Improved method for approximating the hazard rate for convolution Poisson random variables

In this study, we investigate the performance of the saddlepoint approximation of the probability mass function and the cumulative distribution function for the weighted sum of independent Poisson random variables. The goal is to approximate the hazard rate function for this complicated model. The better performance of this method is shown by numerical simulations and comparison with a performance of other approximation methods.


Introduction
The distribution of a linear combination of random variables appears in many applied problems and has attracted the attention of many scholars; specifically, the linear combination of two independent random variables has been investigated by many researchers (see e.g. Amari & Misra [1], Di Salvo [3], Almasi et al. [4], Al Mutairi and Low [5] among others). Currently, the demand for these models has increased due to their importance and wide range of applications.
This study improves on the methods of hazard rate function estimation for a linear combination of random variables by using the saddlepoint approximation.
It is well known that the Poisson distribution is stable; that is, the distribution of the sum of two independent Poisson random variables has the Poisson distribution. This means that if two random variables 1~P ( 1 ) and 2~P ( 2 ) are independent, that the distribution of their sum 1 + 2~P ( 1 + 2 ). At the same time, rescaling of a Poisson random variable is not Poisson anymore. Obviously, random variable = 0.5 , where ~P ( ) is not integer valued and hence not Poisson. Thus, the exact distribution of a linear combination of independent Poisson random variables is not always known and, even if it is, in most cases it is very difficult to obtain.
Therefore, methods of approximation are very important and can provide a useful information about the distribution of these complicated models. The saddlepoint approximation method can replace the exact answer with high accuracy. This adds to its many advantages in terms of saving time and efforts.
Remind that the cumulant generating function (CGF) of a random variable is defined as a logarithm of its moment generating function ( ) = ln . The saddlepoint technique requires calculating the cumulant generating function for the linear combination of Poisson random variables as ( ) = ∑ =1 ( − 1). Note that in Butler [6] Sections 1.1.2 and 1.1.5 (formulae (1.13) and (1.14)), the saddlepoint approximation is defined only for continuous and integer-valued distributions where ̂ the unique solution for the saddlepoint equation ′ () = .
The distribution of the linear combination of Poisson random variables is nor continuous, nor integer valued, but hence we need a continuity correction for the saddlepoint approximation. For more detailed summary of this arguments, see Daniels [6] and Daniels [7], Section 2 for more details. The first continuity correction for the right tail is given as where is the mean of the random variable , Φ and are the standard normal distribution and density functions, respectively, and ̃1 = s ()√2(̂− ()),̃1 = (1 − −̂) √ ′′ (). The second continuity correction is given as where − is the unique solution for the saddlepoint equation ′ () = − 0.5 = − , and The third continuity correction suggested in Butler [5], Section 1, p. 17-18, is the same as formula ̂2 but with ̃2 is replaced by ̃3 =√ ′′ ().
where 1~P ( 1 ) and 2~P ( 2 ). The CGF is To show the performance of this method, we consider the sum of two independent Poisson random variables with means 1 and 2 , which is also a Poisson random variable with mean hence, the saddlepoint is ̂= −1.098612289 and the PMS ̂( ) from equation (1) is The exact ( ) is given by calculating P (3) when = 1 (1) = 3 −3 = 0.1493612051.
Next we find the first continuity correction given in equation (2 with the absolute error of 0.007.
We note that saddlepoint approximations give accurate results in calculating the mass function and distribution function and can replace the exact answers. We can compare this with other approximation methods, for example, the normal approximation for the right tail when = 1 with the absolute error 0.07. There is also another methods of approximation, namely, the Haldane approximation. Pentikäinen [8] proved the accuracy of that method under certain circumstances. Suppose is a random variable with mean , variance 2 , and the coefficient of skewness . The Haldane type A approximation for the right tail is given by (1 − ℎ)(1 − 3ℎ) 2 , see Borowiak [9]. We get ̂= 0.8849 with absolute error of 0.06.
From the calculations above, we conclude that the saddlepoint approximation for the distribution function is more accurate than the normal and Haldane approximations.

The hazard rate function based on the saddlepoint approximation
The hazard rate function for a discrete distribution taking values in ℤ + = {0,1,2,3, … } is defined as (see, e.g., Daly [10]). Consider the same example of the sum of two independent Poisson random variables with means 1 and 2 as in the previous section. Equations (3) and (5)  The discussion above pertains the special case of the linear combination of two independent not identically Poisson random variables with constant 1 = 2 = 1. We consider this case because we know already the exact solution for this sum, which gives us a possibility to compare the exact solution with the methods of approximation. Now we consider other positive real values for 1 and 2 , which gives the linear combination of independent Poisson random variables. It has a complicated distribution and the exact mass and density functions in most cases are unknown or difficult to obtain. Hence, the methods of approximation are very important. Table 0 shows the exact values, saddlepoint continuity corrections, and the normal approximations of the hazard rate function for the linear combination = 3 1 + 5 2 , where 1~P (2) and
For example, suppose that 1~P (1), 2~P (2), 3~P (5), 4~P (3) with constants 1 = 2 = 3 = 4 = 1. Consider the value = 1. This also gives us a good chance to investigate the accuracy of this methods because the exact solution is already known as P (11). From the saddlepoint equation with the absolute error of 0.00002. The corresponding normal approximation for = 1 is ( ≥ 1) = 1 − Φ( 1−11 11 ) = 0.8159 with the absolute error of 0.184. This means that the saddlepoint approximation is closer to the exact value than the normal approximation.
From the calculations above we can derive the saddlepoint approximations for the hazard rate for the first continuity correction when = 1 as   From Table 2 we see that the saddlepoint method of approximation is doing well with high level of accuracy. To derive the exact mass and distribution function intractable calculations are needed. The saddlepoint provides a fairly quick and accurate method of calculating for tail probability.

Conclusion
The saddlepoint approximation technique is an efficient tool that can be used to derive an accurate approximation to the hazard rate function for complicated models, such as a weighted sum of independent not identically Poisson distributed random variables.