A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

Let $X_1,X_2,...,X_n$ be a sequence of independent or locally dependent random variables taking values in $\mathbb{Z}_+$. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum $\sum_{i=1}^nX_i$ and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This "smoothness factor" is of order $\mathrm{O}(\sigma ^{-2})$, according to a heuristic argument, where $\sigma ^2$ denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ201 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Long Strange Segments, Ruin Probabilities and the Effect of Memory on Moving Average Processes

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.Comment: 29 pages, minor changes and a few typo correction from last versio

A Test Statistic for Weighted Runs

A new test statistic based on success runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic $\chi^{2}$ test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute $p$-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.Comment: 20 pages, 4 figures. Match published paper as close as possibl

On the length of the longest head run

We evaluate the accuracy of approximation to the distribution of the length of the longest head run in a Markov chain with a discrete state space. An estimate of the accuracy of approximation in terms of the total variation distance is established for the first time

Poisson approximation of the mixed Poisson distribution with infinitely divisible mixing law

In this work, explicit upper bounds are provided for the Kolmogorov and total variation distances between the mixed Poisson distribution with infinitely divisible mixing law and the Poisson distribution. If μ and σ2 are the mean and variance of the mixing distribution respectively, then the bounds provided here are asymptotically equal to σ2 / (2 μ sqrt(2 π e)) and σ2 / (μ sqrt(2 π e)) for the Kolmogorov and the total variation distance respectively when μ → ∞ and σ2 is fixed. Finally, as an application, the Poisson approximation of the negative Binomial distribution is considered. © 2009 Elsevier B.V. All rights reserved

A New Method for Bounding the Distance Between Sums of Independent Integer-Valued Random Variables

Let X1, X2,..., Xn and Y1, Y2,..., Yn be two sequences of independent random variables which take values in ℤ and have finite second moments. Using a new probabilistic method, upper bounds for the Kolmogorov and total variation distances between the distributions of the sums ∑i=1n Xi and ∑i=1n Yi are proposed. These bounds adopt a simple closed form when the distributions of the coordinates are compared with respect to the convex order. Moreover, they include a factor which depends on the smoothness of the distribution of the sum of the Xi's or Yi's, in that way leading to sharp approximation error estimates, under appropriate conditions for the distribution parameters. Finally, specific examples, concerning approximation bounds for various discrete distributions, are presented for illustration. © 2009 Springer Science+Business Media, LLC

Bounds for the distance between the distributions of sums of absolutely continuous i.i.d. convex-ordered random variables with applications

Let X 1, X 2, ... and Y 1, Y 2,... be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(X i) = E(Y i), i = 1, 2,.... In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums Σ n i=1 X i and Σ n i=1 Y i In the case where the distributions of the X is and the Y is are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations. © Applied Probability Trust 2009