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Exercise 1

Let X1,X2,Xn be a random sample of size n from a distribution with probability density function

f(x,θ)=1θex/θ,x>0, θ>0

Note that, the moments of this distribution are given by

E[Xk]=0xkθex/θ=k!θk.

This will be a useful fact for Exercises 2 and 3.

(a) Obtain the maximum likelihood estimator of θ, ˆθ. (This should be a function of the unobserved xi and the sample size n.) Calculate the estimate when

x1=0.50, x2=1.50, x3=4.00, x4=3.00.

(This should be a single number, for this dataset.)

(b) Calculate the bias of the maximum likelihood estimator of θ, ˆθ. (This will be a number.)

(c) Find the mean squared error of the maximum likelihood estimator of θ, ˆθ. (This will be an expression based on the parameter θ and the sample size n. Be aware of your answer to the previous part, as well as the distribution given.)

(d) Provide an estimate for P[X>4] when

x1=0.50, x2=1.50, x3=4.00, x4=3.00.

Exercise 2

Let X1,X2,Xn be a random sample of size n from a distribution with probability density function

f(x,α)=α2xex/α,x>0, α>0

(a) Obtain the maximum likelihood estimator of α, ˆα. Calculate the estimate when

x1=0.25, x2=0.75, x3=1.50, x4=2.5, x5=2.0.

(b) Obtain the method of moments estimator of α, ˜α. Calculate the estimate when

x1=0.25, x2=0.75, x3=1.50, x4=2.5, x5=2.0.

Exercise 3

Let X1,X2,Xn be a random sample of size n from a distribution with probability density function

f(x,β)=12β3x2ex/β,x>0, β>0

(a) Obtain the maximum likelihood estimator of β, ˆβ. Calculate the estimate when

x1=2.00, x2=4.00, x3=7.50, x4=3.00.

(b) Obtain the method of moments estimator of β, ˜β. Calculate the estimate when

x1=2.00, x2=4.00, x3=7.50, x4=3.00.

Exercise 4

Let X1,X2,Xn be a random sample of size n from a distribution with probability density function

f(x,λ)=λxλ1,0<x<1,λ>0

(a) Obtain the maximum likelihood estimator of λ, ˆλ. Calculate the estimate when

x1=0.10, x2=0.20, x3=0.30, x4=0.40.

(b) Obtain the method of moments estimator of λ, ˜λ. Calculate the estimate when

x1=0.10, x2=0.20, x3=0.30, x4=0.40.