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MAST20005/MAST90058 random sample

创建日期：2024-06-03 16:30:48

所属分类：数学mathematics

标签： MAST20005

random sample

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MAST20005/MAST90058: Week 4 Problems

1. Following on from Week 3 Tutorial Question 9, assume we have a sample of size 20 from
this distribution and observed sample mean x¯ = 0.5. Calculate an approximate standard
error of the MM Estimator.
2. (a) Assume we obtain a random sample size n from a continuous distribution fX(x; θ)
and we have found a unique MLE Estimator Θˆ for θ i.e the likelihood has a unique
maximum. A researcher asks us for an estimate of another parameter ϕ = g(θ)
where the function g is 1 to 1 and so invertible. Prove the invariance property of
the MLE - namely that Φˆ = g(Θˆ), where Φˆ is the MLE for ϕ. You may find it
convenient to use the following mathematical notation: argmaxθ∈Sθ h(θ) = value of
the θ argument that maximises h(θ) over the parameter space Sθ (Sθ is just set of
possible θ values).
(b) Consider X ∼ Poisson(λ). Find the MLE for ϕ = Pr(X = 0).
3. Consider X ∼ Exponential(mean = θ).
(a) Find the Method of Quantiles estimator Θˆ0.5 for θ based on the population median.
(b) Find the Method of Quantiles estimator Θˆ0.25 for θ based on the population first
Quartile.
(c) Find the asymptotic variances of Θˆ0.5 and Θˆ0.25 and compare them - what factors
affect these variances?
(d) Realising that scaling up by an estimator increases its variance, you are motivated
to find a Method of Quantile estimator Θˆq based on cq which requires no scaling to
estimate θ i.e where E(Cˆq) = θ. Find the required value of q and calculate var(Θˆq).
(e) Finally can you find a Θˆq with an even smaller variance than your unscaled estima-
tor? Hint:To find the minimum of a complicated function try graphing it in R over
relevant intervals to approximate the minimum.
4. Consider a random sample of size n from a population X with cdf FX(x). Define the
random variables Hq = Fˆ (cq), 0 < q < 1, where Fˆ is the empirical cdf. Note that Hq
represents the random height of the empirical cdf at the point x = cq.
(a) Find the means and variances of H0.1 and H0.75
(b) Find q such that the variance of Hq is maximised.
5. Let X(1) < · · · < X(5) be the order statistics of 5 independent observations from an
exponential distribution that has a mean of θ = 3.
(a) Find the pdf of the sample median X(3).
(b) Compute the probability that X(4) < 5.
(c) Determine Pr(1 < X(1)).
6. Let X1, X2, · · · , Xn be a random sample of size n drawn from an exponential distribution
with mean 8. Find the probability Pr(X(1) > 5 | X(1) > 2). What is the distribution of
X(1)?
7. We take a random sample of size n, which is odd, from a population distribution which
is symmetric around a number a i.e. fX(a + x) = fX(a − x) for all x. Prove that the
Sample Median pdf is also symmetric around a.
1
8. We take a sample of size n from a population distribution which is symmetric around a
number a. Prove that the pdf’s of X(1) and X(n) are mirror images of each other about
the line x = a; specifically that fX(1)(a+ x) = fX(n)(a− x).
9. Let X(1) < · · · < X(n) be the order statistics of n independent observations from the
uniform distribution Uniform(0, 1).
(a) Find the pdf of X(1).
(b) Verify that E(X(1)) = 1n+1 and use Question 8 to find E(X(n)).
(c) Find the expected value of the Sample Range X(n) −X(1).
(d) Can you find var(X(n)−X(1)) using just the pdf of X(1)? If yes, do so - if not, explain
why?
10. Consider the COX2 mouse data from lectures. Here are some descriptive statistics from
an analysis of them in R:
> x <- c(10.39, 10.43, 9.99, 11.17, 8.91,
+ 11.20, 11.38, 7.74, 10.61, 11.11)
> summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
7.74 10.09 10.52 10.29 11.15 11.38
> sd(x)
[1] 1.159109
Assume that these measurements are a random sample from a normal distribution.
(a) Provide two estimates of the population mean, one based on the sample mean and
one on the sample median.
(b) What is the standard error of each of these estimates?

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