Take Home Midterm, Statistics 512, Spring 2005



Take Home Midterm, Statistics 550, Fall 2008

This is a take home midterm exam and is due Wednesday, October 15th by 5 pm (put in my mailbox in the statistics department or e-mail to me). You can consult any references but cannot speak with anyone (except for the instructor) about the exam. If you do not understand a question, send the instructor an e-mail (dsmall@wharton.upenn.edu ) or speak with him. Good luck!

1. Consider the model that [pic] is an iid random sample from a mixture of normal distributions

[pic]

with parameter space [pic]

(a) Show that this parameterization is not identifiable.

(b) Provide an identifiable parameterization for this model.

2. (a) Suppose you have a Beta (4,4) prior distribution on the probability [pic] that a coin will yield a ‘head’ when spun in a specified manner. The coin is independently spun ten times and ‘heads’ appears fewer than 3 times. You are not told how many heads were seen, only that the number is less than 3. Calculate your exact posterior density for [pic] and sketch it.

Hint: Use the following result from calculus: If [pic] and [pic], then

[pic],

where the gamma function has the property that [pic]. The proof of this result is in W. Rudin, Principles of Mathematical Analysis (1976), pp. 193-194. The gamma function and the beta distribution are described in Section B.2.2 of Bickel and Doksum.

(b) Richardson (1944, Journal of the Royal Statistical Society) presented data on the number of wars in the world that started in each of the years 1500, 1501, …, 1931. The frequency table of the number of years in which different numbers of wars started is the following:

|Number of wars started in the year |Number of such years |

|0 |223 |

|1 |142 |

|2 |48 |

|3 |15 |

|4 |4 |

|>4 |0 |

Let [pic] be the number of wars that started in year [pic], [pic]. Richardson considered the model that [pic] are iid Poisson[pic]. Consider an exponential prior distribution for [pic] with a mean of 1, i.e., [pic]. Show that the posterior distribution for [pic] is in the Gamma family of distributions and find the posterior mean and variance.

Hint: See Appendix B.2.2 and Exercise B.2.4 on page 526 of Bickel and Doksum for properties of the Gamma family of distributions (you may use the results of the exercise without proof).

3. Suppose we observe data [pic]from a distribution [pic], where we do not know the true [pic] but only know that [pic]. We wish to choose an action [pic]for a decision problem. The principle of gamma-minimaxity is a Bayes-frequentist compromise for choosing decision procedures that was discussed in Notes 3. A decision procedure [pic]is gamma-minimax over a class of prior distributions [pic]if

[pic],

where [pic]is the class of decision procedures under consideration and [pic]is the Bayes risk of the decision procedure [pic] for the prior distribution [pic].

(a) Assume that a Bayes rule for the prior [pic] belongs to [pic]. Show that if [pic], that is [pic]consists of one prior, then the Bayes decision procedure for the prior distribution [pic] is gamma-minimax.

(b) Assume that a minimax decision procedure belongs to [pic]. Show that if [pic], then the minimax decision procedure is gamma-minimax.

(c) Suppose that [pic] are iid Bernoulli with unknown probability of success [pic]. Consider three point estimators for [pic], (I) [pic]; (II) [pic]; and (III) [pic]; these are posterior means of [pic] for Beta(1,3), Beta(2,2) and Beta(3,1) priors respectively. Suppose that your loss function is the squared error loss function. Which of the three estimators -- [pic]or [pic] -- is best according to the minimax criterion? Which of the three estimators -- [pic]or [pic] -- is best according to the gamma-minimax criterion with [pic]?

4. (a) Suppose [pic]and [pic]are independent. [pic]has pdf [pic]where [pic] is unknown and [pic]has pdf [pic] where [pic]is unknown. Suppose [pic] is sufficient for [pic]based on [pic]alone and [pic] is sufficient for [pic]based on [pic]alone. Now, you are given data [pic]with unknown parameter [pic], where [pic]. Is [pic]sufficient for [pic]? Prove or provide a counterexample.

(b) Suppose that [pic] are iid [pic], i.e., each [pic] has a normal distribution with mean [pic] and variance [pic] where [pic] is an unknown positive number. Show that [pic] is a minimal sufficient statistic for [pic]. Show that [pic] is a sufficient but not a minimal sufficient statistic for [pic].

5. (a) Let [pic]be a random variable with pdf [pic], where [pic]is unknown and suppose that the family of distributions [pic], [pic]is an exponential family. Consider an event [pic] for which [pic] for all [pic]. Is the family of conditional distributions [pic] an exponential family? Prove or provide a counterexample.

(b) The logarithmic series model was introduced by R.A. Fisher to study the distribution of butterflies on the Malayan Peninsula and has been used in various biological, demographic and economic applications. The model is

[pic].

Show that the logarithmic series model is an exponential family. Use Theorem 1.6.2 to find the moment generating function of the logarithmic series ([pic]) distribution and calculate the mean and variance of a logarithmic series ([pic]) distribution.

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