On the Estimation of Parameter of Weighted Sums of Exponential Distribution

The random variable , with and ？？being independent exponentially distributed random variables with mean one, is considered. Van Leeuwaarden and Temme (2011) attempted to determine good approximation of the distribution of . The main problem is estimating the parameter that has the main state in applicable research. In this paper we show that estimating the parameter by using the relation between and mode is available. The mean square error values are obtained for estimating by mode, moment method, and maximum likelihood method. 1. Introduction The exponential distribution is one of the most applicable distributions in survival models and phenomena with memoryless property. Random size sample from the exponential distribution and considering the discrete distribution are the basis for creating new distributions. Proschan [1] showed that combinations distributions with constant failure rate have decreasing hazard function. So, in recent years, new distributions are introduced based on generalization and correction of the exponential distribution. For example, Adamidis et al. [2] find a new bivariate distribution, exponential-geometric distribution, with decreasing failure rate that is used in survival models. This distribution combines with exponential and geometric distributions. In the case that parameter of exponential distribution has decreasing-increasing failure rate the new distribution is called exponential-weibull that in fact is a generalized of exponential-geometric distribution introduced in 2010 by Silva et al. [3]. In all these studies trough estimating the parameters algorithm EM has been used. Consider as the random variable with and being independent exponentially distributed random variables with mean one. In special case distribution function of the random variable is a simple alternating series. For the case accurs in various contexts, such as linear combinations of order statistics, noise in radio receivers, and run models, Kingman and Volkov [4]. For all the random variable follows the central limit theorem and leads to random variable normal standard. The result of normal approach of would be useful for in terms of X’s, which are near to mean . For the random variable is independent from and identity distribution and perfectly shows that the random variable has exponential distribution. One example of extreme value theory is that has Gumbel distribution as . Because the random variable is Kolmogorov distribution. In general case, the distribution function is a series with alternating sign that Van Leeuwarden and Temme [5] derived an