Justify your answers whenever possible to ensure full credit, not just the final answer you arrive at.If you’re using any theorem for any step, please write out which theorem you’re using.You will receive most of the credit for well-argued derivations.

STAT 3445Q                                                    Midterm exam 2 – Page 2 of 6                                        3/27 – 3/28 noon

1. LetX1,…,Xnbe i.i.d.Gamma(α,2), whereα >0 is unknown. The PDF ofXi’s isgiven byf(x) ={1Γ(α)2αxα−1exp(−x2), x >0,0,elsewhere.

(a) (10 points) Find the MOM estimator forα, denoted aŝα. Carefully argue that̂αis an unbiased and consistent estimator forα.

(b) (10 points) Is ̄Xconsistent forα? If not, which parameter is it consistent for?

(c) (10 points) Now define another estimator forαas: (X1+X2)/4 + 1/n. Prove thatthis estimator is not consistent forα. (hint: use the definition of consistency toprove.)

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2. (10 points) SupposeX1,…,Xnis an i.i.d. sample from a distribution with pdf:α(α+ 1)xα−1(1−x),0< x <1,whereαis unknown parameter. Show that 2 ̄X/(1− ̄X) is consistent forα.

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3. SupposeX1,…,Xnis an i.i.d. sample from a uniformly distributed population betweenθand 2θ, whereθ >0 is unknown. The population distribution PDF isf(x) ={1θ, θ < x <2θ,0,elsewhere.DenoteX(1)= min(X1,…,Xn),X(n)= max(X1,…,Xn).

(a) (5 points) ShowU= (X(1)+X(n),X(n)) is jointly sufficient forθ.

(b) (5 points) We know that “sufficiency” is defined based on the so called “informa-tion” concept. Can you explain what we mean by sayingUdefined above is sufficientforθusing your own words?

(c) (5 points) If I define a statisticsT= (X1,…,Xn), which is just a n dimensionalvector consisting of original sample. IsTsufficient forθ? IsTminimal sufficientforθ? Please give the answer and explain the reason.

STAT 3445Q                                                Midterm exam 2 – Page 5 of 6                                            3/27 – 3/28 noon

4. SupposeX1,…,Xnis an i.i.d. sample of sizenfrom aPoisson(λ) population, whereλ >0 is an unknown parameter. The probability mass function of the populationdistribution isP(X=x) ={λxe−λx!,x= 0,1,2,…,0,otherwise.

(a) (5 points) ShowU=∑ni=1Xiis sufficient forλ.

(b) (10 points) Suppose we want to estimateP(Xi= 1) =λe−λ. Define a estimatorˆθ=I(X1= 1) this way:Iis still an indicator function, and the resultingˆθwill justbe a binary random variable that only takes values 0 or 1. The probability to havevalue 1 will be the probability thatI(X1= 1), which is equivalent toX1= 1.It’s easy to show thatˆθis unbiased forλe−λ. Starting from this naive unbiasedestimatorˆθ, use the definition of Rao-Blackwell Theorem and sufficient statisticU=∑ni=1Xito get an improved unbiased estimator forλe−λ.(hint: from the Rao-Blackwell theorem, you need to work on a conditional expec-tation ofˆθ. Note that even with condition, the possible values forˆθare still 0 and1. So the conditional distribution ofˆθis still binary, but with a probability thatyou need to calculate. You can follow the example on the Note 12 where we havea Bernoulli population distribution.)

(c) (5 points) Explain what the “improvement” means in part (b). And is this im-proved unbiased estimator MVUE forλe−λ? And why?

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5. On Note 16, we discussed about the confidence interval for large sample. This is builton central limit theorem (CLT).

(a) (5 points) CLT is talking about the asymptotic property of sample mean ̄Xunderlarge sample. But we also include the CI for population proportionpin Note 16,which is constructed using sample proportion ˆp. why CLT also works for sampleproportion?

(b) (15 points) The two-sided CI for two population proportion differencep1−p2isalready shown in Note 16 without detailed derivation. Please derive this conclu-sion using pivot method by yourself. (I derived the CI for one population meanin the lecture video. You may use that as a template.) To formally state theproblem, please see below: There are two population distributions: Binary(p1) andBinary(p2), wherep1andp2are unknown parameters, showing the population pro-portions. Two set of samplesX1,…,Xn1andY1,…,Yn2are two sets of samplefrom the two populations, respectively. Construct the two-sided CI with confidencelevel 1−αusing pivot method and explain the meaning of this CI: ̄X− ̄Y+zα/2√ ̄X(1− ̄X)n1+ ̄Y(1− ̄Y)n2, ̄X− ̄Y−zα/2√ ̄X(1− ̄X)n1+ ̄Y(1− ̄Y)n2,wherezα/2is the lower tail quantile.

(c) (5 points) If now you have the value for ̄X= ̄x, and ̄Y= ̄y, where ̄xand ̄yare twoconstants. After plug in the CI above, you’ll get: ̄x− ̄y+zα/2√ ̄x(1− ̄x)n1+ ̄y(1− ̄y)n2, ̄x− ̄y−zα/2√ ̄x(1− ̄x)n1+ ̄y(1− ̄y)n2.Can you still use the same explanation in part (b) to explain this CI after plug in?Why?(Blank page