J. E. Kolassa, Series approximation methods in statistics, Lecture Notes in Statistics, volume 88, Springer-Verlag, 1994. 16  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... in decomposable combinatorial structures (cf. 21] Thus we investigate the asymptotic behaviour of #m (#) as # and m runs through its possible values (depending on #) When # is bounded and the asymptotic behaviour of #m (#) can be easily derived by the usual saddlepoint method, cf. [10, 24, 26, 50]. For completeness, we include the resulting formula at the end of 2.2. Let us first list some known asymptotic estimates of #m (#) in the literature. They are not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other types ....

J. E. Kolassa, Series approximation methods in statistics, Lecture Notes in Statistics, volume 88, Springer-Verlag, 1994. 16

....being to shift the mean) and the implicit use of the saddle point method, see the next section for more details. His results together with his method have since been generalized in many directions by many authors, for detailed information, applications, and references, consult the monographs [33, 35, 3, 37, 25]. 1.1 Main result n g n be a sequence of random variables with distribution functions W n (x) A number of such sequences related to combinatorial structures and arithmetical functions have moment generating functions satisfying the same algebraic and analytic schemes. This fact allows then a ....

J. E. Kolassa, Series Approximation Methods in Statistics, Lecture Notes in Statistics, volume 88, Springer-Verlag, 1994.

....being to shift the mean) and the implicit use of the saddle point method, see the next section for more details. His results together with his method have since been generalized in many directions by many authors, for detailed information, applications, and references, consult the monographs [33, 35, 3, 37, 25]. 1.1 Main result n be a sequence of random variables with distribution functions W n (x) A number of such sequences related to combinatorial structures and arithmetical functions have moment generating functions satisfying the same algebraic and analytic schemes. This fact allows then a ....

J. E. Kolassa, Series Approximation Methods in Statistics, Lecture Notes in Statistics, volume 88, Springer-Verlag, 1994.

.... decomposable combinatorial structures (cf. 21] Thus we shall investigate the asymptotic behaviour of Pi m ( as 1 and m runs through its possible values (depending on ) When is bounded and m 1, the asymptotic behaviour of Pi m ( can be easily derived by the usual saddle point method, cf. [10, 24, 26, 50]. For completeness, we shall include the resulting formula at the end of x 2.2. Let us first list some known asymptotic estimates of Pi m ( in the literature. They are not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other ....

J. E. Kolassa, Series approximation methods in statistics, Lecture Notes in Statistics, volume 88, Springer-Verlag, 1994.

....be specified directly, and we have some flexibility over its form. A convenient choice for our purposes is a Gram Charlier expansion, in which a normal density is augmented with additional terms capturing the effects of skewness and kurtosis. Johnson, Kotz, and Balakrishnan (1994, pp 25 30) and Kolassa (1994, ch 3) describe the underlying statistical theory. This approach was introduced to finance by Jarrow 5 and Rudd (1982) and has since been applied by Abken, Madan, and Ramamurtie (1996) Brenner and Eom (1997) Knight and Satchell (1997) Longstaff (1995) and Madan and Milne (1994) Our ....

Kolassa, John, 1994, Series Approximation Methods in Statistics , New York: SpringerVerlag.

....directly, and wehave some #exibilityover its form. A particularly convenientchoice for our purposes is a Gram Charlier expansion, in which a normal density is augmented with additional terms capturing the e#ects of skewness and kurtosis. Johnson, Kotz, and Balakrishnan #1994, pp 25 30# and Kolassa #1994, ch 3# describe the underlying statistical theory. This approachwas introduced to #nance by Jarrow and Rudd #1982#, and has since been applied by Abken, Madan, and Ramamurtie #1996#, Brenner and Eom #1997#, and Longsta# #1995#. Our application di#ers from Jarrow and Rudd s in approximating the ....

Kolassa, John, 1994, Series Approximation Methods in Statistics , New York: SpringerVerlag.

....(8) originally due to Barndorff Nielsen [4] is asymptotically equivalent to (7) Approximations (5) and (7) were derived in Daniels [13] and Lugannani and Rice [29] respectively, using the saddlepoint technique of asymptotic analysis. Daniels [15] exemplifies the derivation of (7) Both Kolassa [27] and Field and Ronchetti [23] provide rigorous derivations of (5) using the saddlepoint method. General discussions of the saddlepoint method can be found in Bleistein and Handelsman [19] or Courant and Hilbert [11] The approximations can also be derived from the Edgeworth expansion, cf. ....

Kolassa, J.E. (1994). Series Approximation Methods in Statistics. Springer-Verlag, New York.. (A very detailed derivation of saddlepoint and Edgeworth expansions that includes a lot of background material. Very helpful for understanding how higher order approximations are derived.)

....distributions to give an exact saddlepoint approximation to the Wilcoxon MannWhitney null distribution. In other words, we compute the exact coe#cient of 1 N of an inverted Edgeworth approximation to the upper tail area of this distribution; cf. e.g. p. 26 of Jensen (1985) or Section 5. 1 of Kolassa (1997). Froda and van Eeden 1994 also derive the coe#cient of 1 N 2 ) Moreover, this explicit saddlepoint approximation allows for specific proofs of the rate of convergence of the remainder terms. In particular, we prove a uniformity result for our saddlepoint expansion, up to N 1 6 , which is ....

....for this distribution is obtained as a special case of one of the results needed for the saddlepoint proof. For reviews and general results on saddlepoint methods in statistics we refer the reader to Daniels (1954) Barndor# Nielsen and Cox (1979) Reid (1988) Field and Ronchetti (1990) Kolassa (1997), Reid (1996) and Jensen (1995) comment more extensively on the special case of approximating tail areas. In the present paper, a saddlepoint expansion for F is obtained from the Edgeworth expansion of the exponentially tilted F . More specifically, for fixed u # ( #,#) let V (x; u) be this ....

Kolassa, J. E. (1997). Series Approximation Methods in Statistics, 2nd edition. Lecture notes in statistics no. 88, Springer-Verlag, Berlin.

.... recent applications can be found e.g. in Spady(1991) Butler, Huzurbazar, and Booth(1992) Wood, Booth, and Butler (1993) and Wang(1993) There is also a considerable literature on saddlepoint approximations for the exponential family and transformation models; cf. BarndorffNielsen (1983,1986) Kolassa(1994), Reid(1996) The recent book by Jensen(1995) gives a good account of these results. In the techniques presented above it is assumed that the underlying distribution F is known. A recent new development is the empirical saddlepoint approximation. It is obtained by replacing the true underlying ....

KOLASSA J. E. (1994), Series Approximation Methods in Statistics, Springer Verlag, New York.

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Kolassa, J.E. (1997)" Series Approximation Methods in Statistics", Springer.

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