699 research outputs found
Relaxation of monotone coupling conditions: Poisson approximation and beyond
It is well-known that assumptions of monotonicity in size-bias couplings may
be used to prove simple, yet powerful, Poisson approximation results. Here we
show how these assumptions may be relaxed, establishing explicit Poisson
approximation bounds (depending on the first two moments only) for random
variables which satisfy an approximate version of these monotonicity
conditions. These are shown to be effective for models where an underlying
random variable of interest is contaminated with noise. We also give explicit
Poisson approximation bounds for sums of associated or negatively associated
random variables. Applications are given to epidemic models, extremes, and
random sampling. Finally, we also show how similar techniques may be used to
relax the assumptions needed in a Poincar\'e inequality and in a normal
approximation result.Comment: 19 page
The Conway-Maxwell-Poisson distribution: distributional theory and approximation
The Conway-Maxwell-Poisson (CMP) distribution is a natural two-parameter
generalisation of the Poisson distribution which has received some attention in
the statistics literature in recent years by offering flexible generalisations
of some well-known models. In this work, we begin by establishing some
properties of both the CMP distribution and an analogous generalisation of the
binomial distribution, which we refer to as the CMB distribution. We also
consider some convergence results and approximations, including a bound on the
total variation distance between a CMB distribution and the corresponding CMP
limit.Comment: 26 page
Poisson and Gaussian approximations of the power divergence family of statistics
Consider the family of power divergence statistics based on trials, each
leading to one of possible outcomes. This includes the log-likelihood ratio
and Pearson's statistic as important special cases. It is known that in certain
regimes (e.g., when is of order and the allocation is asymptotically
uniform as ) the power divergence statistic converges in
distribution to a linear transformation of a Poisson random variable. We
establish explicit error bounds in the Kolmogorov (or uniform) metric to
complement this convergence result, which may be applied for any values of ,
and the index parameter for which such a finite-sample bound is
meaningful. We further use this Poisson approximation result to derive error
bounds in Gaussian approximation of the power divergence statistics.Comment: 14 page
On strong stationary times and approximation of Markov chain hitting times by geometric sums
Consider a discrete time, ergodic Markov chain with finite state space which
is started from stationarity. Fill and Lyzinski (2014) showed that, in some
cases, the hitting time for a given state may be represented as a sum of a
geometric number of IID random variables. We extend this result by giving
explicit bounds on the distance between any such hitting time and an
appropriately chosen geometric sum, along with other related approximations.
The compounding random variable in our approximating geometric sum is a strong
stationary time for the underlying Markov chain; we also discuss the
approximation and construction of this distribution
On geometric-type approximations with applications to risk and related processes
We explore two aspects of geometric approximation via a coupling approach to
Stein's method. Firstly, we refine precision and increase scope for
applications by convoluting the approximating geometric distribution with a
simple translation selected based on the problem at hand. Secondly, we apply
our geometric and geometric-type approximations to risk processes and related
areas, including the approximation of Poisson processes with a random time
horizon and Markov chain hitting times. Particular attention is given to
geometric approximation of random sums, for which explicit bounds are
established. These are applied to give simple approximations, including error
bounds, for the infinite-horizon ruin probability in the compound binomial risk
process.Comment: 12 page
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