I consider parametric models with a scalar parameter ofinterest and multiple nuisance parameters. The likelihood ratio statistic is frequently used in statistical inference. The standard normal approximation to the likelihood ratio statistic generally has error of order O(n-1/2), where "n" denotes the sample size. When "n" is small, the normal approximation may not be adequate to do accurate inference. In practice, the true error is more important than asymptotic order. The intention of this

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... dy is to find an approximation which is relatively easy to apply, but which is accurate under small sample size settings. Saddlepoint approximations are well-known for higher order accuracy properties and remarkably good relative error properties. There are several saddlepoint approximations. I look for one that is flexible inapplication while keeping a satisfactory convergence rate.I evaluate, via Monte Carlo, the accuracies of several saddlepoint approximations, and of some classical methods, when these approximations are used to approximate p-values for hypotheses about a scalar parameter. Based on the results, I find that DiCiccio and Martin's (1993) approximations are interesting and deserve more research. Approximations of DiCiccio and Martin (1993) involve exploiting the parallels between Bayesian and frequentist inference, and can be constructed from general log-likelihood functions with relatively easy calculation, while keep the accuracy property.Two difficulties arise in the application of these approximations. One is the instability around a singularity. The other and far more significant is the construction of the prior density functions utilized in these approximations. These prior density functions arealso called matching priors.To make DiCiccio and Martin's (1993) approximations applicable in practice, I successfully resolve the above two problems. I remove the instability and fix the numerical difficulties in applying these approximations. The matching prior is the solution to a first order partial differe [...]