Signed binomial approximation of binomial mixtures via differential calculus for linear operators

We obtain sharp estimates in signed binomial approximation of binomial mixtures with respect to the total variation distance. We provide closed form expressions for the leading terms, and show that the corresponding leading coefficients depend on the zeros of appropriate Krawtchouk polynomials. The special case of P?olya-Eggenberger distributions is discussed in detail. Our approach is based on a differential calculus for linear operators represented by stochastic processes, which allows us to give unified proofs.

[1]  K. Takeuchi,et al.  On sum of 0–1 random variables I. Univariate case , 1987, Contributions on Theory of Mathematical Statistics.

[2]  P. Deheuvels,et al.  A Semigroup Approach to Poisson Approximation , 1986 .

[3]  B. Roos,et al.  An expansion in the exponent for compound binomial approximations , 2006 .

[4]  A. Barbour,et al.  Poisson Approximation , 1992 .

[5]  B. Roos Improvements in the Poisson approximation of mixed Poisson distributions , 2003 .

[6]  Fernando López-Blázquez,et al.  Binomial approximation to hypergeometric probabilities , 2000 .

[7]  W. Ehm Binomial approximation to the Poisson binomial distribution , 1991 .

[8]  Bivariate distributions generated from Po´lya-Eggenberger Urn models , 1990 .

[9]  Aihua Xia,et al.  Approximating the number of successes in independent trials: Binomial versus Poisson , 2002 .

[10]  M. Agarwal,et al.  Generalized Polya-Eggenberger model of order k via lattice path approach , 2002 .

[11]  ESTIMATION OF A PROPORTION USING SEVERAL INDEPENDENT SAMPLES OF BINOMIAL MIXTURES , 2005 .

[12]  May-Ru Chen,et al.  A new urn model , 2005 .

[13]  Harshinder Singh,et al.  Stochastic comparisons of Poisson and binomial random variables with their mixtures , 2003 .

[14]  Michel Loève,et al.  Probability Theory I , 1977 .

[15]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[16]  Moshe Shared,et al.  On Mixtures from Exponential Families , 1980 .

[17]  Norman L. Johnson,et al.  Urn models and their application , 1977 .

[18]  B. Roos Binomial Approximation to the Poisson Binomial Distribution: The Krawtchouk Expansion , 2001 .

[19]  Noel A Cressie,et al.  Spatial Mixture Models Based on Exponential Family Conditional Distributions , 2000 .

[20]  Alberto Lekuona,et al.  Sharp estimates in signed Poisson approximation of Poisson mixtures , 2005 .

[21]  Taylor's formula and preservation of generalized convexity for positive linear operators , 2000, Journal of Applied Probability.

[22]  Spario Y. T. Soon BINOMIAL APPROXIMATION FOR DEPENDENT INDICATORS , 1996 .