Asymptotic Expansions

Asymptotic expansions of functions are useful in statistics in three main ways. Firstly, conventional asymptotic expansions of special functions are useful for approximate computation of integrals arising in statistical calculations. An example given below is the use of Stirling's approximation to the gamma function. Second, asymptotic expansions of density or distribution functions of estimators or test statistics can be used to give approximate conndence limits for a parameter of interest or p-valuesfor an hypothesis test. Use of the leading term of the expansion as an approximation leads to conndence limits and p-values based on the limiting form of the distribution of the statistic, whereas use of further terms often results in more accurate inference. Third, asymptotic expansions for distributions of estimators or test statistics may be used to investigate properties such as the eeciency of an estimator or the power of a test. The rst two of these are discussed in this entry, which is a continuation of Serring 35]. The third is discussed in the entry Asymptotics, Higher Order. An asymptotic expansion of a function is a re-expression of the function as a sum of terms adjusting a base function, expressed as follows f n (x) = f 0n (x)f1 + b 1n g 1 (x) + b 2n g 2 (x) +. .. + b kn g k (x) + O(b k+1;n)g (n ! 1): (1)