Ex1: Suppose random variable X has a Bernoulli distribution for which the pa- rameter 0 is unknown (0 < 0 < 1). We shall determine the Fisher information I(8) in X. The point mass function of X is f(r0) = 0"(1 – 0)- for z = 1 or r 0.
Ex1: Suppose random variable X has a Bernoulli distribution for which the pa- rameter 0 is unknown (0 < 0 < 1). We shall determine the Fisher information I(8) in X. The point mass function of X is f(r0) = 0"(1 – 0)- for z = 1 or r 0.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:Ex1:
Suppose random variable X has a Bernoulli distribution for which the pa-
rameter 0 is unknown (0 < 0 < 1). We shall determine the Fisher information I(6) in
х.
The point mass function of X is
f(r|0) = 0"(1 – 0)'-- for r = 1 or r = 0.
Suppose that X - N(u,o*), and u is unknown, but the value of o is given.
Ex2:
find the Fisher information I() in X.
Ex3:
Suppose a random sample X1, ., X, from a normal distribution N(,0),
with u given and the variance e unknown. Calculate the lower bound of variance for any
estimator, and compare to that of the sample variance S.
Let X.. X, are independent with geometric distribution
EX4:
P(X, = r) = p(1- p)--1 for r = 1,2,.. Let 8 = p-.
(6) Find the maximum likelihood estimator for p. Considering n= 1, is it unbiased?
(i) Show that è = X is the maximum likelihood estimator for 0. Is it unbiased?
(ii) Compute the expected Fisher information for 0.
(iv) Does é attain the Cramer-Rao lower bound?
EX5: Let X1, X2, ., Xn- Poisson(2). Find CRLB of the MLE of A and prove it is an
...
efficient estimator.
Give the asymptotic distribution of vn(X – A).
Give the asymptotic distribution of n( -)
EX6: If X1, X2, ., X, has an exponential distribution with parameter Let T, and
T2 are unbiased estimates of l and respectively. Find CRLB of T, and T2.
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