In this paper, through the monotonicity property of reversed hazards ratio, a new stochastic ordering is introduced. Two well-known parametric families of distributions are proved to be ordered with respect to their parameters, according to the new proposed stochastic order. Other situations, where the new order is applicable, are described in details. A number of elementary and basic properties of the order, for example, preservation under increasing transformations are derived. Stochastic comparisons of the parallel systems are made using the new stochastic order. In parallel with the obtained results, some examples will explain the concepts.

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived .

The methods of estimation of nonparametric regression function are quite common in statistical application. In this paper, the new Bayesian wavelet thresholding estimation is considered. The new mixture prior distributions for the estimation of nonparametric regression function by applying wavelet transformation are investigated. The reversible jump algorithm to obtain the appropriate prior distributions and value of thresholding is used. The performance of the proposed estimator is assessed with simulated data from well-known test functions by comparing the convergence rate of the proposed estimator with respect to another by evaluating the average mean square error and standard deviations. Finally by applying the developed method, density function of galaxy data is estimated.

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