Saddlepoint Approximations

ABRIL, JUAN CARLOS

International Encyclopedia of Statistical Science

Springer

Lugar: Berlin; Año: 2010; p. 1267 - 1269

It is often required to approximate to the distribution of some statistics whose exact distribution cannot be conveniently obtained. When the .rst few moments are known, a common procedure is to .t a law of the Edgeworth type having the same moments as far as they are given. This method is often satisfactory in practice, but has the drawback that error in the .tail. regions of the distribution are sometimes comparable with the frequencies themselves. Notoriously, the Edgeworth approximation can assume negative values in such regions. The characteristic function of the statistic may be known, and the di¢ - culty is then the analytical one of inverting a Fourier transform explicitly. It is possible to show that for some statistics a satisfactory approximation to its probability density, when it exists, can be obtained nearly always by the method of steepest descents. This gives an asymptotic expansion in powers of n��1, where n is the sample size, whose dominant term, called the saddle- point approximation, has a number of desirable features. The error incurred by its use is O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. point approximation, has a number of desirable features. The error incurred by its use is O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. point approximation, has a number of desirable features. The error incurred by its use is O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. point approximation, has a number of desirable features. The error incurred by its use is O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. point approximation, has a number of desirable features. The error incurred by its use is O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. n��1, where n is the sample size, whose dominant term, called the saddle- point approximation, has a number of desirable features. The error incurred by its use is O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. normal normal approximation. O(n��1) as against the more usual O(n��1=2) associated with the normal normal approximation.