NONUNIFORM ESTIMATES IN THE APPROXIMATION BY THE IRWIN LAW

An approximation of a cumulative distribution function by the Irwin cumulative distribution function is considered. The approximating distribution function can be cumulative distribution function of sums (products) of independent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method. The cumulant method is used by introducing special cumulants which satisfy the V. Statulevičius type condition. The main result is a nonuniform bound for the difference between the cumulative distribution functions in terms of special cumulants.


Introduction
The approximation of cumulative distribution functions of sums (products) of independent (dependent) random variables by normal and Poisson distributions is analysed in many papers.In this paper, we consider the approximation by the Irwin law, because the Irwin law is interesting in probability theory and mathematical statistics.
In 1937, J. O. Irwin [2] considered a random variable Y = X 1 − X 2 , where X 1 ∼ P (λ), X 2 ∼ P (λ) are independent and P (λ) is Poisson distribution.In 1946, J. G. Skellam [10] and in 1952, A. de Castro and A. Prekopa [8] generalized this random variable Y , where X 1 ∼ P (λ 1 ) and X 2 ∼ P (λ 2 ) are independent; therefore, this distribution of random variable Y was named the Skellam distribution.Later, in 1962, J. Strakee and J. J. D. van der Gon [11] presented tables of the cumulative distribution function of the Skellam distribution to four digits for some combinations of values of the two parameters.In 2006, D. Karlis and I. Ntzoufras [4] started to examine the random variable Y , when X 1 ∼ P (λ 1 ) and X 2 ∼ P (λ 2 ) are correlated.The Skellam distribution estimates of the parameters obtained by the Of course, the estimates of the parameter will exist if S 2 − Y > 0. And only in 2000, D. Karlis and I. Ntzoufras [3] discussed in detail the properties of the Skellam distribution and obtained the maximum likelihood estimates.The Skellam distribution estimates of the parameters obtained by the maximum likelihood method are is a modified Bessel function of the first kind, and Later, in 2006, D. Karlis and I. Ntzoufras [4] derived Bayesian estimates and used the Bayesian approach for testing the equality of the two parameters of the Skellam distribution.
The Skellam distribution has many applications in different fields.D. Karlis and I. Ntzoufras [5] applied the Skellam distribution for modeling the difference of the number of goals in football games.They also modeled the difference in the decayed, missing and filled teeth (DMFT) index before and after treatment in article [4].This index is one of the most common methods in oral epidemiology for assessing dental caries prevalence as well as dental treatment needs among populations and has been used for about 75 years.Y. Hwang, J. Kim and I. Kweon [1] introduced the Skellam distribution as a sensor noise model for CCD (charge-coupled device) or CMOS (complementary metal-oxide-semiconductor) cameras.This is derived from the Poisson distribution of photons that determine the sensor response.They showed that the Skellam distribution can be used to measure the intensity difference of pixels in the spatial domain, as well as in the temporal domain.In addition, they showed that Skellam parameters are linearly related to the intensity of the pixels.On applications of the Skellam law in economics, see [6].
We use the cumulant method for the approximation by the Irwin law.The cumulant method is widely described in the L. Saulis' and V. Statulevičius' monography [9].

Irwin Law
We define the Irwin law as an instance of the Skellam distribution: The characteristic function of the Irwin law is Denote m = s +|k|, then we obtain −1) .
Applying the expansion we obtain the Irwin law cumulants By the expansion Nonuniform estimates in the approximation by the Irwin law we obtain the Irwin law moments In particular, the Irwin law moments are

Special cumulants in approximation by the Irwin law
To approximate the symmetric cumulative distribution function F(x) of a random variable X by the cumulative distribution function of the Irwin law, we use special cumulants where It is easy to find connection between Γ k and Γ m = d m d(it) m ln f X (t) t=0 , m = 1, 2, 3, . . ., k, k = 1, 2, 3, . ..: where !N! are the Catalan numbers, when N ∈ N and 1 A the indicator function of a set A.
In particular,  It is easy to make sure that for the Irwin law

Nonuniform estimates of the remainder term
In this section, we will find nonuniform estimates of the remainder term in the approximation by the Irwin law.For that purpose, we will use a lemma: Lemma 4.1 [7] If X and Y are integer-valued random variables with cumulative distribution functions F(x), G(x) and characteristic functions f (t), g(t), then Assume that a random variable X with the symmetric cumulative distribution function F(x) and a characteristic function f (t) have all finite moments EX k , k = 1, 2, 3, . ... Then ln f (t) can be represented in the form We denote the Irwin law cumulative distribution function with parameter λ by G(x).
Theorem 4.1 Let cumulants Γ k of integer-valued random variable X with the symmetric cumulative distribution function F(x) and the variance DX = 2λ, satisfy the V. Statulevičius type condition: then where Proof of Theorem 4.1.From the inequality |e w − 1| |w| e |w| , w ∈ C, we obtain f (t) − g(t) = e ln f (t) − e ln g(t) = g(t) e ln f (t)−ln g(t) − 1 Notice that Γ 1 = 2λ and ln g(t) = 2λz, where z = cost − 1; therefore, from (9), we obtain and Nonuniform estimates in the approximation by the Irwin law Since g(t) = e 2λ(cost−1) = e −4λ sin 2 t 2 , (14) from ( 12), (13) we obtain that From ( 11), we obtain 1 − 1 Now we estimate ( f (t) − g(t)) .We have Since from the estimates (13), ( 14), ( 15), ( 17) and ( 18), we obtain that  Theorem 4.1 is proved.We will apply the results of Theorem 4.1 for the approximation of cumulative distribution functions of sums of independent generalized Rademacher random variables by the cumulative distribution function of the Irwin law.