Let μ be the expected value of a random variable andX n the corresponding sample mean of n observations. If the transformed expectation f (μ) is to be estimated by f X n then the delta method is a widely used tool to describe the asymptotic behaviour of f X n . Regarding bias and variance, however, conventional theorems require independent observations as well as boundedness conditions of f being violated even by "simple" functions such as roots or logarithms. It is shown that asymptotic expansions for bias and variance still hold if restrictive boundedness conditions are replaced by considerably weaker requirements upon the global growth of f . Moreover, observations are allowed to be dependent.
In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p
1,p
2,…, p
n
is often approximated by a Poisson distribution with parameter λ = p
1 + p
2 + p
n
. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p
1,p
2,…,p
n
.
Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.
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