MAST90083 Computational Statistics & Data Mining Bootstrap Methods
Tutorial & Practical 8: Bootstrap Methods
Question 1
In this question we explore the bias reduction performance of a bootstrap estimator of a third
power of the mean of a population assumed to be a scalar θ0 = θ(F0) = µ
3 where
µ =
∫
xdF0(x)
Let X = {x1, …, xn} be a sample drawn from F0 where xi ∈ R, used for the estimation of
θ0.
1. Provide the form of the nonparametric estimator obtained from the empirical distribution
F1.
2. Derive the expression of the bias b1 = E
(
θ̂ − θ0
)
.
3. Derive the expression of the bootstrap estimate of b1.
4. Use this expression to derive the bootstrap bias-reduced estimate θ̂1 of θ
5. Derive the expression of the bias b2 = E
(
θ̂1 − θ0
)
6. Compare b1 and b2
Question 2
In this question we explore the bias reduction performance of a bootstrap estimator of a third
power of the mean of a Normal population N(µ, σ2) when the parameters are estimated using
the maximum likelihood estimator from a set X = {x1, …, xn} ∼ F0, xi ∈ R
x̄ =
1
n
n∑
i=1
xi and σ̂
2 =
1
n
n∑
i=1
(xi − x̄)
2
1. Provide the form of the nonparametric estimator θ̂ obtained from the empirical distri-
bution F1 and its associated bias b1.
2. Derive the expression of the bootstrap bias-reduced estimate θ̂1 of θ
3. Derive the expression of the bias associated with θ̂1, b2 and compare it with b1
Question 3
Suppose that we estimate the distribution of θ̂−θ by the bootstrap distribution θ̂∗− θ̂. Denote
the α−percentile of θ̂∗− θ̂ by H−1(α). Derive the interval for θ that results from inverting the
relation
Ĥ−1(α) ≤ θ̂ − θ ≤ Ĥ−1(1− α)
1