Question

Determine the form of a minimal sufficient statistic for a sample of size n from the Uniform[0,θ] model where θ> 0

Answer #1

Suppose that (X1, · · · , Xn) is a random sample from uniform
distribution U(0, θ).
(a) Prove that T(X1, · · · , Xn) = X(n) is minimal sufficient
for θ. (X(n) is the largest order statistic, i.e., X(n) = max{X1, ·
· · , Xn}.)
(b) In addition, we assume θ ≥ 1. Find a minimal sufficient
statistic for θ and justify your answer.

Show that the sum of the observations of a random sample of size
n from
a gamma distribution that has pdf f(x; θ) = (1/θ)e^(−x/θ), 0 < x
< ∞, 0 < θ < ∞,
zero elsewhere, is a sufficient statistic for θ. Use Neyman's
Factorization Theorem.

Let Y1, · · · , yn be a random sample of size n from a beta
distribution with parameters α = θ and β = 2. Find the sufficient
statistic for θ.

Suppose a random sample of size n is drawn from the pdf
f(sub)Y(y;θ) =1/θ, 0 ≤ y ≤ θ.
Find a sufficient statistic for θ.

Let Y1,Y2,...,Yn denote a
random sample of size n from a population with a uniform
distribution on the interval (0,θ). Let Y(n)=
max(Y1,Y2,...,Yn) and U =
(1/θ)Y(n) .
a) Show that U has cumulative density function
0 ,u<0,
Fu (u) = un ,0≤u≤1,
1 ,u>1

Suppose that X1,..., Xn form a random sample from the
uniform distribution on the interval [0,θ], where the value of the
parameter θ is unknown (θ>0).
(1)What is the maximum likelihood estimator of θ?
(2)Is this estimator unbiased? (Indeed, show that it underestimates
the parameter.)

Let X be the mean of a random sample of size n from a N(θ, σ2)
distribution,
−∞ < θ < ∞, σ2 > 0. Assume that σ2 is known. Show that
X
2 − σ2
n is an
unbiased estimator of θ2 and find its efficiency.

Let X1, X2,...,Xn be a random sample from Bernoulli
(p). Determine a sufficient statistic for p and derive the UMVUE
and MLE of T(p)=p^2(1-p)^2.

a. If ? ̅1 is the mean of a random sample of size n from a
normal population with mean ? and variance ?1 2 and ? ̅2 is the
mean of a random sample of size n from a normal population with
mean ? and variance ?2 2, and the two samples are independent, show
that ?? ̅1 + (1 − ?)? ̅2 where 0 ≤ ? ≤ 1 is an unbiased estimator
of ?.
b. Find the value...

Suppose that X1,...,Xn ∼ U(0,θ); that is, a sample of N
observations from a random variable with a uniform distribution
where the lower bound is 0 and the upper bound θ is unknown. Find
the maximum likelihood estimate of θ, also demonstrating this in R.
Draw the pdf and the likelihood, and explain what they represent,
in R.

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