Question

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.

Answer #1

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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...

Let X¯ be the sample mean of a random sample X1, . . . , Xn from
the exponential distribution, Exp(θ), with density function f(x) =
(1/θ) exp{−x/θ}, x > 0. Show that X¯ is an unbiased point
estimator of θ.

Let X1, X2, ..., Xn be a random sample (of size n) from U(0,θ).
Let Yn be the maximum of X1, X2, ..., Xn.
(a) Give the pdf of Yn.
(b) Find the mean of Yn.
(c) One estimator of θ that has been proposed is Yn. You may
note from your answer to part (b) that Yn is a biased estimator of
θ. However, cYn is unbiased for some constant c. Determine c.
(d) Find the variance of cYn,...

Let Y1,
Y2, …, Yndenote a random sample of size
n from a population whose density is given by
f(y) = 5y^4/theta^5
0<y<theta
0 otherwise
a) Is an unbiased estimator of
θ?
b) Find the MSE of Y bar
c) Find a function of that is an
unbiased estimator of θ.

Let we have a sample of 100 numbers from exponential
distribution with parameter θ
f(x, θ) = θ e- θx , 0
< x.
Find MLE of parameter θ. Is it unbiased estimator? Find unbiased
estimator of parameter θ.

Let X_1,…, X_n be a random sample from the Bernoulli
distribution, say P[X=1]=θ=1-P[X=0].
and
Cramer Rao Lower Bound of θ(1-θ)
=((1-2θ)^2 θ(1-θ))/n
Find the UMVUE of θ(1-θ) if such exists.
can you proof [part (b) ] using (Leehmann Scheffe
Theorem step by step solution) to proof
[∑X1-nXbar^2 ]/(n-1) is the umvue , I have the key
solution below
x is complete and sufficient.
S^2=∑ [X1-Xbar ]^2/(n-1) is unbiased estimator of θ(1-θ) since
the sample variance is an unbiased estimator of the...

Suppose Y1,··· ,Yn is a sample from a
exponential distribution with mean θ, and let Y(1),···
,Y(n) denote the order statistics of the sample.
(a) Find the constant c so that cY(1) is an unbiased
estimator of θ.
(b) Find the suﬃcient statistic for θ and MVUE for θ.

Let Xl, n be a random
sample from a gamma distribution with parameters a = 2 and p =
20.
a) Find an
estimator , using the method of maximum likelihood
b) Is the estimator obtained in part a) is unbiased and
consistent estimator for the parameter 0?
c) Using the
factorization theorem, show that the estimator found in part a) is
a sufficient estimator of 0.

Let X1,...,Xn be a random sample from the pdf f(x;θ) = θx^(θ−1)
, 0 ≤ x ≤ 1 , 0 < θ < ∞ Find the method of moments estimator
of θ.

5. Let a random sample, X1, X2, ..., Xn of size n = 10 from a
distribution that is N(μ1, σ2 ) give ̄x = 4.8 and s 2+ 1 = 8.64 and
a random sample, Y1, Y2, ..., Yn of size n = 10 from a distribution
that is N(μ2, σ2 ) give y ̄ = 5.6 and s 2 2 = 7.88. Find a 95%
confidence interval for μ1 − μ2.

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