[SOLVED] Bayesian 1. Consider again a shifted exponential distribution but with 0 unknown.

1. Consider again a shifted exponential distribution but with 0 unknown.
fx1ex 1x
Suppose we observe n observations from this distribution.2pt
a. Derive the MLEs for bothand .2pt
b. Show that the MLEs are jointly sufficient.
c. Derive the expected values of the estimators. Are they unbiased?2pt
d. Give a onetoone function of the MLEs that are unbiased.15pt
2. Suppose that we have bivariate data x, y as we did in the linear model, but here Y is binary, either 0 or 1. We might think to model the probability p that Y1, but it is clear a linear model should not be used, as we have constraints on p that the linear model would not respect. Consider the model
log pi xi 1pi
where piP Yi1.2pt
a. Assuming independence of the Yi, we know they have a bernoullipi distribution. Write down the likelihood for .2pt
b. Compute the probability of a success if Xi0.2pt
c. Write down the first derivative of the loglikelihood, and realize that you wouldnt be able to solve it
something like NewtonRaphson would need to be used.15pt
3. I have a large jar of coins at home. One day I decide that I would like to know how much money the jar contains, so I begin to count it. I count up the total number of quarters 47, and decide that I have no interest in continuing this. Consider the hypothesis that:
i change is always given optimally, i.e. the fewest number of coins, so for 32 cents in change, one quarter, one nickel and two pennies is optimal rather than three dimes and 2 pennies
ii the change amount on any purchase is a discrete uniform distribution over the possibilities from .00 to .99
iii I always pay in cash only no change
a. Under this hypothesis, estimate the total amount of money in the jar.2pt
b. Construct a 95 percent confidence interval for the total number of transactions that produced this jar
of change, as well as a CI for the total amount of money in the jar.2pt c. I finally count the money, and it yields 58 pennies, 25 dimes and 10 nickels. Do a goodness of fit test for the model described above in iiii.
4. The following data set of n20 was generated from either a standard normal distribution or a tdistribution with unknown degrees of freedom .
4.31.3 0.40.80.1 3.10.00.60.60.1
4.20.31.7 1.30.2 0.51.0 0.4 2.2 0.8
a. Considering what we know about the proportion of points falling within 2 standard deviations from a normal distribution, compute a goodness of fit 2 based on multinomial data test for the hypothesis that this data is standard normal.2pt
1

b. Use the binomial distribution to compute the true pvalue the above relies on the asymptotic distri bution of 2 log , which may not be very good for n small.2pt
c. The variance of a tdistribution withdegrees of freedom can be shown to be. Using this 2
information, estimateusing a frequentist approach either MLE or MoM, which one would better utilize the given information?.2pt
d. Using a noninformative improper prior of p1 , and recalling that the pdf of a tdistribution is
given as
fx
1 x2
2 112,
n12 2
write down the kernel everything except constants, since they involve the integral in the denominator
of Bayes Theorem of the posterior distribution of .2pt
e. Compute the asymptotic variance of the estimate in c using the delta method as its a function of
a sample mean of some random variable. So that we all have the same notation, define kEXk and leave your answer in terms of this quantity.2pt f. Using a plug in method everywhere you see ain the variance, plug in , compute an approximate 95 percent confidence interval forbased on your estimate in d. You might want to just approximate higher order moments from a tdistribution using simulation, the calculus isnt so easy. 2pt
5. In the last week of the class, we showed that the only thing necessary to describe the distribution of the mean function x in a Bayesian setting was the dot products of pairs of observation vectors bf phix. We then showed that the dot product based on infinite length observation vectors of a particular type corresponding to adding infinitely many normalbumps, centered everywhere actually had a very computationally friendly form. This function, called a kernel, is what allows us to computationally work with infinite or maybe just very large dimensional data vectors easily. Consider data that is in R2 as opposed what we did in class in R. So our data is of the form xx1 , x2 . Show that the kernel function
Kx, yxyx2y2
for x, yR2 corresponds to augmenting the data space with a single extra feature, x3x21x2, so
that xR3.2pt
6. Write at most 6 sentences discussing the relative merits of frequentist vs bayesian point estimation.
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