MAST20005/MAST90058: Assignment 1
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MAST20005/MAST90058: Assignment 1
2022
Instructions: See the LMS for the full instructions, including the submission policy and how to submit your assignment. Remember to submit early and often: multiple submission are allowed, we will only mark your final one. Late submissions will receive zero marks.
Problems:
1. Let X1 , . . . , Xn be a random sample from the binomial distribution Bi(m, p), where m is given.
(a) Show that pˆ = 
/m is an unbiased estimator of p.
(b) Show that var(pˆ) = p(1 - p)/(nm).
(c) Find a value c so that cpˆ(1 - pˆ) is an unbiased estimator of var(pˆ) = 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 
and s for this sample.
ii. Find 匝(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 θˆ = 1 - f0 /n, where f0 is the number of observed 0’s in the sample.
iii. Calculate a standard error of θˆ.
3. Let X1 , . . . , Xn be a random sample from the inverse Gaussian distribution, IG(µ, λ), whose pdf is:
f (x l µ, λ) = ╱ 
、1/2 exp { 
} , x > 0,
where µ = 匝(X1 ) is the mean and λ is called the shape parameter.
(a) Given that var(X1 ) = µ3 /λ, find the method of moments estimator (MME) of µ and
.
(b) Show that the maximum likelihood estimator (MLE) of λ is
= n
(c) Schwartz and Samanta (1991) proved that nλ/ 
~ χn(2) − 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:
3)48 o)3o o)43 ()84 o)4o o)(4 ()o7 o)3o o)8o o)33 o)33 3)o6 ()6( 3)47 o)67 o)(( o)63 o)÷8 o)39 ()o8 o)3( ()48 o)3÷ 3)3o o)o6 ()(7
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 qgnydauss一( and rgnydauss一( , respectively, in the R package statmoè .]
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:
3÷÷8 346(÷ 36÷33 ((÷6÷ (4÷(( ÷869 36÷( 6683 3739÷ (437o 3836 36÷o6 36÷34 ((÷9÷ (9733 339o 347o 88÷9 (3÷66 (3÷6÷
(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, σ2 ), 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.
2022-08-24