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(a) Show that ˆp =¯X/m is an unbiased estimator of p.
(b) Show that var(ˆp) = p(1 − p)/(nm).
(c) Find a value c so that cpˆ(1 − pˆ) is an unbiased estimator of var(ˆp) = p(1 − p)/(mn).
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.
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.
f(x | µ, λ) = λ 2πx3 1/2exp −λ(x − µ)22µ2x , x > 0, where µ = E(X1) is the mean and λ is called the shape parameter.
(a) Given that var(X1) = µ3/λ, fifind the method of moments estimator (MME) of µ and λ.
(b) Show that the maximum likelihood estimator (MLE) of λ is ˆλ =n P i(Xi−1 − ¯X−1).
(c) Schwartz and Samanta (1991) proved that nλ/ˆλ ∼ χ2n−1 . Use this result to derive a 100 · (1 − α)% confifidence 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
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.
[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.]
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. Brieflfly 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 fifitted log-normal distribution.
Include a reference line. Comment on the fifit of the model to the data.