Reference no: EM132880930

Question 1: Suppose that your are the CEO of a chain store called "729" . It consists of o convenience stores. Let yi be the revenue made by the ith store, and z, be the floor area of the ith store for i = 1, 2, ... , o. Assume that x1, ... , xn are known and deterministic unless otherwise specified. We are interested in knowing the association between floor area and revenue. Consider the following model

[yi |xi,θ] ~ N(θxi, h(xi)),            i = 1,2,....n, 

θ ~ N(μ, τ²)

for some μ ∈ R and r ∈ R⁺, where h(.) is a known function. Equation (1.1) allows Var(yi, | xi, θ) to vary with xi. It is called heteroscedasticity, which generalizes the standard constant noise variance assumption, known as homoscedasticity, in classical regression. Indeed, it is a very realistic assumption as the variability of revenue is likely to be increasing with the size of the store. See the graphical visualization below. This model is also useful for handling observations having different qualities

The aim of this question is to learn the relationship in terms of θ.
1. Find the posterior distribution [θ | x1:n y1:n] .
2. Construct an appropriate Bayesian estimator ^θB and a 95% credible interval of θ.
3. Give an empirical Bayes est etc ^θ_EB of θ.
4. Because of some operational reasons, the CEO fails to obtain the exact floor areas of some of the stores. Denote

Ii = 1(the floor area of the ith superstore is known),       i = 1, ......n.

5. Comment the following statement:

"The empirical Bayes estimator ^θEB emprically bases on the data in choosing be hpperporameters, whereas the Bayes estimator θo is less data-driuen. So, ^θEB must 6e better."

In this part, you may discuss whatever you think relevant and insightful to understand and interpret the statement. For example, you may (i) comment on the correctness and validity of the statement, (ii) propose better ways to state the assertion, (iii) use different statistical principles and philosophies to interpret the statement, (iv) perform simulation experiments to verify your claim(s), (v) visualize the results, etc. Your answer must be within 2 pages.

Question 2: The total number of enrollments in Stat 4010, Stat 4003 (Spring 2020), and Stat 3005 (Fall 2020 is n1 = 231 . Let (Si, Gi) ∈ {0, 100] x {A, A-, B+, B, B-, C+ , C , C -, D, D+, F } be the mid-term score and the coiavsponding grade of the ith student for i = 1, . . . , n1. All values of si's are observed. Unfortunately, owing to the possibility of pass/fail grading option (P/F option) in the previous three semesters, some of the Gi's are not fully observed.

1. We may regard Hi as a coarsened version of Gi . A "coarsening mechanism" is a generalization of a "missing mechanism* . Similar to Definition 9.2 in the lecture note, we define
» coarsening completely at random (CCAR) if P(Ii = r|Si = s, Gi = g) = Φ(r) for all r, s,g;
» coarsening at random (CAR) if P(Ii = r | Si = s,Gi = g) = Φ(r, s,gr + h (1-r)) for all r,s,g and
» coarsening not at random (CNAR) if otherwise,

where Φ is some function, and h = 1(g = 1). Which coarsening mechanism do you prefer to assume? Why? (Use no more than about 100 words).

2. Exam now on, assume that

P(Ii = 1|Si, Gi, Φ0, Φ1 = pnorm(Φ0, + Φ1Si), i = 1,....n1

for some free parameters Φ0, Φ1, v R. Propose a prior Π(μ, σ, θ1,...., θ10, β, Φ₀, Φ₁). Explain your choice.

3. Use an appropriate MCMC sampler to draw sample from the posterior

[μ, σ, θ1,......θ10, β,G_Ei|S1n1, G_E0, H_Ei, I_1:n1 ]

Briefly diggnne your MCMC sample.

4. Use an appropriate Bayesian method to recover the missing grade Gi for i ∈ E₁

5. Predict the grades of the students this year, i.e., G_n1 +1, .. . , G_n1+n2.

6. In this part, you may do whatever you think relevant and insightful for under- standing Question 2. T6 earn the full credit, you must use the knowledge you learned in Stat 4010. Fbr example, you may
» conduct a simulation experiment (by simulating an artificial dataset having the same structure as coarseninggrade.csv) to check the accuracy, and precision of your ntimators and predictors,
» discuss the pre and oons of the models and (RB) from a Bayesian point of view,
» propose improved or alternative Bayesian-related models,
» use a hierarchical model to handle the covariate major, year, course and ID,
» perform additional statistical inference,
» compute empirical Bays estimates,
» conduct detailed and intensive oonvergence diagnosis on the MCMC sample,
» derive some theoretical results.

Attachment:- mock test.rar