Abstract
Calculating the exact critical value of the test statistic is important in nonparametric statistics. However, to evaluate the exact critical value is difficult when the sample sizes are moderate to large. Under these circumstances, to consider more accurate approximation for the distribution function of a test statistic is extremely important. A distribution-free test for stochastic ordering in the competing risks model has been proposed by Bagai et al. (1989). Herein, we performed a saddlepoint approximation in the upper tails for the Bagai statistic under finite sample sizes. We then compared the saddlepoint approximations with the Bagai approximation and investigate the accuracy of the approximations. Additionally, the orders of errors of the saddlepoint approximations were derived.
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References
Anderson TW, Darling DA (1952) Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. Ann Math Stat 23: 193–212
Anderson TW, Darling DA (1954) A test of goodness of fit. J Am Stat Assoc 49: 765–769
Bagai I, Deshpandé JV, Kochar SC (1989) Distribution free tests for stochastic ordering in the competing risks model. Biometrika 76: 775–781
Bean R, Froda S, van Eeden C (2004) The normal, Edgeworth, saddlepoint and uniform approximations to the Wilcoxon–Mann–Whitney null-distribution: a numerical comparison. J Nonparametric Stat 16: 279–288
Butler RW (2007) Saddlepoint approximations with applications. Cambridge University Press, Cambridge
Chen Q, Giles DE (2008) General saddlepoint approximations: application to the Anderson–Darling test statistic. Commun Stat Simul Comput 37: 789–804
Daniels HE (1954) Saddlepoint approximations in statistics. Ann Math Stat 25: 631–650
Daniels HE (1987) Tail probability approximations. Int Stat Rev 55: 37–48
Easton GS, Ronchetti E (1986) General saddlepoint approximations with applications to L statistics. J Am Stat Assoc 81: 420–430
Froda S, van Eeden C (2000) A uniform saddlepoint expansion for the null-distribution of the Wilcoxon–Mann–Whitney statistic. Can J Stat 28: 137–149
Gibbons JD, Chakraborti S (2003) Nonparametric statistical Inference. 4. Dekker, New York
Giles DE (2001) A saddlepoint approximation to the distribution function of the Anderson–Darling test statistic. Commun Stat Simul Comput 30: 899–905
Goutis C, Casella G (1999) Explaining the saddlepoint approximation. Am Stat 53: 216–224
Huzurbazar S (1999) Practical saddlepoint approximations. Am Stat 53: 225–232
Jensen JL (1995) Saddlepoint approximations. Oxford University Press, Oxford
Kolassa JE (2006) Series approximation methods in statistics. Springer, New York
Lugannani R, Rice SO (1980) Saddlepoint approximation for the distribution of the sum of independent random variables. Adv Appl Probab 12: 475–490
Murakami H (2009a) A saddlepoint approximation to a distribution-free test for stochastic ordering in the competing risks model. REVSTAT 7: 189–201
Murakami H (2009b) Saddlepoint approximations to the limiting distribution of the modified Anderson–Darling test statistic. Commun Stat Simul Comput 38: 2214–2219
Reid N (1988) Saddlepoint methods and statistical inference (with discussion). Stat Sci 3: 213–238
Sinclair CD, Spurr BD (1988) Approximations to the distribution function of the Anderson–Darling test statistic. J Am Stat Assoc 83: 1190–1191
Sinclair CD, Spurr BD, Ahmad MI (1990) Modified Anderson–Darling test. Commun Stat Theory Methods 19: 3677–3686
Terrell GR (2003) A stabilized Lugannani-Rice formula. Symposium of Interface, Computing Science and Statistics
Wood ATA, Booth JG, Butler RW (1993) Saddlepoint approximations to the CDF of some statistics with nonnormal limit distributions. J Am Stat Assoc 88: 680–686
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Murakami, H. A numerical comparison of the normal and some saddlepoint approximations to a distribution-free test for stochastic ordering in the competing risks model. Comput Stat 25, 633–643 (2010). https://doi.org/10.1007/s00180-010-0193-5
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DOI: https://doi.org/10.1007/s00180-010-0193-5