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.
Let X i , i ≥ 1, describe the lifetimes of items with finite mean µ = E (X i ) which are successively placed in service. In order to estimate the replacement rate 1 /µ or related quantities, the random variables X i are usually assumed to be independent and identically distributed.It is shown that a nonparametric estimation of the replacement rate and other reciprocal functions of renewal theory is possible while using a delta method with weakened requirements upon the global growth of f which also allows dependent observations and respects the unboundedness of the analyzed reciprocal functions. Moreover, results on the moments and, furthermore, on corresponding simulations are included.
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