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Information Journal Paper

Title

Stochastic Comparisons of Convolution of Independent Random Variables in the Scale Model

Pages

  25-40

Abstract

 Introduction Convolutions of independent random variables often arise in many applied areas, including applied probability, reliability theory, actuarial science, nonparametric goodness-of-fit testing, and operations research. Since the distribution theory is quite complicated when the Convolution involves independent and non-identical random variables, it is of great interest to investigate the stochastic properties of Convolutions and derive bounds and approximations on some characteristics of interest in this setup. The results in this work give responses that under the new condition, the Convolution of two random variables is ordered according to some stochastic orders such as Likelihood Ratio Order, Hazard Rate Order and reversed Hazard Rate Order. In general cases, let X1, X2 and X,1,X,2 be independent random variables that X1 , lr X2 and X,1 , lr X,2. Then it is not necessarily true that X1 + X2 , lr X,1 + X,2. However, if these random variables have log-concave densities, then it is true. This paper compared the random variables from Scale models according to Likelihood Ratio Order, Hazard Rate Order and reversed Hazard Rate Order. Random variable X be said to belong to the scale family of distributions if it has the distribution function. The density function F(, x) and , f(, x), respectively, where F is an absolutely continuous distribution function with density function f and ,is the scale parameter. Material and Methods The comparison of essential characteristics associated with lifetimes of technical systems is an exciting topic in reliability theory since it usually enables us to approximate complex systems with simpler designs and subsequently obtain various bounds for important ageing characteristics of the complex system. A convenient tool for this purpose is the theory of stochastic orderings. Results and Discussion This paper deals with some stochastic comparisons of Convolution of random variables comprising scale variables. Sufficient conditions are established for these Convolutions’,Likelihood Ratio Ordering and Hazard Rate Ordering. The results established in this paper generalize some known results in the literature. Several examples are also presented for more illustrations. Conclusion Convolutions of independent random variables occur quite frequently in probability and statistics, stochastic activity networks, optics, acoustics, electrical engineering, physics, the area of digital signal and insurance mathematics. Therefore, their stochastic properties are essential and have been discussed extensively in the literature. We obtained sufficient conditions to compare the Convolution of random variables from the Scale model concerning likelihood ratio and Hazard Rate Order. Recently Amini-Seresht and Barmalzan (٢, ٠, ٢, ٠, ) have studied ordering properties of parallel and series systems consisting of outlier scale components. They provided some sufficient conditions on the parameter vectors for the likelihood ratio, hazard rate, reversed hazard rate and mean residual lifetime orders between the lifetimes of the series and parallel systems, respectively. Therefore, a generalization of the present work to the case random variables with an outlier Scale model framework will be of interest. We are working on this problem and hope to report these findings in a forthcoming paper.

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    APA: Copy

    Amini Seresht, E., Barmalzan, GH., & Nasiroleslami, E.. (2022). Stochastic Comparisons of Convolution of Independent Random Variables in the Scale Model. JOURNAL OF STATISTICAL SCIENCES, 16(1 ), 25-40. SID. https://sid.ir/paper/1021459/en

    Vancouver: Copy

    Amini Seresht E., Barmalzan GH., Nasiroleslami E.. Stochastic Comparisons of Convolution of Independent Random Variables in the Scale Model. JOURNAL OF STATISTICAL SCIENCES[Internet]. 2022;16(1 ):25-40. Available from: https://sid.ir/paper/1021459/en

    IEEE: Copy

    E. Amini Seresht, GH. Barmalzan, and E. Nasiroleslami, “Stochastic Comparisons of Convolution of Independent Random Variables in the Scale Model,” JOURNAL OF STATISTICAL SCIENCES, vol. 16, no. 1 , pp. 25–40, 2022, [Online]. Available: https://sid.ir/paper/1021459/en

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