Asymptotic Laplace-Transforms
Hermann Dinges · Birkhäuser Boston eBooks · 1991
In 1954 Henry Daniels published his pioneering paper “Saddlepoint approximations in statistics”. Daniels studies asymptotic expansions for the densities of the averages $$\frac{1}{n}\,({Y_1} + \cdots \,{Y_n})$$ when the Y j , are i.i.d. with short tails. In the discrete case it is assumed that L(Y) is concentrated on ℤ and not on any proper subgroup of ℤ. Here one is interested in the weights $${P_r} + ({Y_1} + \cdots {Y_n}\, = k)\,$$ for $$\frac{k}{n}$$ in some neighborhood of εY. In the continuous case it is assumed that,C(Y), the law of Y, satisfies Cramér’s condition. Using techniques similar to those in Daniels’ paper Lugannani and Rice in 1980 obtained uniform asymptotic expansions for the distribution functions.