Question

(a) Let Y_{1},Y_{2},··· ,Y_{n} 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 Y_{1},··· ,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.

Answer #1

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 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 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, . . ., 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...

. 2. Let Y1,Y2,...,Yn be i.i.d. draws from a distribution of
mean µ.
A test of H0 : µ ≥ 5 versus H1 : µ < 5
using the usual t-statistic yields a p-value of 0.03.
a. Can we reject the null at 5% significance level (or α =
0.05)? Explain?
b. How about at 1% significance level (or α = 0.01)? Explain?
[Draw a figure to explain, if helpful.]

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.

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 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, 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 denote a random sample from a Weibull
distribution with parameters m=3 and unknown alpha:
f(y)=(3/alpha)*y^2*e^(-y^3/alpha) y>0
0 otherwise
Find the MLE of alpha. Check when its a maximum

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