Question

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.

Answer #1

(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.

Let Y1,Y2,...,Yn denote a
random sample of size n from a population with a uniform
distribution on the interval (0,θ). Let Y(n)=
max(Y1,Y2,...,Yn) and U =
(1/θ)Y(n) .
a) Show that U has cumulative density function
0 ,u<0,
Fu (u) = un ,0≤u≤1,
1 ,u>1

Please type out your answer.
Let Y1, . . . , Yn be a random sample from the gamma
distribution with parameters α and β, where α is known. Find the
maximum likelihood estimator of β. Compute its mean and
variance.

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...

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 (X1, Y1), . . . ,(Xn, Yn), be a random sample from a
bivariate normal distribution with parameters µ1, µ2, σ2 1 , σ2 2 ,
ρ. (Note: (X1, Y1), . . . ,(Xn, Yn) are independent). What is the
joint distribution of (X ¯ , Y¯ )?

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, 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)

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