Exercises
Exercise 3.1 We have a srs of size n from a population with mean μ and variance σ2.
- Prove that ∑ni=1aiXi is an unbiased estimator of μ if ∑ni=1ai=1.
- Among all the unbiased estimators of this form, find the one with minimum variance and compute it.
Exercise 3.2 Let (X1,…,Xn) be a srs from a U(0,θ), θ>0. Find out whether the following estimators are unbiased for the population mean and, in case of a positive answer, find their biases:
ˉX,X1,X(n),X(1),0.5X(n)+0.5X(1).
Exercise 3.3 The consumption of a certain good in a family with four members during the summer months is a rv with U(α,α+1) distribution. Let (X1,…,Xn) be a srs of consumption of the same good for different families.
- Show that the sample mean is biased for α and that its bias is 1/2.
- Compute the MSE of ˉX as an estimator of α.
- Obtain from ˉX an unbiased estimator of α and provide its MSE.
Exercise 3.4 Let (X1,…,Xn) be a srs from an Exp(λ).
- Find an unbiased estimator of λ based on X(1).
- Compare the previous estimator with ˉX and decide which one is better.
- The lifetime of a light bulb in days is usually modeled by an exponential distribution. The following data are lifetimes for light bulbs: 50.1, 70.1, 137, 166.9, 170.5, 152.8, 80.5, 123.5, 112.6, 148.5, 160, 125.4. Estimate the average duration of a light bulb using the two previous estimators.
Exercise 3.5 Assume that the rv’s Y1,…,Yn are such that
Yi=βxi+εi,i=1,…,n,
where the xi’s are constants and the εi are iid rv’s distributed as N(0,σ2).
- Prove that ∑ni=1Yi/∑ni=1xi is an unbiased estimator of β.
- Prove that (1/n)∑ni=1Yi/xi is also unbiased.
- Compute the variances of both estimators.
Exercise 3.6 Let (X1,…,Xn1) be a srs of X with E[X]=μ1 and Var[X]=σ2, and let (Y1,…,Yn2) be a srs of Y with E[Y]=μ2 and Var[Y]=σ2, with X and Y independent.
- Prove that S2:=wS′21+(1−w)S′22 is a consistent estimator of σ2, for any w∈(0,1).
- Show that
ˉX−ˉY−(μ1−μ2)S√1n1+1n2d⟶N(0,1).
Exercise 3.7 The number of independent arrivals to an emergency service during a day follows a Pois(λ) distribution. In order to estimate λ and forecast the amount of required personnel in the service, we observe the arrivals during n days. We know that T=∑ni=1Xi is a sufficient statistic. Determine:
- k>0 such that the estimator Tk=kT is unbiased for λ.
- The condition that the sequence kn must satisfy for obtaining a sequence of estimators that is consistent in probability, if the sample size n is allowed to grow.
Exercise 3.8 Consider a sample as in Exercise 3.2.
- Prove that X(n) is consistent in probability for θ.
- Check whether Yn=2ˉXn is also consistent in probability for θ.
Exercise 3.9 Let (X1,…,X10) be a srs from a distribution with mean μ and variance σ2. Consider the following two estimators of μ:
ˆμ1=X1+12X2−12X3,ˆμ2=X4+15X5−110X10.
Which one is better in terms of MSE? Are they consistent in probability?
Exercise 3.10 Let (X1,…,Xn) be a srs from a distribution with mean μ and variance σ2. Consider the following estimators of μ:
ˆμ1=X1+2X2+3X36,ˆμ2=X1+4X2+X36,ˆμ3=32X1+12X2+X3+⋯+Xnn.
- Which ones are unbiased?
- Among the unbiased ones, which is the most efficient?
- Find an unbiased estimator of μ different from ˉX that is more efficient than the previous unbiased estimators.
- Which of them is consistent in squared mean?
Exercise 3.11 Let (X1,…,Xn) be a srs of a rv with Pois(λ) distribution. Check that
ˉX−λ√ˉX/nd⟶N(0,1).
Exercise 3.13 Let (X1,…,Xn) be a srs from a geometric distribution, whose pmf is
p(x;θ)=θ(1−θ)x−1,x=1,2,…, θ∈(0,1).
Prove that ∑ni=1Xi is sufficient for θ.
Exercise 3.14 Let (X1,…,Xn) be a srs from a population with pdf
f(x;θ)=2xθe−x2/θ1{x>0},θ>0.
Show that ∑ni=1X2i is sufficient for θ and that ∑ni=1Xi is not.
Exercise 3.15 Let (X1,…,Xn) be a srs from a continuous Bernoulli distribution, whose pdf is
f(x;λ)=c(λ)λx(1−λ)1−x,x∈[0,1], λ∈(0,1),
with c(λ)=21{λ=1/2}+2tanh−1(1−2λ)1−2λ1{λ≠1/2}. Find a sufficient statistic for λ.
Exercise 3.16 Let X1 be a normal rv with mean zero and variance σ2. Is |X1| a sufficient statistic for σ2?
Exercise 3.17 Let (X1,…,Xn) be a srs from a particular Weibull distribution with pdf
f(x;θ)=2xθe−x2/θ,x>0, θ>0.
Find a minimal sufficient statistic for θ.
Exercise 3.18 Let (X1,…,Xn) be a srs from a Laplace distribution Laplace(μ,σ), whose pdf is
f(x;μ,σ)=12σe−|x−μ|/σ,x∈R, μ∈R, σ>0.
Find a minimal sufficient statistic for σ assuming that μ is known.
Exercise 3.19 Let (X1,…,Xn) be a srs from a beta distribution Beta(α,β), whose pdf is
f(x;α,β)=Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1,x∈(0,1), α>0, β>0.
Find a minimal sufficient statistic for:
- α, assuming that β is known.
- β, assuming that α is known.
Exercise 3.20 Compute the Fisher information for X∼Exp(λ). Then, derive an efficient estimator for λ.
Exercise 3.21 Is it possible to compute the Fisher information for the following distributions?
- U(0,θ).
- The pdf f(x;θ)=θ/x2, x≥θ>0.
What do these two distributions have in common?
Exercise 3.22 Let (X1,…,Xn) be a srs of a rv X∼N(μ,σ2). Show that ˉX is an efficient estimator of μ.
Exercise 3.23 Let (X1,…,Xn1) be a srs of X∼N(μ1,σ2) and let (Y1,…,Yn2) be a srs of Y∼N(μ2,σ2), with X and Y independent. As an estimator of σ2, we consider a linear combination of the sample quasivariances S′21 and S′22, that is, wS′21+(1−w)S′22, for 0≤w≤1.
- Prove that this estimator is unbiased for any value of w.
- Obtain the value of w that provides the most efficient estimator.
Exercise 3.24 Consider a srs (X1,…,Xn) from a Cauchy(θ,σ) distribution. Use a simulation like that in Figure 3.1 to check empirically if ˆθn is a consistent estimator for θ. Consider these estimators:
- ˆθn=ˉX.
- ˆθn=med{X1,…,Xn}.
- ˆθn=T0.10 (0.10-trimmed mean).
Explore in each case N=5 simulations for each of the three choices of (θ,σ) and a sequence of n’s that increase at least until n=104. What are your conclusions?
Exercise 3.25 The Fisher information (3.7) is an expectation. Therefore, from a srs (X1,…,Xn) from a continuous rv with pdf f(⋅;θ), it can be approximated as the sample mean
ˆI(θ):=1nn∑i=1(∂logf(Xi;θ)∂θ)2.
The construction is analogous for a discrete rv.
Consider a Pois(λ) distribution. Use a simulation to check empirically how ˆI(λ) approximates I(λ)=1/λ by:
- Plotting the curve λ↦I(λ) over the grid
lambda <- seq(1, 10, by = 0.5)
. - For each
lambda
, simulating (X1,…,Xn)∼Pois(λ) and plotting the curve λ↦ˆI(λ).
Use n=10,100,1000. What are your conclusions? What if in Step 2 the simulation was done from (X1,…,Xn)∼Pois(λ0), λ0=2, for each lambda
?