Question

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)

Answer #1

Let Y1,Y2,Y3,...,Yn denote a random sample from a Poisson
probability distribution.
(a) Show that sample mean, ˆ θ1 = ¯ Y and the sample variance, ˆ
θ2 = S2 are both unbiased estimators of θ.
(b) Calculate relative eﬃciency of the two estimators in (a).
Based on your calculation, Which of the two estimators in 3a would
you select as a better estimator?

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...

Problem 3. Let Y1, Y2, and Y3 be independent, identically
distributed random variables from a population with mean µ = 12 and
variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 +
Y3) denote the average of these three random
variables.
A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an
unbiased estimator of µ?
B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?
C. Consider a different estimator...

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 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.

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¯ )?

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 < Y3 <
Y4 be the order statistics of a random sample of size n
= 5 (Y1 < Y2 < Y3 <
Y4 <Y5). from the distribution having pdf
f(x) = e−x, 0 < x < ∞, zero elsewhere. Find P(Y5 ≥ 3).

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 15 minutes ago

asked 25 minutes ago

asked 34 minutes ago

asked 38 minutes ago

asked 47 minutes ago

asked 47 minutes ago

asked 52 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago