INTRODUCTION
Saddlepoint approximations are powerful tools for obtaining accurate terms for distribution functions that are difficult to obtain in closed form. Saddlepoint approximations almost surpass other techniques with respect to computational cost, although it does not necessarily surpass other techniques with respect to accuracy.
RANDOMSUM DISTRIBUTIONS
Randomsum distributions have many natural applications and appear frequently in probability theory and applications. For example, these distributions have a wide range of applications in branching processes (Neyman, 1939), renewal processes, damage processes (Rao et al., 1980), stopped random walks (Malinovskii, 1994) and risk theory (Esscher, 1932; Jensen, 1995; Gurland, 1957). One of the most important families of random sums is the family of Poisson random sums, in which N is a Poisson (λ) random variable and the X_{i}’^{s} are independent and identically distributed. The random variable Y_{N} is said to have a random distribution if Y is of the following form (Johnson et al., 2005) Eq. 1:
where, the number of terms N is uncertain, the random variables X_{i} are independent and identically distributed (with a common distribution V) and each X_{i} is independent of N. If N = 0, then we have Y = 0. Although, this is implicit in the definition, we want to call attention to this point for clarity. The distribution function of Y_{N} is given by the following Eq. 2:
where, for n≥1, G_{n}(y) is the distribution function of the independent sum . We can also express Y_{N} in terms of convolutions (Eq. 3):
where, f is the common distribution function for X_{i} and f^{*n} is the nfold convolution of f. If the common distribution X is discrete, then the random sum Y_{N} is discrete. On the other hand, if X is continuous and if P[N = n]>0, then the random sum Y has a mixed distribution. The mean of the random sum Y_{N} is as follows (Eq. 4):
The expected value of the mean has a natural interpretation. It is the product of the expected number of events N and the expected individual distribution X. This makes intuitive sense. The variance of the random sum is as follows (Eq. 5):
The momentgenerating function of the random sum Y_{N} is defined as follows (Eq. 6) (Hogg and Tanis, 1983):
where, the function In is the natural log function. The cumulantgenerating function is defined as follows in Eq. 7:
SADDLEPOINT APPROXIMATION TO DENSITIES AND MASS FUNCTIONS
The most basic saddlepoint approximation was introduced by Daniels (1954) and is fundamentally a formula for approximating the density and mass function from an associated momentgenerating function. Saddlepoint approximations are constructed by assuming the existence of the Momentgenerating Function (MGF) or equivalently, the Cumulantgenerating Function (CGF), of a random variable. For improvements to the saddlepoint methodology and associated techniques, the reader is referred to Daniels (1954, 1987) for details concerning density and mass approximation, Skovgaard (1987) for a conditional version of this approximation, Reid (1988) for applications to inference, Borowiak (1999) for discussion of a tailarea approximation with a uniform relative error and Terrell (2003) for a stabilized lugannanirice formula.
Suppose a random variable X has density function f(x) identified for all real values of x. The MGF is defined as follows (Hogg and Craig, 1978) (Eq. 8):
Wherever this expectation exists, M_{x}(0) always exists and is equal to 1. We shall assume that M_{x}(s) converges over the largest open neighborhood (a, b) at zero. The cumulantgenerating function CGF is given by the following Eq. 9:
For a continuous random variable X with CGF K_{x}(s) and unknown density f(x), the saddlepoint approximation density of f(x) is given by the following Eq. 10:
where,ŝ = ŝ(x) denotes the unique solution to the saddlepoint equation K’(ŝ) = x and K”_{x}(ŝ) is the second derivative (Daniels, 1954). This approximation is useful for values of x that are within the point of support {x: f(x)>0} = χ.
The saddlepoint approximation for univariate cumulative distribution functions F(x) is given by the following Eq. 11:
where, the continuous random variable X has CDF F(x) and CGF CGFK_{x}(s) with mean μ = E(x) and and are defined as follows (Eq. 12):
ŵ and are functions of x and saddlepoint ŝ , where ŝ is the implicitly defined function of x given by the unique solution to K’_{X}(ŝ) = x. The symbols φ and Φ denote the standard normal density and CDF, respectively and sgn(ŝ) captures the sign (±) of ŝ (Butler, 2007).
APPLICATIONS OF THE RANDOMSUM POISSONWEIBULL DISTRIBUTION
The Weibull distribution is a continuous probability distribution with the following probability density function (Eq. 13):
where, k>0 is a shape parameter and η is a scale parameter of the distribution. The mean and the variance of a Weibull random variable can be expressed as follows:
The MGF is defined as follows:
The randomsum PoissonWeibull variable has the following form (Eq. 14):
where, the sample size N follows a Poisson (λ) distribution and the X_{i}^{,s} are i.i.d. random variables that follow a Weibull distribution. The sum Y_{N} is said to have a randomsum PoissonWeibull distribution. The exact calculation of this distribution is very complex and difficult. The saddlepoint approximation method overcomes this problem. This method is based on the momentgenerating function for the random sum. The cumulantgenerating function for N is given by the following Eq. 15:
For X_{i}^{,s} that are i.i.d. random variables following a Weibull distribution, the cumulantgenerating function is defined as follows (Eq. 16):
This leads to the cumulantgenerating function for the randomsum PoissonWeibull distribution, which takes the following form (Eq. 17):
The saddlepoint equation is as follows (Eq. 18):
Saddlepoint computation involves first and second derivatives. The saddlepoint solution is a root of the first derivative. The unique real root ŝ = ŝ(x) can be determined numerically. The second derivative of the cumulantgenerating function is given by the following Eq. 19:
Using Eq. 10, the saddlepoint density function for the randomsum PoissonWeibull distribution can be expressed in the following form (Eq. 20):
where, K”Y_{N}(ŝ) is given by Eq. 19 and KY_{N}(ŝ) is the cumulantgenerating function at the value of saddlepoint ŝ.
Because of its flexible shape and its ability to model a wide range of models, the Weibull distribution has been used successfully in many applications, such as the modeling of wind speeds.
Wind speeds in most places in the world can be modeled using a Weibull distribution. This statistical tool tells us how often winds of different speeds will be observed at a location with a certain average (mean) wind speed. Knowing this helps us to choose a wind turbine with the optimal cutin speed (the wind speed at which the turbine starts to generate usable power) and the cutout speed (the speed at which the turbine hits the limit of its alternator and can no longer put out power with further increases in wind speed).
The shape of the Weibull distribution depends on a parameter called (helpfully) shape. In Northern Europe and most other locations around the world, the value of shape is approximately 2. The shape parameter will typically range from 13. For a given average wind speed.
Let N be the number of wind events that occur in one country or region during a given time period and let X be the amount of energy a wind turbine produces over a given time. Y_{N} gives the total amount of energy a wind turbine produces over a given time period.
Now, let as given in Eq. 11, with ŵ and as given in Eq. 12 and ŝ as given in Eq. 19. Let (k = 1 shape parameter ), λ = 5 and η = 4 when x = 0.02. Then, we can determine the value of the saddlepoint to be:
ŝ= 7.65569415
The cumulantgenerating function value is:
K_{Y}(ŝ) = 4.841886117
The second derivative of the cumulantgenerating function is:
K"_{Y}(ŝ) = 0.005059644256
and:
ŵ= 3.06227766, = 0.544558531
Table 1:  Compares the exact probabilities with saddlepoint approximations for PoissonWeibull distribution 


Fig. 1:  Comparison of the exact CDF for a PoissonWeibull distribution (solid line) vs. its saddlepoint approximation (dashed line) 
Then, the saddlepoint cumulative distribution function value for the randomsum PoissonWeibull distribution when x = 0.02 is:
We can use the empirical distribution function to determine the exact cumulative distribution function for the PoissonWeibull distribution by using the MATLAB program to simulate 10^{6} independent values of Y_{N}, where N is Poisson (5) and the X_{i}’^{s} are generated from a Weibull (1, 4) distribution. Table 1 compares the exact probabilities with saddlepoint approximations for PoissonWeibull distribution, for each X, the first value of each cell of Table 1 is the exact PoissonWeibull distribution, the second is the saddlepoint approximations and the third value is the relative error.
Figure 1 shows that the saddlepoint approximation of the CDF has the same accuracy as the exact CDF. The mean squared error of the saddlepoint approximation is MSE = 0.0^{7}83707, which indicates that the saddlepoint approximation is almost exact. From the CDF, we can drive various statistical measures of central tendency, such as the mean, the median and the mode. The median is the inverse of the CDF at 0.5. Using the MATLAB program, we find that Φ^{1}(0.5) = 18. From Eq. 4, we find that the mean = 20. Because the distribution is asymmetric, we use the formula below to drive the mode:
Mode = 3median2mean = 5440 = 14
CONCLUSION
This study introduced saddlepoint approximations to the cumulative distribution function for randomsum PoissonWeibull distributions in continuous settings. We discussed approximations to randomsum random variables with dependent components assuming the presence of a momentgenerating function. Measures of central tendency were derived. We used an empirical distribution function to calculate the exact CDF value by simulation of one million independent values of Y_{N}. A numerical example of a continuous distribution function for a PoissonWeibull distribution was presented. We found that the saddlepoint approximation for CDF yields the same accuracy as the exact CDF and that the mean squared error of the saddlepoint approximation was close to zero.
ACKNOWLEDGMENTS
This study was supported financially by Taibah University, Al MadinahM., Kingdom of Saudi Arabia.