stat609-29 - Lecture 29 Convergence concepts Asymptotic...

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beamer-tu-log Lecture 29: Convergence concepts Asymptotic approach In statistical analysis or inference, a key to the success of finding a good procedure is being able to find some moments and/or distributions of various statistics. In many complicated problems we are not able to find exactly the moments or distributions of given statistics. When the sample size n is large, we may approximate the moments and distributions of statistics, using asymptotic tools, some of which are studied in this course. In an asymptotic analysis, we consider a sample X = ( X 1 ,..., X n ) not for fixed n , but as a member of a sequence corresponding to n = n 0 , n 0 + 1 ,... , and obtain the limit of the distribution of an appropriately normalized statistic or variable T n ( X ) as n . The limiting distribution and its moments are used as approximations to the distribution and moments of T n ( X ) in the situation with a large but actually finite n . UW-Madison (Statistics) Stat 609 Lecture 29 2014 1 / 10
beamer-tu-log This leads to some asymptotic statistical procedures and asymptotic criteria for assessing their performances. The asymptotic approach is not only applied to the situation where no exact method (the approach considering a fixed n ) is available, but also used to provide a procedure simpler (e.g., in terms of computation) than that produced by the exact approach. In addition to providing more theoretical results and/or simpler procedures, the asymptotic approach requires less stringent mathematical assumptions than does the exact approach. Definition 5.5.1 (convergence in probability) A sequence of random variables Z n , i = 1 , 2 ,... , converges in probability to a random variable Z iff for every ε > 0, lim n P ( | Z n Z | ≥ ε ) = 0 . A sequence of random vectors Z n converges in probability to a random vector Z iff each component of Z n converges in probability to the corresponding component of Z . UW-Madison (Statistics) Stat 609 Lecture 29 2014 2 / 10
beamer-tu-log This leads to some asymptotic statistical procedures and asymptotic criteria for assessing their performances. The asymptotic approach is not only applied to the situation where no exact method (the approach considering a fixed n ) is available, but also used to provide a procedure simpler (e.g., in terms of computation) than that produced by the exact approach. In addition to providing more theoretical results and/or simpler procedures, the asymptotic approach requires less stringent mathematical assumptions than does the exact approach. Definition 5.5.1 (convergence in probability) A sequence of random variables Z n , i = 1 , 2 ,... , converges in probability to a random variable Z iff for every ε > 0, lim n P ( | Z n Z | ≥ ε ) = 0 . A sequence of random vectors Z n converges in probability to a random vector Z iff each component of Z n converges in probability to the corresponding component of Z .

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