关于有删失数据存在的常用生存模型中极大似然比的精确和渐进分布

The likelihood ratio test is commonly used in hypothesis test. Due to the complicated form of the likelihood ratio, it prevents us from finding its exact distribution. Under some regularities, usually the chi-square distribution can be used to approximate it. This approximation works well with a large sample size. However, for a small sample, inaccuracy may occur. For this reason, it is important and necessary to find the exact distribution of the likelihood ratio for a small sample..In this project, we will start from the exponential distribution, one of the most widely used lifetime distributions in reliability. We would like to derive the exact distribution of the likelihood ratio for testing the scale parameter of one exponential population based on different censored data. The null hypothesis here is \theta=\theta_0. For exponential distribution, if the data is a complete, Type-II censored or progressively Type-II censored sample, then the exact distribution is already known. However, for some other general censored samples, the current methodology cannot be used. Here, we will present a more general way to find the exact distribution which can not only work for these special cases, but also for a more general censored form. After obtaining the exact distribution, we will show how to utilize it to obtain the asymptotic distribution, which will be efficient in computation when sample size is large. Through the similar approach, its power function can be determined. The result will then be extended to a two-parameter exponential distribution. Note, the support of a random variable from a two-parameter exponential distribution depends on the unknown parameter, therefore the asymptotic distribution of the likelihood ratio may not be chi-square distribution anymore. .Next, we will discuss the likelihood ratio test for two exponential populations based on combined censored data. It is the nature of interest to test the equality of two populations, i.e., if these two populations follow the same exponential distribution. However, to the best of our knowledge, there is no work on the exact distribution of likelihood ratio in this field so far. Many researchers have taken a lot of effort on this problem, but can only find some asymptotic distributions to approximate it. They shared their experience and mentioned the problems need to be overcame. Based on their work, we will derive both exact and asymptotic distributions of the likelihood ratio and then use them to determine the power function. By using the power function, we will also work on the optimal design..Finally, we will extend the idea to several widely used survival models, for example, accelerating life test, competing risk model, etc. On the other hand, instead of using the exponential assumption, we will use some other popular distributions, for example, gamma distribution, Weibull distribution, Laplace distribution, etc..During this project, Monte Carlo simulation will be carried out to check the accuracy of the results derived.

在假设检验中，极大似然比检验是最常用的手段。在数据量足够大且满足一定条件时，可用卡方分布进行拟合。但在数据量稀少时，盲目使用卡方分布拟合，会造成一定的误差。在大量文献中，研究者们通过不同的渐进分布来逼近它，却无法攻破精确分布的难题。本项目致力于完善这一空缺，为使研究陈果更有意义，我们会考虑不同数据删失的情况。.我们从常用的一参及二参指数分布入手，检验它的参数是否等于给定的值。除了在数据齐全及一些个别特殊删失情况下，极大似然比的精确分布都是未知的，甚至有时连卡方分布的渐进都会失效。我们将会给出一个更通用的方法来确定极大似然比的精确及渐进分布，并推导出相应的功效函数分布。此后我们将会对两组联合删失的数据进行讨论，研究其是否来自同一分布，这是一极为常见的假设检验，同样该检验极大似然比的精确分布至今未被攻克，也是我们期待解决的问题。最后我们会把这些理论应用到一些常见的生存模型并拓展到其他分布中。

在假设检验中，极大似然比检验是最常用的手段。在数据量足够大且满足一定条件时，可用卡方分布进行拟合。但在数据量稀少时，盲目使用卡方分布拟合，会造成一定的误差。在大量文献中，研究者们通过不同的渐进分布来逼近它，却无法攻破精确分布的难题。本项目致力于完善这一空缺，为使研究陈果更有意义，我们会考虑不同数据删失的情况。.我们从常用的指数分布入手，检验它的参数是否等于给定的值。除了在数据齐全及一些个别特殊删失情况下，极大似然比的精确分布都是未知的，甚至有时连卡方分布的渐进都会失效。我们给出了一个更通用的方法来确定极大似然比的精确及渐进分布，并推导出相应的功效函数分布。此后我们对两组联合删失的数据展开了讨论，研究其是否来自同一指数分布，这是一极为常见的假设检验，这也是检验极大似然比的精确分布第一次被攻克。在此基础上，我们还将所得理论拓展到了一些常用模型，比如加速寿命试验。这些假设检验在生物统计，可靠性研究等常见问题中有广泛使用，因此我们的结论具有很强的应用性。同时我们还把部分结论推广到了其他分布，例如，拉普拉斯分布。

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