Canadian Journal of Pure & Applied Sciences

Abstract: Approximations are very important because it is sometimes not possible to obtain an exact representation of the probability distribution function (PDF) and the cumulative distribution function (CDF). Even when true (exact) representations are possible, approximations, in some cases, simplify the analytical treatments. In this paper, we extend the known saddlepoint tail probability approximations to univariate cases, including univariate conditional cases. Our first approximation (the weighted random sum applies to unknown and very difficult statistics (we discuss the approximations within the random sum Poisson-Exponential random variables).We evaluate the performance of the saddlepoint approximation using simulations. Our second approximation (convolutions of Gamma random variables, ), are difficult to obtain. These computations are also compared with the exact and normal approximations. We find that the saddlepoint methods provide very accurate approximations for the CDFs probabilities that surpass other methods of approximation, such as normal approximation.The third approximation, including conditional saddlepoint approximations, uses the double saddlepoint. To demonstrate the methods of conditioning in statistical inference, we find a mid p-value using a conditionalsaddlepoint approximation for percentile modified linear rank tests. We show that in the double saddlepoint case, the saddlepoint approximations demonstrate better accuracy than the normal approximation while sharing the same accuracy.