Statistics in Computer Science
(Final) Homework Assignment 5
Stanislav Katina, Mojmir Vinkler
December 5, 2016
Exercise 1. Log-normally distributed random variable X ∼ lnN(µ, σ2
) has probability density
function
fX(x) =
1
σ
√
2π
exp −
(ln(x) − µ)2
2σ2
.
Assume that σ is fixed and find its Fisher information I(µ).
We have the following random sample distributed log-normally with σ2
= 1
4.856, 0.487, 0.580, 0.839, 0.721, 2.416, 0.715, 0.361, 0.703, 5.829
Plot scaled log-likelihood function of the sample with µ on x-axis and scaled log-likelihood
l∗
(µ|x) = l(µ|x) − max(l(µ|x)) on y-axis. Compare l∗
(µ|x) with its quadratic approximation
calculated using Taylor approximation ln(L(µ|x)
L(µ|x)
) ≈ −1
2
I(µ)(µ − µ)2
.
Exercise 2. Load dataset one-sample-mean-skull-mf.txt with variable skull.L that represents
skull length of ancient Egyptian male population in mm. Let’s assume that the skull length
is normally distributed, i.e. N(µ, σ2
).
1. Test null hypothesis that mean of skull length is equal to 177.568 mm (H0 : µ = 177.568)
on significance level α = 0.05.
2. Calculate empirical 100 × (1 − α)% confidence interval for population mean of skull
length with 1 − α = 0.95.
For (1) use Wald test statistic TW and likelihood ratio test statistics ULR. For (2) use
empirical Wald confidence interval and empirical likelihood confidence interval.
Exercise 3. Statistical graphics
1. Take some figures combining raster graphics and statistical graphics from the press (A)
three good examples and (B) three bad examples and describe the reasoning behind
your choice, i.e. why the figure is good or bad. See PDF Statistical Graphics in Czech
and World Press for an inspiration.
2. Take one from good or bad examples from above assignment and use the numbers to
create your own figure. This figures should look better than an example from the press.
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3. Create figure in as superimposition of raster graphics and statistical graphics. Describe
what can be seen on the figure. To import any raster graphics to and its
processing, use one or more of the following libraries - rasterImage, library(png), library(pixmap),
library(rimage), library(tiff) library(raster), library(rasterVis), library(maps),
library(mapsdata) or library(mapstools).
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