Let (x₁, Y₂) (i = 1,...,n) be independent following the Poisson regression model (without intercept) Y~ Poission (x₁) (This is often used to model number of events occurred given a covariate. For example, the number of traffic accidents in a month as Y; at the i-th intersection vs the number of cars going through the same intersection in the same period, as x;) (a) Since E(Y;) = Xx;, we can write Y; = Xx; + e; where E(e;) = 0. Find the least square estimator for A. (b) Write the likelihood function L(X) (treat £₁,...,n as non-random) (c) Find MLE for A. (d) Suppose we use Gamma(a, ß) with pdf Ba T(a) T(X) = -Aª-le-BA for X>0 as the prior distribution. Find the posterior distribution given observations (y₁, ₁), ..., (Yn, xn). And find the corresponding Bayes estimator for A. [Hint: The prior is a conjugate prior so the posterior distribution is also a Gamma distribution. The mean of the Gamma(a, b) distribution is a/b]
Let (x₁, Y₂) (i = 1,...,n) be independent following the Poisson regression model (without intercept) Y~ Poission (x₁) (This is often used to model number of events occurred given a covariate. For example, the number of traffic accidents in a month as Y; at the i-th intersection vs the number of cars going through the same intersection in the same period, as x;) (a) Since E(Y;) = Xx;, we can write Y; = Xx; + e; where E(e;) = 0. Find the least square estimator for A. (b) Write the likelihood function L(X) (treat £₁,...,n as non-random) (c) Find MLE for A. (d) Suppose we use Gamma(a, ß) with pdf Ba T(a) T(X) = -Aª-le-BA for X>0 as the prior distribution. Find the posterior distribution given observations (y₁, ₁), ..., (Yn, xn). And find the corresponding Bayes estimator for A. [Hint: The prior is a conjugate prior so the posterior distribution is also a Gamma distribution. The mean of the Gamma(a, b) distribution is a/b]
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![Let (xi, Yi) (i = 1,...,n) be independent following the Poisson regression model
(without intercept)
Y~ Poission (x₁)
(This is often used to model number of events occurred given a covariate. For example,
the number of traffic accidents in a month as Y; at the i-th intersection vs the number
of cars going through the same intersection in the same period, as x₂)
(a) Since E(Y;) = Ari, we can write Y₁ = λr; + e; where E(e;) = 0. Find the least
square estimator for A.
(b) Write the likelihood function L(A) (treat x₁,...,En as non-random)
(c) Find MLE for A.
(d) Suppose we use Gamma(a, ß) with pdf
Ba
π(X) = -Aa-le-BA for A>0
r(a)
as the prior distribution. Find the posterior distribution given observations (y₁, ₁),..., (Yn, In).
And find the corresponding Bayes estimator for A. [Hint: The prior is a conjugate
prior so the posterior distribution is also a Gamma distribution. The mean of the
Gamma(a, b) distribution is a/b]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d1e8e77-1952-4091-84bd-24cd16182052%2F4906c784-14b5-414d-9ce3-25db3e347df6%2Foiwajl_processed.png&w=3840&q=75)
Transcribed Image Text:Let (xi, Yi) (i = 1,...,n) be independent following the Poisson regression model
(without intercept)
Y~ Poission (x₁)
(This is often used to model number of events occurred given a covariate. For example,
the number of traffic accidents in a month as Y; at the i-th intersection vs the number
of cars going through the same intersection in the same period, as x₂)
(a) Since E(Y;) = Ari, we can write Y₁ = λr; + e; where E(e;) = 0. Find the least
square estimator for A.
(b) Write the likelihood function L(A) (treat x₁,...,En as non-random)
(c) Find MLE for A.
(d) Suppose we use Gamma(a, ß) with pdf
Ba
π(X) = -Aa-le-BA for A>0
r(a)
as the prior distribution. Find the posterior distribution given observations (y₁, ₁),..., (Yn, In).
And find the corresponding Bayes estimator for A. [Hint: The prior is a conjugate
prior so the posterior distribution is also a Gamma distribution. The mean of the
Gamma(a, b) distribution is a/b]
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