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Let X1,X2,…Xn be a random sample of size n from a distribution with probability density function
f(x,θ)=1θe−x/θ,x>0, θ>0
Note that, the moments of this distribution are given by
E[Xk]=∫∞0xkθe−x/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.
Let X1,X2,…Xn be a random sample of size n from a distribution with probability density function
f(x,α)=α−2xe−x/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.
Let X1,X2,…Xn be a random sample of size n from a distribution with probability density function
f(x,β)=12β3x2e−x/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.
Let X1,X2,…Xn be a random sample of size n from a distribution with probability density function
f(x,λ)=λxλ−1,0<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.