Maximum likelihood of a statistical model

  • #1
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Homework Statement


I look at the distribution ##(Y_1,Y_2,...,Y_n)##
where
##Y_i=μ+(1+φ x_i)+ε_i## where ##-1<φ<1## and ##-1<x_i<1## . x's are known numbers. ε's are independent and normally distributed with mean 0 and variance 1.

I need to find the the maximum likelihood estimator for μ and φ

Homework Equations




The Attempt at a Solution


I get to the Log likelihood funcion: ##L(Y_1,Y_2,...,Y_n;μ,φ)=∑μ+(1+φ x_i)+ε_i##
When I differentiate it get:
##d L / dμ =∑1/(μ+(1+φ x_i)+ε_i), d L / dφ =∑x/(μ+(1+φ x_i)+ε_i##. Right?
These equations doesn't allow me to set ##d L / dμ=0,d L / dφ=0## because you can't divide by zero. What is going wrong for me?
 

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
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I get to the Log likelihood funcion: ##L(Y_1,Y_2,...,Y_n;μ,φ)=∑μ+(1+φ x_i)+ε_i##

Before you get to the log liklihood function, what function are you using for the liklihood function?
if the ##Y_i## represent the sample data, the expression for the liklihood function should have the variables ##Y_i## in it. Otherwise, you'd be doing a computation that ignores the data.
 
  • #3
Ray Vickson
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Homework Statement


I look at the distribution ##(Y_1,Y_2,...,Y_n)##
where
##Y_i=μ+(1+φ x_i)+ε_i## where ##-1<φ<1## and ##-1<x_i<1## . x's are known numbers. ε's are independent and normally distributed with mean 0 and variance 1.

I need to find the the maximum likelihood estimator for μ and φ

Homework Equations




The Attempt at a Solution


I get to the Log likelihood funcion: ##L(Y_1,Y_2,...,Y_n;μ,φ)=∑μ+(1+φ x_i)+ε_i##
When I differentiate it get:
##d L / dμ =∑1/(μ+(1+φ x_i)+ε_i), d L / dφ =∑x/(μ+(1+φ x_i)+ε_i##. Right?
These equations doesn't allow me to set ##d L / dμ=0,d L / dφ=0## because you can't divide by zero. What is going wrong for me?

This does not look like any likelihood function I have ever seen.

Start with the basics: what is the probability density ##f_i(y)## of the random variable ##Y_i##, that is, what is the function ##f_i(y)## in the statement ##P(y < Y_i < y + dy) = f_i(y)\, dy ?## In terms of the density functions ##f_1(y_1), f_2(y_2), \ldots, f_n(y_n)##, what is the likelihood of an observed event ##\{ Y_1 = y_1, Y_2 =y_2, \ldots, Y_n = y_n\}?## For given ##\{y_i\}## you want to maximize that likelihood function.
 
  • #4
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Before you get to the log liklihood function, what function are you using for the liklihood function?
if the ##Y_i## represent the sample data, the expression for the liklihood function should have the variables ##Y_i## in it. Otherwise, you'd be doing a computation that ignores the data.
I use ##L(Y_1,...Y_n;μ,φ) =Y_1*Y_2*...*Y_n##
 
  • #5
Stephen Tashi
Science Advisor
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I use ##L(Y_1,...Y_n;μ,φ) =Y_1*Y_2*...*Y_n##

That isn't correct. The problem does not say that ##Y_i## is a random variable with an exponential distribution. The problem says that ##\epsilon_i## is a random variable with a normal distribution.

The only random variable with a known distribution in the equation ##Y_i = \mu + (1 + \phi x_i) + \epsilon_i## is the variable ##\epsilon_i##.

The liklihood that ##\epsilon_i = \alpha_i## is ##\frac{1}{\sqrt{2\pi}} e^{- \frac{\alpha_i^2}{2}} ##

Solve the equation ##Y_i = \mu + (1 + \phi x_i) + \alpha_i ## for ##\alpha_i## to express that liklihood in terms of ##Y_i##.
 
  • #6
Ray Vickson
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I use ##L(Y_1,...Y_n;μ,φ) =Y_1*Y_2*...*Y_n##

No: you do not multiply the random variables together; if anything, you multiply their probability distributions.

So, to return to my question in #3: what is (a formula for) the probability density function ##f_i(y)## of the random variable ##Y_i?##. Until you can answer that question you will get absolutely nowhere with this problem!
 
  • #7
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0
No: you do not multiply the random variables together; if anything, you multiply their probability distributions.

So, to return to my question in #3: what is (a formula for) the probability density function ##f_i(y)## of the random variable ##Y_i?##. Until you can answer that question you will get absolutely nowhere with this problem!
Can I get the prob. distribution from the formula of ##Y_i##.
 
  • #8
Ray Vickson
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Can I get the prob. distribution from the formula of ##Y_i##.

Yes, that is exactly what you need to do.
 

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