Rules

See the Section about Rules for Quizzes and Homeworks on the General Info page.

Your work handed into Moodle should be a plain text file with R commands and comments that can be run to produce what you did. We do not take your word for what the output is. We run it ourselves.

Note: Plain text specifically excludes Microsoft Word native format (extension .docx). If you have to Word as your text editor, then save as and choose the format to be Text (.txt) or something like that. Then upload the saved plain text file.

Note: Plain text specifically excludes PDF (Adobe Portable Document Format) (extension .pdf). If you use Sweave, knitr, or Rmarkdown, upload the source (extension .Rnw or .Rmd) not PDF or any other kind of output.

If you have questions about the quiz, ask them in the Moodle forum for this quiz. Here is the link for that https://ay16.moodle.umn.edu/mod/forum/view.php?id=1338881.

You must be in the classroom, Armory 202, while taking the quiz.

Quizzes must uploaded by the end of class (1:10). Moodle actually allows a few minutes after that. Here is the link for uploading the quiz https://ay16.moodle.umn.edu/mod/assign/view.php?id=1338883.

Homeworks must uploaded before midnight the day they are due. Here is the link for uploading the homework. https://ay16.moodle.umn.edu/mod/assign/view.php?id=1338886.

Quiz 5

Problem 1

The following R command 


load(url("http://www.stat.umn.edu/geyer/s17/3701/data/q5p1.rda"))
loads two R objects
logl
an R function that calculates the log likelihood and up to two derivatives for data from the zero-truncated Poisson distribution. This is copied exactly from the solution to Problem 4 on Homework 3.
x
an R numeric vector that is independent and identically distributed (IID) zero-truncated Poisson data.

Note that this logl function has multiple arguments 

> args(logl)
function (theta, x)
neither of which vectorizes. Here we have a vector of IID data, but our function only works for sample size 1. Moreover it returns a list (components function value, gradient, and hessian) rather than a number so it cannot be vectorized using the R function Vectorize. Nevertheless, a function that calculates log likelihood for vector data can be easily made using the theory in section about log likelihood for IID data in the course notes (the log likelihood for the whole data is the sum of the values for logl applied to each term in the data, and similarly for derivatives if you want to use them, which you probably don't because there is no extra credit for doing so).

You do not have to use the R function logl loaded here to do this problem. For example, you could cut-and-paste the code into an editor and modify it somewhat to do what this problem requires. But my solution will use it. Also this function is well tested, so it is perhaps better not to modify it.

Find the MLE for these data. A fairly good starting point for these data would be the MLE for the untruncated data model, which is log(mean(x)).

This starting point is inconsistent, but it can be shown that the log likelihood for this model has a unique local maximizer which is also the global optimizer, so any local maximum is the "right" local maximum. It doesn't actually matter where you start (in this particular problem).

Possibly helpful references

Problem 2

This problem continues where the preceding problem left off. Make a 90% (note not 95%) confidence interval for the true unknown θ using a Wald interval (point estimate plus or minus critical value times standard error) using observed Fisher information to calculate standard error, which is described in Section 3.2.4.4.8.1 of the course notes on models, part II and exemplified in Section 5.4.2 of the course notes on models, part II.

Problem 3

This problem also continues where problem 1 left off. Make a 90% (note not 95%) confidence interval for the true unknown θ using a Wilks interval (a level set of the log likelihood), which is described in Section 3.2.4.4.8.2 of the course notes on models, part II and exemplified in Section 5.4.4 of the course notes on models, part II.

Homework 5

Homework problems start with problem number 4 because, if you don't like your solutions to the problems on the quiz, you are allowed to redo them (better) for homework. If you don't submit anything for problems 1–3, then we assume you liked the answers you already submitted.

Problem 4

The R command 


load(url("http://www.stat.umn.edu/geyer/s17/3701/data/q5p4.rda"))
loads one R object, which is a function named sally. As 

> args(sally)
function (x) 
NULL
says, it has one argument x that is required to be a numeric vector of length 10.

The the object of this problem is to maximize (note not minimize) this function.

I will tell you that the maximum is somewhere in the box in 10-dimensional space consisting of points all of whose coordinates are less than 10 in absolute value. You may, of course, look at the function, but it is quite complicated and not at all obvious where the maximum is. I will also tell you that it is a smooth function that it likely to have many local maxima.

  1. Use the R function optim with method = "SANN" to maximize sally.
  2. Use the R function optim with method = "BFGS" starting at the result of the preceding step to see if it is possible to go uphill further.

In both steps it is possible to make optimize maximize rather than minimize, as explained in help(optim).

In the simulated annealing step, most of the parameters for simulated annealing explained in the Details section of help(optim) seem OK but the default number of steps is too short. The longer you run it, the better its chance of finding the global maximum. Use at least 106 steps. Use more if you have time. It is not required that you find the global maximum to get full credit.

Problem 5

minimize: x2 + y4 + z6 + sin(x + y + z)
subject to: x2 + y2 + z2 ≥ 4

Problem 6

This is the same problem as problem 5 except that you are required to supply functions that calculate the derivatives of the objective function (the function to minimize) and the constraint function (the function required to be ≥ 0) to the R function doing the minimization.

Also these derivatives must be analytic derivatives (not calculated by the R package numDeriv or any other R code that does derivatives by finite differences). You may use the R functions D or deriv to calculate these derivatives, but I just did them by hand.