Exercises

Exercise 3.1 We have a srs of size n from a population with mean μ and variance σ2.

  1. Prove that ni=1aiXi is an unbiased estimator of μ if ni=1ai=1.
  2. 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.

  1. Show that the sample mean is biased for α and that its bias is 1/2.
  2. Compute the MSE of ˉX as an estimator of α.
  3. 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).

  1. Find an unbiased estimator of λ based on X(1).
  2. Compare the previous estimator with ˉX and decide which one is better.
  3. 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).

  1. Prove that ni=1Yi/ni=1xi is an unbiased estimator of β.
  2. Prove that (1/n)ni=1Yi/xi is also unbiased.
  3. 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.

  1. Prove that S2:=wS21+(1w)S22 is a consistent estimator of σ2, for any w(0,1).
  2. Show that

ˉXˉY(μ1μ2)S1n1+1n2dN(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:

  1. k>0 such that the estimator Tk=kT is unbiased for λ.
  2. 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.

  1. Prove that X(n) is consistent in probability for θ.
  2. 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+12X212X3,ˆμ2=X4+15X5110X10.

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.

  1. Which ones are unbiased?
  2. Among the unbiased ones, which is the most efficient?
  3. Find an unbiased estimator of μ different from ˉX that is more efficient than the previous unbiased estimators.
  4. 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

ˉXλˉX/ndN(0,1).

Exercise 3.12 Prove the law of the large numbers from Theorem 3.2 using Theorem 3.1.

Exercise 3.13 Let (X1,,Xn) be a srs from a geometric distribution, whose pmf is

p(x;θ)=θ(1θ)x1,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)=2xθex2/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(1λ)1x,x[0,1], λ(0,1),

with c(λ)=21{λ=1/2}+2tanh1(12λ)12λ1{λ1/2}. 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)=2xθex2/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|xμ|/σ,xR, μR, 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)xα1(1x)β1,x(0,1), α>0, β>0.

Find a minimal sufficient statistic for:

  1. α, assuming that β is known.
  2. β, assuming that α is known.

Exercise 3.20 Compute the Fisher information for XExp(λ). Then, derive an efficient estimator for λ.

Exercise 3.21 Is it possible to compute the Fisher information for the following distributions?

  1. U(0,θ).
  2. The pdf f(x;θ)=θ/x2, xθ>0.

What do these two distributions have in common?

Exercise 3.22 Let (X1,,Xn) be a srs of a rv XN(μ,σ2). Show that ˉX is an efficient estimator of μ.

Exercise 3.23 Let (X1,,Xn1) be a srs of XN(μ1,σ2) and let (Y1,,Yn2) be a srs of YN(μ2,σ2), with X and Y independent. As an estimator of σ2, we consider a linear combination of the sample quasivariances S21 and S22, that is, wS21+(1w)S22, for 0w1.

  1. Prove that this estimator is unbiased for any value of w.
  2. 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:

  1. ˆθn=ˉX.
  2. ˆθn=med{X1,,Xn}.
  3. 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):=1nni=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:

  1. Plotting the curve λI(λ) over the grid lambda <- seq(1, 10, by = 0.5).
  2. For each lambda, simulating (X1,,Xn)Pois(λ) and plotting the curve λˆI(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?