Question

Suppose Y_{1},··· ,Y_{n} 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.

Answer #1

Let Y1, Y2, . . ., Yn 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(Y2) = 2θ2.
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...

(a) Let Y1,Y2,··· ,Yn be i.i.d.
with geometric distribution P(Y = y) = p(1−p)y-1 y=1, 2,
........, 0<p<1. Find a suﬃcient statistic for p.
(b) 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 Y1, . . . , Yn is a sample from an Exponential
distribution with mean β.
(a)Find the distribution of Sn =Y1 +···+Yn
(b) Find E[Sn] and V [Sn]

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 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 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, ..., Yn be IID Poisson(λ) random variables. Argue that
Y¯ , the sample mean, is a sufficient statistic for λ by using the
factorization criterion. Assuming that Y¯ is a complete sufficient
statistic, explain why Y¯ is the minimum variance unbiased
estimator.

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, . . . , 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 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.

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