A Saddlepoint Approximation to a Distri- Bution-free Test for Stochastic Order- Ing in the Competing Risks Model

• The approximation for the distribution function of a test statistic is extremely important in statistics. A distribution-free test for stochastic ordering in the competing risks model has been proposed by Bagai et al. (1989). Herein, we performed a standard saddlepoint approximation in the tails for the Bagai statistic under finite sample sizes. We then compared the saddlepoint approximation with the Bagai approximation to obtain the exact critical value. The table of the critical values was extended by using the saddlepoint approximation. Additionally, the orders of errors of a saddlepoint approximation were derived. Key-Words: • saddlepoint approximation; moment generating function; Bagai statistic; distribution-free test. AMS Subject Classification: • 62F03, 62G10.