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Quiz 6

Problem 1

This is a simulation study (course notes about simulation).

We are going to study almost but not exactly the same model as in (Section 5 of those course notes). The model for this problem is normal with mean θ and variance θ4 (not θ2 as in the example in the notes).

The estimators we want to compare are the sample mean (calculated by mean(x) if the data are x) and the signed square root of the sample sd (calculated by sign(mean(x)) * sqrt(sd(x)) if the data are x). Just those two estimators.

Use sample size n = 15.

Use simulation truth parameter value θ = e (that is, theta <- exp(1)).

Use at least 104 simulations.

Like in the course notes, make a table comparing the MSE of these estimators (with MCSE as appropriate). You do not have to format these nicely in a table using Rmarkdown. As long as you print each number (one estimate of MSE and one estimate of MCSE of MSE for each estimator, which is four numbers) with a clear indication (maybe a comment, maybe a row or column label in a matrix) of what it is, that's enough.

The R code that makes the tables at the ends of Section 5.5 of those course notes and Section 5.6 of those course notes is hidden (not in the HTML) but can be seen in the Rmarkdown source of those course notes.

Be sure to use the principle of common random numbers.

Problem 2

The R command 


x <- read.csv(url("http://www.stat.umn.edu/geyer/5102/data/prob7-1.txt"))$x
assigns one R object x, which is the data vector for this problem.

These same data were also used Section 5 of the course notes on models, Part II and Section 9 of those course notes, but unlike in those notes we are going to investigate different estimators.

The estimators of location and scale we are going to use are the 25% trimmed mean (calculated by mean(x, trim = 0.25) if the data are x) and the interquartile range (calculated by IQR(x) if the data are x), respectively. Both are highly robust, tolerating up to 25% gross outliers or errors in the data.

Calculate a 95% bootstrap t confidence interval using these data and these estimators (the trimmed mean is the estimator of the parameter of interest and the IQR is the sdfun).

Use the method of the first example in Section 6.1.4 of the course notes on the bootstrap.

Do not use the method of the second example in that section where the sdfun argument is omitted.

You could also use the method of Section 6.1.3 of those notes, but why? It is a lot more code to fuss with.

Use at least 104 bootstrap samples.

Problem 3

This problem is about two-parameter maximum likelihood, which was covered in Sections 6–9 of the course notes on models, Part II.

In particular, it is about Wald-type confidence intervals, exemplified in Section 9.3.1 of those course notes and Section 9.3.2 of those course notes.

However, in order to not use a distribution already used in the course notes, we are going to use the logistic distribution (which gives logistic regression its name).

The R function dlogis calculates the PDF of this distribution and works like all other d functions for distributions R knows about. The help page help("dlogis") gives a mathematical equation for the PDF but you shouldn't need it for this problem (you can use it if you think you need to).

This is a symmetric distribution. The population median is the center of symmetry, which is the location parameter. The interquartile range (IQR) of the standard logistic distribution is 


> qlogis(0.75) - qlogis(0.25)
[1] 2.197225
times the scale parameter (the number above is for the default value of the scale parameter, which is one). We can see this from 

> foo <- rexp(5)
> foo
[1] 1.3009717 0.1212703 0.2030711 0.6550454 0.9039590
> (qlogis(0.75, scale = foo) - qlogis(0.25, scale = foo)) / foo
[1] 2.197225 2.197225 2.197225 2.197225 2.197225
Hence the sample median is a good estimator of location and the IQR divided by qlogis(0.75) - qlogis(0.25) is a good estimator of scale.

The R command 


x <- read.csv(url("http://www.stat.umn.edu/geyer/s17/3701/data/q6p3.csv"))$x
assigns one R object x, which is the data vector for this problem.

  1. Use these starting values to find MLE of both parameters (location and scale) for these data assuming they are IID logistic distributed.
  2. Provide Wald-type 95% confidence intervals of both parameters (location and scale) using observed Fisher information to compute standard errors for the point estimates.