Abstract
We consider the approximation of the convolution product of not necessarily identical probability distributions q j I + p j F, (j=1,...,n), where, for all j, p j =1−q j ∈[0, 1], I is the Dirac measure at point zero, and F is a probability distribution on the real line. As an approximation, we use a compound binomial distribution, which is defined in a one-parametric way: the number of trials remains the same but the p j are replaced with their mean or, more generally, with an arbitrary success probability p. We also consider approximations by finite signed measures derived from an expansion based on Krawtchouk polynomials. Bounds for the approximation error in different metrics are presented. If F is a symmetric distribution about zero or a suitably shifted distribution, the bounds have a better order than in the case of a general F. Asymptotic sharp bounds are given in the case, when F is symmetric and concentrated on two points.
Similar content being viewed by others
References
Arak, T.V., Zaitsev, A.Yu. (1988). Uniform limit theorems for sums of independent random variables. (Translated from Russian). In Proceedings of the Steklov Institute of Mathematics, 174(1). Providence: American Mathematical Society.
Barbour A.D., Holst L., Janson S. (1992). Poisson approximation. Clarendon Press, Oxford
Barbour A.D., Choi K.P. (2004). A non-uniform bound for translated Poisson approximation. Electronic Journal of Probability 9:18–36
ČekanaviČius V. (1995). On smoothing properties of compound Poisson distributions. Lithuanian Mathematical Journal 35:121–135
ČekanaviČius, V. (2002). On multivariate compound distributions. Teoriya Veroyatnostei i ee Primeneniya, 47, 583–594 (English translation in Theory of Probability and its Applications, 47, 493–506).
ČekanaviČius V., Roos B. (2004). Two-parametric compound binomial approximations. Lietuvos Matematikos Rinkinys 44:443–466
ČekanaviČius V., Vaitkus P. (2001). Centered Poisson approximation via Stein’s method. Lithuanian Mathematical Journal 41:319–329
Choi K.P., Xia A. (2002). Approximating the number of successes in independent trials: Binomial versus Poisson. Annals of Applied Probability 12:1139–1148
Ehm W. (1991). Binomial approximation to the Poisson binomial distribution. Statistics & Probability Letters 11:7–16
Hengartner W., Theodorescu R. (1973). Concentration functions. Academic Press, New York
Ibragimov I.A., Linnik Yu.V. (1971). Independent and stationary sequences of random variables. Wolters-Noordhoff, Groningen
Le Cam L. (1965). On the distribution of sums of independent random variables. In: Neyman J., Le Cam L. (eds) Bernoulli–Bayes–Laplace anniversary volume. Springer, Berlin Heidelberg New York, pp. 179–202
Michel R. (1987). An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. ASTIN Bulletin 17:165–169
Presman, É.L. (1985). Approximation in variation of the distribution of a sum of independent Bernoulli variables with a Poisson law. Teoriya Veroyatnostei i ee Primeneniya, 30, 391–396. (English translation in Theory of Probability and its Applications, 30, 417–422.)
Roos B. (1999). Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution. Bernoulli 5:1021–1034
Roos, B. (2000). Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion. Teoriya Veroyatnostei i ee Primeneniya, 45, 328–344. (See also in Theory of Probability and its Applications, 45, 258–272.)
Roos, B. (2001a). Multinomial and Krawtchouk approximations to the generalized multinomial distribution. Teoriya Veroyatnostei i ee Primeneniya, 46, 117–133. (See also in Theory of Probability and its Applications, 46, 103–117.)
Roos B. (2001b). Sharp constants in the Poisson approximation. Statistics & Probability Letters 52:155–168
Roos B. (2003). Improvements in the Poisson approximation of mixed Poisson distributions. Journal of Statistical Planning and Inference 113:467–483
Roos B. (2005). On Hipp’s compound Poisson approximations via concentration functions. Bernoulli 11:533–557
Soon S.Y.T. (1996). Binomial approximation for dependent indicators. Statistica Sinica 6:703–714
Szegö G. (1975). Orthogonal polynomials (4th ed). American Mathematical Society, Providence
Takeuchi K., Takemura A. (1987a). On sum of 0–1 random variables. I. Univariate case. Annals of the Institute of Statistical Mathematics 39:85–102
Takeuchi K., Takemura A. (1987b). On sum of 0–1 random variables. II. Multivariate case. Annals of the Institute of Statistical Mathematics 39:307–324
Tsaregradskii, I.P. (1958). On uniform approximation of the binomial distribution by infinitely divisible laws. Teoriya Veroyatnostei i ee Primeneniya, 3, 470–474. (English translation in Theory of Probability and its Applications, 3, 434–438.)
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article can be found at http://dx.doi.org/10.1007/s10463-006-0094-0
About this article
Cite this article
Čekanavičius, V., Roos, B. Compound Binomial Approximations. Ann Inst Stat Math 58, 187–210 (2006). https://doi.org/10.1007/s10463-005-0018-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-005-0018-4