied the finite difference method which is biased. Another strand of literature has made use of the Malliavin calculus within a jump diffusion framework. However, the existing papers have provided the price sensitivities by conditioning on some of the stochastic part of the complicated random process. The current paper provides price sensitivities in jump diffusion model without conditioning on any stochastic part in the model. These estimates are shown to be unbiased. Thus, the solution that is provided in this paper is expected to induce decision making under uncertainty more precise.

It is shown that the (empirically determined) mode of the kernel estimate uniformly converges to the conditional mode function under the ?-mixing condition over an increasing sequence of compact sets which increases to d.