B8: Statistics
1.
Problem Sheet 1
Let X1 , X2 , . . . , X be a random sample from a distribution with
mean µ and finite non
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zero variance σ 2 and let X = n−1 X and S 2 = (n − 1)−1
X − X be the sample
X − X 2 = (X − µ)2 − n X − µ2 to show
mean and variance. Use the identity
that E S 2 = σ2 .
State the Weak Law of Large Numbers and use again the identity given above to deduce
that S 2 → σ2 .
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P
2.
Let X be the mean of a random sample X1 , X2 , . . . , X from a distribution with mean µ
and variance V (µ). Show that the variance of h(X ) is approximately constant, where
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h(µ) =
µ
V (u)− 1/2du.
Show that
(i) V (µ) = µ for a Poisson distribution;
(ii) V (µ) = µ(1 − µ)/ m for the proportion X / m of successes when X ∼ B (m, µ);
(iii) V (µ) = µ2 ν when X has a gamma distribution with mean µ and index ν .
Find variance stabilising transformations for these distributions.
3.
A random variable θ , based on a random sample of size n > 1, is said to be an unbiased
estimator of a parameter θ if E (θ ) = θ for
θ is unbiased for θ and that V (θ ) = O n−all1θ. ∈ Θ, the parameter space. Suppose that
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(i) Use the δ -method to show that if h(θ ) is unbiased for h(θ), where h is continuously
differentiable, then h(x) = a + bx, where a and b are constants.
(ii) If X1 , X2, . . . , X is a random sample from N µ, σ 2 , show that no unbiased estimator of e can be based on the sample mean X .
(iii) Show that E e = exp µ + σ 2 2n and E e /2 = exp σ2 2n + o n−1 ,
where S 2 = =1 (X − µ)2 n. Hence find an estimator of e that is unbiased
apart from terms of smaller order than n−1 .
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X
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4.
Let X , X , .. . , X be a random sample from a uniform distribution on the interval
θ − , θ + . Show that the joint density of the sample minimum and maximum, X
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1
2
2
n
1
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(1)
and X( ) , is
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(u, v) = n(n − 1)(v − u) − , θ − 12 < u < v < θ + 12 .
The sample
rangeis R = X − X , and a natural estimator of θ is the mid-range,
T = X +X
2. Show that the conditional density of T given R is
f (t|r; θ) = (1 − r)− , 0 < r < 1, θ − 12 + 2r < t < θ + 12 − 2r .
f
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X(1) ,X( )
(n)
(1)
2
(1)
(n)
1
Calculate the mean and variance of T conditional on R. Discuss the adequacy of T given
R as an estimator of θ when a) r → 0, b) r → 1.
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