Saddlepoint approximations to the distribution of the estimator of the serial correlation coefficient in the case of AR(1) process with unitary root

ABRIL, JUAN CARLOS; ABRIL, MARÍA DE LAS MERCEDES; MARTÍNEZ, CARLOS I.

Londres

Workshop; Seminario de la LSE; 2016

The London School of Economics and Political Science

Usually, we approximate the distribution of a statistic whose exact distribution cannot be obtained conveniently. When the first moments are known, a common procedure is to fit Pearson or Edgeworth type laws with those moments. Both methods are usually satisfactory in practice but they have the disadvantage that the errors on the "tails" of the distribution are often comparable with the frequencies themselves. Edgeworth approach in particular, can take negative values on these regions. If we know the characteristic function of the statistic, then the difficulty is of analytic type, and to achieve its distribution we must explicitly make the inversion of this characteristic function using the Fourier transform. It can be shown that for a large number of statistics, it is almost always possible to get a satisfactory approximation to its probability density, when it exists, through the steepest descents method. This will result in an asymptotic expansion in powers of n⁻¹ whose dominant term, called saddlepoint approximation, has a number of very desirable features. The error incurred by using this approach is O(n⁻¹) in contrast of the usual O(n^{-1/2}) that arises from using the normal approximation. The steepest descents method was used for the first time in a systematic way by Debbye for greater order functions (see Watson, 1948) and was used again by Darwin and Fowler (see for example, Fowler, 1936) in mechanical statistics where it is considered to be an essential tool. Besides the work of Jeffreys (1948) and other isolated applications by other authors (like Cox, 1948) this technique was largely ignored in the development of our science. Only in 1954 Daniels introduced it once again but it was left behind until the beginning of the 1970 decade when it begins a new period of research on this technique in different areas. We begin our paper introducing saddlepoint approximations and their related asymptotic expansion for the probability density of the mean X of a sample of size n. We will establish the general conditions under which the relative error of the approximation is O(n⁻¹) uniform for all x with its corresponding results for the asymptotic expansion. After that, we generalize our findings for other kinds of statistics.