H. E. Daniels, Tail probability approximations, International Statistical Review , 55, 37--48, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....analytically inherent in the underlying Fourier integrals is a simple pole and a saddlepoint. From this perspective, the appearance of normal distribution in our problems is not so unexpected because of the appearance of a double pole and a moving saddlepoint in the integral representation (cf. [17, 2, 29]) In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. 1, 29] Recurrence relation. It is more convenient to work with E nk : Q ....

H. E. Daniels, Tail probability approximations, International Statistical Review , 55, 37--48, 1987.

....b 0 = b 1 = 12(1 . Note that each b j has a removable singularity at r = 1. It should be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddlepoint and a simple pole (one being allowed to approach the other) see [47, 7, 36, 29, 11, 24] and [50, pp. 356 360] In particular, error bounds for (3) are discussed in [29, 39, 40] 4. By the definition of #m (#) 4) which is itself an asymptotic expansion for m = o(#) First of all, from (4) we have roughly # j m # , 5) and we expect that the last ....

H. E. Daniels, Tail probability approximations, International Statistical Review, 55:37--48 (1987).

....; b 1 = 12(1 Gamma r) Note that each b j has a removable singularity at r = 1. It should be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddle point and a simple pole (one being allowed to approach the other) see [47, 7, 36, 29, 11, 24] and [50, pp. 356 360] In particular, error bounds for (3) are discussed in [29, 39, 40] 4. By the definition of Pi m ( 4) which is itself an asymptotic expansion for m = o( First of all, from (4) we have roughly e 1 Gamma m= 5) and we expect that the last expression ....

H. E. Daniels, Tail probability approximations, International Statistical Review, 55:37--48 (1987).

....analytically inherent in the underlying Fourier integrals is a simple pole and a saddlepoint. From this perspective, the appearance of normal distribution in our problems is not so unexpected because of the appearance of a double pole and a moving saddlepoint in the integral representation (cf. [17, 2, 29]) In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. 1, 29] Recurrence relation. It is more convenient to work with E nk : Q ....

H. E. Daniels, Tail probability approximations, International Statistical Review , 55, 37-48, 1987.

....noncentral 2 distribution. It is based on the second order Wiener germ approximation to the cdf. Theoretical account is given in Dinges (1989) The notion of a Wiener germ seems not to be widely known amongst applied statisticians who are much more acquainted with the saddlepoint approximation (Daniels (1987)) for tail areas. In fact, the first order Wiener germ approximation has the same asymptotic order of the error like the well known Lugannani Rice approximation (see Dinges (1989) The second order Wiener germ approximation brings about a small further improvement. The advantage of the Wiener ....

Daniels, H. E. (1987),' Tail probability approximations',International Statistical Review, 55, 37--48.

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