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    188宝金博页面版: statisticalinference课件5

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    内容提示： 5Lecture 5: Likelihood Inference. First order asymp-totics5.1Why asymptoticsWe realized that finding the UMVUE for a fixed sample size n could be difficult in somecases especially when the CR bound is not attainable. Finding them requires some art,and there is no easy to follow constructive algorithm for their determination. While weknow that sometime the MLE could be biased or, even if not unbiased, could not attainthe CR bound when outside the exponential family setting, still they are typically easy-to-...

    文档格式：PDF | 页数：7 | 浏览次数：90 | 上传日期：2013-03-21 12:21:21 | 文档星级：
    5Lecture 5: Likelihood Inference. First order asymp-totics5.1Why asymptoticsWe realized that finding the UMVUE for a fixed sample size n could be difficult in somecases especially when the CR bound is not attainable. Finding them requires some art,and there is no easy to follow constructive algorithm for their determination. While weknow that sometime the MLE could be biased or, even if not unbiased, could not attainthe CR bound when outside the exponential family setting, still they are typically easy-to-construct and, as seen on many examples, usually the UMVUE are just a bias-correctedMLE. Indeed, the UMVUE for the probability of success in n independent Bernoullitrials was¯X(1 −¯X)θ of uniform (0, θ) distribution wasthe probability of no occurrence based on n independent Poisson random variables was(1−1the sample size increases. Therefore the UMVUEs are either MLEs or ”almost” MLEs.Hence, it is justified to look for a strong backing of the properties of MLEs in a generalsetting. This can be done using asymptotic arguments, i.e. by looking at the performanceof MLEs when n −→ ∞, i.e. by letting the amount of information become arbitrarilylarge. Statistical folklore says then that “nothing can beat the MLE asymptotically”.nn−1whereas the MLE is¯X(1 −¯X); the UMVUE for the endpointn+1nX(n) whereas the MLE is X(n); the UMVUE forn)n¯ Xwhereas the MLE is exp(−¯X). The bias-correction itself tends to be negligible as5.2Convergence concepts in asymptoticsWe remind some stochastic convergence concepts first.An estimator Tnof the parameter θ is said to be:i) consistent (or weakly consistent) iflimn→∞Pθ(| Tn− θ |> ) = 0for all θ ∈ Θ and for every fixed > 0. We denote this by Tn→ii) strongly consistent if Pθ{limn→∞Tn= θ} = 1 for all θ ∈ Θ.iii) mean-square consistent if MSEθ(Tn) −→n→∞0 for all θ ∈ Θ.Pθ.It is important to note that if the estimator is mean-square consistent then itis also consistent. This relation has probably the most important practical consequence.The reason is that most often we are interested in weak consistency and a common methodthat often works in proving it, is by showing mean- square consistency first. To justify therelation between mean-square consistency and consistency we can use the ChebyshevInequality. It states that for any random variable X and any > 0 it holds for the k-thmoment:P(|X| > ) ≤E(|X|k)k33

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