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）︰102.06.19
考試時限（分鐘）：
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 Final Examination 19 Jun 2014
1. Let X=(X1,X2,....Xn), where X1,X2,....Xn are iid random variables from a
Bernouli(p) distribution. Consider testing:H0:p＝p0 versus p≠p0.
(a)10% Derive an expression for -2logλ(x), where λ(x) is the LRT statistic.
(b)10% Give an approximate level α test.
2. A random sample of X1,....,Xn is drawn from a Gamma(α,β) population with
pdf 1
f(x│α,β)= -------------x^(α-1)e^(-x/β)I (x),α＞0,β＞0.
Gamma(α)β^α (0,∞)
where α is known, β is unknown and I is the indicator function. Consider
testing: H0:β＝β0 versus β≠β0.
(a)10% Derive a Wald statistic and a score statistic.
(b)10% Construct approximate level α tests using the test statistic in (a).
3. (10%) Verify that g(y)=y^0.5 is a variance stabilizing transformation of a
Poisson random variable.
4. Assume the one-way ANOVA null hypothesis is true.
─
k (Yi- Y)^2
(a)10% Show thatΣ --------------- gives an unbiased estimator of σ^2.
i=1 k-1
(b)10% Derive the ANOVA F test.
5. Suppose the random variables Y1,Y2,....Yn satisfy Yi=βxi + ei, i=1,....,n
where x1,x2,....xn are fixed constants,e1,e2,....en are iid Normal(0,σ^2)
errors, and β and σ^2 are unknown constants.
(a)10% Find the MLEs of β and σ.
(b)10%
Show that ΣYi/Σxi and n^(-1)Σ(Yi/xi) are both unbiased estimators of β.
(c)10% Compare the variances of the estimators of β in (a) and (b).
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※ 編輯: d3osef (111.243.102.90), 07/12/2014 03:15:52