Question

Let Y_{1}, Y_{2}, . . ., Y_{n} be a
random sample from a Laplace distribution with density function

f(y|θ) = (1/2θ)e^{-|y|/θ} for -∞ < y < ∞

where θ > 0. The first two moments of the distribution are
E(Y) = 0 and E(Y^{2}) = 2θ^{2}.

a) Find the likelihood function of the sample.

b) What is a sufficient statistic for θ?

c) Find the maximum likelihood estimator of θ.

d) Find the maximum likelihood estimator of the standard deviation of the double exponential distribution.

e) Find the method of moments estimator of θ.

f) Show the maximum likelihood estimator is a MVUE of θ.

Answer #1

Let Y1, Y2, ... Yn be a random sample of an exponential
population with parameter θ. Find the density function of the
minimum of the sample Y(1) = min(Y1, Y2, ..., Yn).

Let Y1, Y2, . . ., Yn be a
random sample from a uniform distribution on the interval (θ - λ, θ
+ λ) where -∞ < θ < ∞ and λ > 0. Find the method of
moments estimators of θ and λ.

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.

1. (a) Y1,Y2,...,Yn form a random sample from a probability
distribution with cumulative distribution function FY (y) and
probability density function fY (y). Let Y(1) = min{Y1,Y2,...,Yn}.
Write the cumulative distribution function for Y(1) in terms of FY
(y) and hence show that the probability density function for Y(1)
is fY(1)(y) = n{1−FY (y)}n−1fY (y). [8 marks]
(b) An engineering system consists of 5 components connected in
series, so, if one components fails, the system fails. The
lifetimes (measured in...

5. Let Y1, Y2, ...Yn (independent and identically distributed. ∼
f(y; α) = 1/6 α8y3 · e^(−α2y3 ), 0 ≤
y < ∞, 0 < α < ∞.
(a) (8 points) Find an expression for the Method of Moments
estimator of α, ˜α. Show all work.
(b) (8 points) Find an expression for the Maximum Likelihood
estimator for α, ˆα. Show all work.

Let Y1, Y2, . . . , Yn denote a random sample from a uniform
distribution on the interval (0, θ). (a) (5 points)Find the MOM for
θ. (b) (5 points)Find the MLE for θ.

Let Y1,Y2.....,Yn be independent ,uniformly distributed random
variables on the interval[0,θ].，Y(n)=max(Y1,Y2,....,Yn)，which is
considered as an estimator of θ. Explain why Y is a good estimator
for θ when sample size is large.

Let Y1, Y2, Y3 ,..,, Yn be a random sample from a normal
distribution with mean µ and standard deviation 1. Then find the
MVUE( Minimum - Variance Unbiased Estimation) for the parameters:
µ^2 and µ(µ+1)

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 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.

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