Abstract
The random sum distribution is a key role in probability theory and its applications as well, these applications could be used in different sciences such as insurance system, biotechnology, allied health science, etc. The statistical significance in random sum distribution initiates when using the applications of probability theory in the real life, where the total quantity X can be only observed, which is included of an unknown random number X of random contributions. Saddlepoint approximation techniques overcome this problem. Saddlepoint approximations are effective tools in getting exact expressions for distribution functions that are not known in closed form. Saddlepoint approximations usually better than the other methods in which calculation costs, but not necessarily about accuracy. This paper introduces the saddlepoint approximations to the cumulative distribution function for random sum Poisson- Exponential distributions in continuous settings. We discuss approximations to random sum random variable with dependent components assuming existence of the moment generating function. A numerical example of continuous distributions from the Poisson- Exponential distribution is presented.
Highlights
Saddlepoint approximations are powerful tools for obtaining accurate expressions for distribution functions which are not known in closed form
Saddlepoint approximations are depend on using the moment generating function (MGF) or, equivalently, the cumulant generating function (CGF), of a random variable
Saddlepoint approximation for cumulative distribution function shares the same accuracy with exact and the mean squared error of the saddlepoint approximation is MSE= 0.06604708 which shows that the saddlepoint approximation is almost exact
Summary
Saddlepoint approximations are powerful tools for obtaining accurate expressions for distribution functions which are not known in closed form. Saddlepoint approximations are depend on using the moment generating function (MGF) or, equivalently, the cumulant generating function (CGF), of a random variable. We will discuss approximations to the random sum variable with dependent components assuming the moment generating function that has been exists. Suppose a continuous random variable X has density function f ( x ) defined for all real values of x. For continuous random variable X with CGF K( s ) and unknown density f ( x ) , the saddlepoint density approximation of f ( x ) is given by (Johnson et al, 2005). Where the continuous random variable X has CDF F( x ) and CGF K( s ) with mean E( x ) and wand uare defined as w sgn( s) 2{sx K (s)}. W , uare function of x and saddlepoint s , where sis the implicitly defined function of x given by the unique solution respectively and sgn(s) to K (s) x captures sign and ( )
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