MAST20005 MAST90058

11am, Friday 26 August 2022

Problems:
1. Let X 1 ,...,X n be a random sample from the binomial distribution Bi(m,p), where m is
given.
(a) Show that ˆ p =
¯
X/m is an unbiased estimator of p.
(b) Show that var(ˆ p) = p(1 − p)/(nm).
(c) Find a value c so that cˆ p(1− ˆ p) is an unbiased estimator of var(ˆ p) = p(1−p)/(mn).
2. A discrete random variable X has the following pmf:
x 0 1 2
p(x) 1 − θ θ/4 3θ/4
A random sample of size n = 30 produced the following observations:
0, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0.
For each of the following quantities, derive a general formula and, where applicable,
calculate it using the given data.
(a) i. Find ¯ x and s for this sample.
ii. Find E(X) and var(X).
iii. Find the method of moments estimate of θ.
iv. Calculate a standard error of this estimate.
(b) i. Find the likelihood function.
ii. Show that the maximum likelihood estimate of θ is
ˆ
θ = 1 − f 0 /n, where f 0 is
the number of observed 0’s in the sample.
iii. Calculate a standard error of
ˆ
θ.
3. Let X 1 ,...,X n be a random sample from the inverse Gaussian distribution, IG(µ,λ),
whose pdf is:
f(x | µ,λ) =
?
λ
2πx 3
? 1/2
exp
? −λ(x − µ) 2
2µ 2 x
?
, x > 0,
where µ = E(X 1 ) is the mean and λ is called the shape parameter.
(a) Given that var(X 1 ) = µ 3 /λ, find the method of moments estimator (MME) of µ and
λ.
(b) Show that the maximum likelihood estimator (MLE) of λ is
ˆ
λ =
n
P
i (X
−1
i
−
¯
X −1 ) .
(c) Schwartz and Samanta (1991) proved that nλ/ ˆ λ ∼ χ 2
n−1 . Use this result to derive a
100 · (1 − α)% confidence interval for λ.
(d) (R) Given the following random sample of size n = 26 from an inverse Gaussian
distribution:
2.48 0.30 0.43 1.84 0.40 0.14 1.07 0.20 0.80 0.23 0.32 2.06 1.61
3.47 0.67 0.11 0.63 0.58 0.29 1.08 0.21 1.48 0.35 3.20 0.06 1.17
i. Compute the method of moments estimate for λ.
ii. Compute the maximum likelihood estimate for λ and give a 95% confidence
interval.
iii. Do a simulation (assuming µ = 1 and λ = 0.5 and using n = 26) to compare
the MME and MLE in terms of their bias and variance. Include a side-by-side
boxplot that compares their sampling distributions.
iv. Repeat the simulation with a larger sample size n = 100. How do the bias and
variance change?
[Hint: Quantiles and random number generation for the IG distribution may be
computed using the functions qinvgauss() and rinvgauss(), respectively, in the
R package statmod.]
4. (R) Let X be a random variable representing distance travelled (in kilometers) until a
tire is worn out. The following are 20 observations of X:
3558 24615 36533 11565 14511 5869 3651 6682 27295 14370
3836 26506 26524 11595 19722 3390 3470 8859 13566 12565
(a) Give basic descriptive statistics for these data and produce a box plot. Briefly
comment on the center, spread and shape of the distribution.
(b) Assuming a log-normal distribution for X, i.e. ln(X) ∼ N(µ,σ 2 ), compute maximum
likelihood estimates for the parameters µ and σ.
(c) Draw a density histogram and superimpose a pdf for a log-normal distribution using
the estimated parameters.
(d) Draw a QQ plot to compare the data against the fitted log-normal distribution.
Include a reference line. Comment on the fit of the model to the data.