課程名稱︰資料科學之統計基礎 (一)
課程性質︰
課程教師︰黃名鉞
開課學院：理學院
開課系所︰數學研究所 應用數學科學研究所 資料科學學位學程
考試日期（年月日）︰2021/01/14
考試時限（分鐘）：180
試題 :
Statistical Foundations of Data Science (I) FINAL EXAM
There are 7 sets of problem on 2 pages with tital of 110 points.
1. (20 pts) Let {X_i}^n_{i=1}, be a random sample from Unif(0, a),
where a>0 is an unknown parameter.
(a) (6 pts) Find the moment estimator for a based on the first moment.
(b) (5 pts) Find the asymptotic normality of the above moment
estimator.
(c) (5 pts) Find the MLE of a.
(d) (5 pts) Show that the above MLE \hat{\theta_{ML}},
satisfies n(a-\hat{\theta_{ML}}\rightarrow^d Exp(1/a)
as n \leftarrow \inf.
2. (10 pts) X is said to follow a mixture normal distribution if X=TA+(1-T)B
for two normal random variables A, B and a Bernoulli random
variable T. Suppose that A~N(0, 1) and B~N(1, 1).
Moreover, T~Bern(p), where 0<=p<=1 is an unknown parameter.
Find the monotone EM sequence for estimating p based on a random
sample {X_i}^n_{i=1}.
3. (10 pts) Prove that min_a E[(X-a)^2]=Var(X) and the equality holds
if and only if a=E(X).
4. (10 pts) Let {X}^n_{i=1} be a random sample from Exp(\lambda),
where \lambda is a Bayes parameter with prior Exp(\lambda_0).
That is,
f_{X|u}(x|\lambda) = \lambda exp(-\lambda x)I(x>0) and
f_{\lambda}(u) = \lambda_0 exp(-\lambda_0 u)I(u>0).
Find the Bayes estimator for \lambda based on {X}^n_{i=1},
with respect to the L^2 loss function d(x,y)=(x-y)^2.
5. (20 pts) Let {X}^{n+2}_{i=1} be a random sample from Bern(p),
where 0<=p<=1 an unknown parameter, and define the function
h(p)=P(\sum^n_{i=1} X_i>X_{n+1}+X_{n+2}; p).
(a) (10 pts) Find the MLE of h(p).
(b) (10 pts) Find the UMVUE of h(p) given the fact that
\sum^{n+2}_{i=1}X_i is a complete statistic for p.
6. (30 pts) Let {X}^n_{i=1} be a random sample from N(\mu, 1),
where \mu is an unknown parameter
(a) (5 pts) Determine c such that the test \pho=I(\hat{\mu}>=c) is a
size \alpha test for testing H_0:\mu<=0 vs. H_A: \mu>0,
where \hat{\mu}=n^{-1}\sum^n_{i=1}X_i.
(b) (10 pts) Show that \phi is a UMP level \alpha test for testing
H_0: \mu<=0 vs. H_A: \mu>0
(c) (5 pts) Determine c such that the test \psi=I(|\hat{\mu}>=c) is a
size \alpha test for testing H_0: \mu=0 vs. H_A: \mu\neq 0.
(d) (10 pts) Show that \psi is NOT a UMP level \alpha test for testing
H_0: \mu=0 vs. H_A: \mu\neq 0.
7. (10 pts) Let {X}^n_{i=1} be a random sample from N(\mu, \sigma),
where \mu and \sigma>0 are unknown parameters.
Find the size \alpha (generalized) likelihood ratio test for
testing H_0:\mu=1 vs. H_A: \mu\neq 1.