課程名稱︰高等統計推論二 課程性質︰選修 課程教師︰鄭明燕 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰102.04.19 考試時限（分鐘）：110 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Advanced Statistical Inference II Midterm Examination 19 Apr 2014 1. A company wants to estimate the proportioin p, 0＜p＜1, of dedective items it produces. Since they rarely produce defective items, n workers are asked to continue inspecting until they each has observed one defective item. For th i=1,2,....,n, let X be the number of items the i worker inspect before s/he has observed one defective item. (a)5% What assumptions are needed so that the model X1,......,Xn i.i.d. distributed as Geometric(p) is appropriate for this setting? ^ n ^ (b)10% Let p1= (n-1)(Σ Xi-1)^(-1). Show that p1 is the uniformly minimum i=1 variance unbiased estimator of p. (c)10% ^ Show that the maximum likelihood estimator of p, denoted as p2, is biased. 2. Suppose that X1,X2,....,Xn is an i.i.d sample from the Uniform(-θ,θ) distribution which has density f(x)=2^(-1)θ^(-1)I (x), where I is (-θ,θ) the indicator funtion. ^ (a)8% Show that the maximum likelihood estimator of θ, denoted as θ1 is ^ biased and has a simple rescaling which is unbiased. Call this rescaling θ2. ^ (b)8% Is θ2 an uniformly minimum variance unbiased estimator of θ ? Why or why not? 3. A random sample X=(X1,....,Xn) is drawn from a Pareto population with pdf θυ^θ f(x│θ,υ)= --------------I (x), θ＞0, υ＞0, where θ and υ are x^(θ+1) [υ,∞) both unknown. (a)5% Show that the LRT of H0:θ=1 versus H1:θ≠1 has critical region of the form:{x:T(x)≦c1 or T(x)≧c2} where 0＜c1＜c2 and ΠXi T=log [-----------]. min(Xi)^n (b)10% Show that, under H0, 2T has a chi-square distribution and find the number of degrees of freedom. 4. Let X1,X2,....Xi be iid Beta(μ,1) and Y1,Y2,....Yi be iid Beta(θ,1). (a)10% Show that a LRT test of H0:θ=μ versusθ≠μ can be based on the statistic ΣlogXi T=--------------------- ΣlogXi + ΣlogYi (b)10% Find the distribution of T when H0 is true, and then show how to get a test of size α. 5. Let X be a single observation from the pdf f(x)=θx^(θ-1) I (x), where (0,1) θ＞0 is unknown. (a)8% Let T= X^θ. Use T as a pivot construct 1-α confidence intervals for θ. (b)8% Among the confidence intervals in (a), find the one that has the minimum length. (c)8% Find the 1-α confidence interval for θ by inverting the acceptance regions of the LRTs H0: θ=θ0 versus H0:θ≠θ0. --

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