Hypothesis testing the homogeneity of means of k univariate normal populations against the hypothesis of one sided ordered means with unknown and equal variances is considered. A new completely method to find the uniformly most powerful test at significance level α is presented based on the multivariate t distribution. Since for more than two populations finding the null distribution of test statistic is not easy, the power of test is computed and then the critical values of test statistic for different significance levels obtained. This testing method is used for real examples. Also testing homogeneity of k mean vectors against two sided ordered mean vectors of multivariate normal populations is considered. Using Monte Carlo simulation the values of classical power of test for two bivariate and trivariate normal distributions at different significance levels are compared.

The collective risk model of insurance company with constant initial capital when process of claims number have the poisson distribution with constant rate is considered. For computing the infinite time ruin probability the stochastic processes and differential equations are used. Also a formula is obtained to compute the Lundberg approximation in finding the approximate of infinite time ruin probability based on the distribution function of claims number. The numerical examples to illustrate these results are given and showed that for any value of initial capital the approximate of our infinite time ruin probability is closer to its real value rather than the ruin probability computed by other authors and has less error.

In this paper, it is shown that using the cliams with Pareto distribution for computing the ruin probabilities could has detriment for the heads of insurance company. With computing the relative error of these cliams it is shown that the estimation of claims mean is not suitable in insurance models. We will show that existance of claims with Pareto distribution in the excess of loss reinsurance model may be detriment for the policyholders of company. Also in this portfolio, with computing the conditional expectation of claims measure show that using the claims with Pareto distribution is not suitable in the estimation of claims. The estimation of conditional expectation of random variable of claims is computed by simulation method for some of the statistical distributions. The results are investigated with real examples.

In this article, we wish to find and select appropriate estimators for statistical population density function using line transect sampling in the present of detection functions with light and heavy tailed distributions. Also it is shown that how the type of detection function could be effective in selection of the best estimator and then we propose a unbiased estimators that has the lower variance than the existed estimators. the simulation results show that if detection functions have heavy tailed distribution, then the new estimators have least mean square error.

In the excess loss reinsurance risk model, the amount of insurance premium paid by the company is influential in the ruin of that company. In this paper, the premium function is presented based on the expected amount of total payments of the reinsurer to the assigning insurer, the constraint on this function is investigated, and for the claims with any arbitrary distribution, the contour plots are drawn and with presenting optimization algorithm, infinite time ruin probability function will be minimum for different values of initial capital and threshold value. Finally, the excess loss reinsurance risk model with non-exponential claims is considered, and the infinite time ruin probability is calculated with numerical examples.