© Hindawi Publishing Corp. A NONUNIFORM BOUND FOR THE APPROXIMATION OF POISSON BINOMIAL BY POISSON DISTRIBUTION

It is well known that Poisson binomial distribution 
can be approximated by Poisson distribution. In this paper, we 
give a nonuniform bound of this 
approximation by using Stein-Chen method.

1. Introduction and main result.Let X 1 ,X 2 ,...,X n be independent, possibly not identically distributed, Bernoulli random variables with P (X i = 1) = 1 − P (X i = 0) = p i and let S n = X 1 + X 2 + ••• + X n .The sum of this kind is often called a Poisson binomial random variable.In the case where the "success" probabilities are all identical, p i = p, S is the binomial random variable Ꮾ(n, p).Let λ = n i=1 p i and let ᏼ λ be the Poisson random variable with parameter λ, that is, P (ᏼ λ = ω) = e −λ λ ω /ω! for all nonnegative integers ω.It has long been known that if p i 's are small, then the distribution of S n can be approximated by a distribution of ᏼ λ (see, e.g., Chen [2]).
In this paper, we investigate the bound of this approximation.As an illustration, we look at the case of p 1 = p 2 = ••• = p n = p.There are at least three known uniform bounds: Kennedy and Quine [6] showed that, for 0 Barbour and Hall [1] showed that and Deheuvels and Pfeifer [5] proved that and [x] is understood to be the integer part of x.
2. Proof of the main result.Stein [9] gave a new technique to find a bound in the normal approximation to a distribution of a sum of dependent random variables.His technique was free from Fourier methods and relied instead on the elementary differential equation where h is a function that is used to test convergence and where Z is the standard normal.Chen [3] applied Stein's ideas in the Poisson setting.Corresponding to the differential equation in the normal case above, one has an analogous difference equation where ᏼ λ (h) = E[h(ᏼ λ )] and f and h are real-valued functions defined on Z + ∪{0}.Let ω 0 ∈ {1, 2,...,n− 1} and define h, h ω 0 : Then we see that the solution f of (2.2) can be expressed in the form ) Let S (i) n = S n − X i for i = 1, 2,...,n.By using the facts that each X j takes on values 0 and 1 and that X j 's are independent, we have which implies, by (2.5), that From (2.4), it follows that (2.8) we have where we have used the facts that λ ∈ (0, 1] and 0 ≤ ω + 1 − k ≤ ω + 1 in the first inequality and the conditions ω ≤ ω 0 − 2 and e −λ ω+1 k=0 (λ k /k!) ≤ 1 in the second inequality.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation