Question

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.

Answer #1

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.

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 X2, ... , Xn denote a random sample
from a discrete uniform distribution over the integers - θ, - θ +
1, ... , -1, 0, 1, ... , θ - 1, θ,
where θ is a positive integer. What is the maximum
likelihood estimator of θ?
A) min[X1, .. , Xn]
B) max[X1, .. , Xn]
C) -min[X1, .. , Xn]
D) (max[X1, .. , Xn] -
min[X1, .. , Xn]) / 2
E) max[|X1| , ... , |Xn|]

Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...

Let X1,..., Xn be a random sample from a
distribution with pdf as follows:
fX(x) = e^-(x-θ) , x > θ
0 otherwise.
Find the sufficient statistic for θ.
Find the maximum likelihood estimator of θ.
Find the MVUE of θ,θˆ
Is θˆ a consistent estimator of θ?

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

Let X1, . . . , Xn be a random sample from a Poisson
distribution.
(a) Prove that Pn i=1 Xi is a sufficient statistic for λ.
(b) The MLE for λ in a Poisson distribution is X. Use this fact
and the result of part (a) to argue that the MLE is also a
sufficient statistic for λ.

Suppose X1 ...... Xn is a random sample from the uniform
distribution on [a; b].
(a) Find the method of moments estimators of a and b.
(b) Find the maximum likelihood estimators of a and b.
please step by step

Let
X1,X2,...,Xn
be i.i.d. (independent and identically distributed) from the
uniform distribution U(μ,μ+1) where μ∈R is
unknown. Find a minimal sufficient statistic for μ
parameter.

Let X1, . . . , Xn be a random sample from the following
pdf:
f(x|θ)= (x/θ)*e^(-x^2/2θ). x>0
(a) Find a sufficient statistic for θ.

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