الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Sometimes, it is required to approximate the distribution of some statistics
whose exact distribution cannot conveniently obtain. Saddlepoint approximation
methods are mathematical techniques applied to give a highly accurate approximation of the density and cumulative distribution functions for a statistic using
its moment generating function. The proposed method is distinguished from the
asymptotic method in being very accurate with small and moderate sample sizes,
and being very accurate in the tails of the distribution. The saddlepoint approximation is more precise than asymptotic method and computationally faster
than the simulation procedure. In general, for a sample of size n, the saddlepoint
approximation is accurate up to O(n
− 3
2 ), when considered over sets of bounded
central tendency, while up to O(n
− 1
2 ) for the central limit theory. Here, the saddlepoint approximations are used to approximate the exact p-value of the weighted
log-rank tests for left-truncated data.
The thesis consists of five chapters:
Chapter 1
This chapter is an introduction for the thesis in which basic concepts and
information needed for the following chapters are presented. First, we propose the
weighted log-rank class of two sample tests. Second, we provide the saddlepoint
approximation methods and their applications.
Chapter 2
In this chapter, the saddlepoint approximation methods is used to approximate
the mid-p-values of the weighted log-rank tests for left-truncated data under the
random allocation design. The accuracy of saddlepoint approximation is verified
through real data examples and simulation studies. The speed and accuracy of
saddlepoint approximation allows us to invert the permutation tests to determine
100(1 − α)% confidence intervals for the treatment effect under Cox proportional
hazard model.
1
Summary
Chapter 3
In this chapter, we derive the saddlepoint approximation for the exact permutation p-values of the weighted log-rank tests for left-truncated data under Wei’s
urn design as a more accurate alternative than the asymptotic approach. Real
data examples are analyzed and simulation studies are applied to show the accuracy and efficiency of the saddlepoint approximation. The weighted log-rank
tests are inverted to determine nominal 95% confidence intervals for the treatment
effect in the presence of left-truncated data.
Chapter 4
If the data are collected from different clinics and we need to reduce selection bias and control the imbalance of group sizes, the appropriate design is the
randomized block design. In this chapter, the permutation distribution of the
weighted log-rank tests for left-truncated data under randomized block design
is derived. Furthermore, the saddlepoint approximation is proposed to approximate the exact p-values of the weighted log-rank tests as a possible alternative to
the normal approximation method. The performance of the proposed method is
shown through real data examples and simulation studies. The test is inverted to
calculate the confidence intervals for the treatment effect.
Chapter 5
The truncated binomial design is one of the commonly used designs for forcing
balance in clinical trials to eliminate experimental bias. In this chapter, we consider the exact distribution of the weighted log-rank class of left-truncated data
under the truncated binomial design.
A double saddlepoint approximation for the p-value of this class is derived under
truncated binomial design. The accuracy of the proposed method over the normal
asymptotic facilitate the inversion of the weighted log-rank tests to determine the
nominal 95% confidence intervals for the treatment effect in the presence of the
left-truncated data.