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Ex1: Suppose random variable X has a Bernoulli distribution for which the pa- rameter 6 is unknown (0 < 0 < 1). We shall determine the Fisher information I(@) in X. The point mass function of X is f(z|0) = 0"(1 – 0)- for r =1 or r 0. Suppose that X N(1,0*), and u is unknown, but the value of o² is given. Ex2: find the Fisher information I(4) in X. Ex3: Suppose a random sample X1, X, from a normal distribution N(u,0), with u given and the variance ở unknown. Calculate the lower bound of variance for any estimator, and compare to that of the sample variance S. EX4: Let X..X, are independent with geometric distribution P(X, = r) = p(1- p)--1 for 1= 1,2,.. Let 8 = p. (i) 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? (iii) Compute the expected Fisher informuation for 0. (iv) Does é attain the Cramer-Rao lower bound? EXS: 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 – 4). Give the asymptotic distribution of n- }) EX6: If X1, X2, ., X has an exponential distribution with parameter Let T1 and ....

Ex1: Suppose random variable X has a Bernoulli distribution for which the pa- rameter 6 is unknown (0 < 0 < 1). We shall determine the Fisher information I(@) in X. The point mass function of X is f(z|0) = 0"(1 – 0)- for r =1 or r 0. Suppose that X N(1,0*), and u is unknown, but the value of o² is given. Ex2: find the Fisher information I(4) in X. Ex3: Suppose a random sample X1, X, from a normal distribution N(u,0), with u given and the variance ở unknown. Calculate the lower bound of variance for any estimator, and compare to that of the sample variance S. EX4: Let X..X, are independent with geometric distribution P(X, = r) = p(1- p)--1 for 1= 1,2,.. Let 8 = p. (i) 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? (iii) Compute the expected Fisher informuation for 0. (iv) Does é attain the Cramer-Rao lower bound? EXS: 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 – 4). Give the asymptotic distribution of n- }) EX6: If X1, X2, ., X has an exponential distribution with parameter Let T1 and ....

MATLAB: An Introduction with Applications
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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Ex1: Suppose random variable X has a Bernoulli distribution for which the pa- rameter 6 is unknown (0 < 0 < 1). We shall determine the Fisher information I(@) in х. The point mass function of X is f(z|0) = 0"(1 – 0)- for r = 1 or r 0. Suppose that X N(1,0*), and u is unknown, but the value of o² is given. Ex2: find the Fisher information I(4) in X. Suppose a random sample X1, X, from a normal distribution N(u,0), Ex3: with u given and the variance é unknown. Calculate the lower bound of variance for any estimator, and compare to that of the sample variance S. EX4: Let X.. X, are independent with geometric distribution P(X, = r) = p(1- p)--1 for z= 1,2,.. Let 8 = p. (i) 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? (iii) Compute the expected Fisher informuation for 0. (iv) Does é attain the Cramer-Rao lower bound? EXS: 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 – 1). Give the asymptotic distribution of yn- }) EX6: If X1, X2, .., X has an exponential distribution with parameter - Let T, and T2 are unbiased estimates of A and respectively. Find CRLB of T, and T2.
Transcribed Image Text:Ex1: Suppose random variable X has a Bernoulli distribution for which the pa- rameter 6 is unknown (0 < 0 < 1). We shall determine the Fisher information I(@) in х. The point mass function of X is f(z|0) = 0"(1 – 0)- for r = 1 or r 0. Suppose that X N(1,0*), and u is unknown, but the value of o² is given. Ex2: find the Fisher information I(4) in X. Suppose a random sample X1, X, from a normal distribution N(u,0), Ex3: with u given and the variance é unknown. Calculate the lower bound of variance for any estimator, and compare to that of the sample variance S. EX4: Let X.. X, are independent with geometric distribution P(X, = r) = p(1- p)--1 for z= 1,2,.. Let 8 = p. (i) 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? (iii) Compute the expected Fisher informuation for 0. (iv) Does é attain the Cramer-Rao lower bound? EXS: 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 – 1). Give the asymptotic distribution of yn- }) EX6: If X1, X2, .., X has an exponential distribution with parameter - Let T, and T2 are unbiased estimates of A and respectively. Find CRLB of T, and T2.
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