【作者】 张东；

【导师】 汤银才；

【作者基本信息】 华东师范大学 ， 应用统计（专业学位）， 2017， 硕士

【摘要】 随机数抽样对于统计学科意义重大。目前为止,常规分布的随机数生成可由计算机软件如R直接操作完成,但对于非常规分布而言尚缺少统计软件的支持。另一方面,自适应性算法具有其根据条件改善算法自身从而提高精度的优良性质,因此可被用于随机数抽样算法之中。本文首先介绍了若干自适应改进下的随机数生成算法:对于概率密度函数具有对数凹函数性质的目标分布,可依据自适应拒绝算法在支撑点处建立切线方程构造目标概率密度函数的分段指数型包络函数加以抽样。对于其他分布,可根据改进自适应拒绝算法,在对数凹区间上建立切线方程,并在对数凸区间上建立割线方程构造包络函数并抽样;亦可使用凹凸分解自适应拒绝算法将有限定义域上的目标分布分解成凹凸两部分并分别构造包络函数;还可根据自适应切片算法建立水平切片进行切片抽样。最后,自适应拒绝Metropolis抽样算法还可实现多元分布抽样。针对上述算法,我们设计并开发一款名为AdapSamp的R功能包来实现自适应随机数抽样。其中,rARS,rMARS,rCCARS,rASS与rARMS函数可分别实现上述功能。经过案例分析,我们得出该功能包所生成的随机数皆来自给定分布的结论。此外,rARMS与rASS函数由于循环和判断次数少于其他函数,因此速度快且具有广泛的实用性。相比而言,rMARS函数则耗时过长导致效率较低。我们开发的新的R包整合了许多优秀的自适应抽样算法,能够解决几乎所有的分布的抽样,是现有R功能包中常规分布抽样函数的有效补充。因其普遍的适用性与使用的便利性,此R包一定会受到广大统计专业人员的欢迎。更多还原

【Abstract】 Sampling plays an important role in statistics.Sampling from conventional distri-butions can be done directly by statistics softwares such as R while it is hard for uncon-ventional distributions.On the other hand,adaptive algorithm can be used for sampling owing to its good property of adapting itself for improving accuracy.We showed several sampling methods related to adaptive algorithm.For distributions whose probability densitry functions are log-concave,Adaptive Rejection Sampling(ARS)algorithm can be used to build tangent lines on support points as envelope functions for sampling.For others,we can use Modified Adaptive Rejection Sampling(MARS)algorithm for building tangent lines on log-concave intervals and secant lines on log-convex intervals as envelope functions.We can also decompose density functions into convex and concave parts and find envelope functions respectively by Concave-Convex Adaptive Rejection(CCARS)Sampling algorithm.Adaptive Slice Sampling(ASS)algorithm is also a good choice by calculating slices.At last.,Adaptive Rejection Metropolis Sampling(ARMS)can realize sampling for multivariate distributions.Then we designed an R package called AdapSamp for all the algorithms above.There are mainly 5 functions in this package:rARS,rMARS,rCCARS,rASS and rARMS.After some experimental analysis,we got a conclusion that samples generated by this package are all from target distributions correctly.Also function rARMS and rASS have a high efficiency and a wide practicability with less loops and judgements while rMARS has a slow speed.The new R package we developed contains many perfect adaptive algorithms and it can solve sampling problems for almost all distributions.Also,it is a worth addition of existing R packages for sampling from conventional distributions.AdapSamp will finally enjoy great popularity from statisticians due to its applicability and convenience.更多还原