Maximum likelihood of a statistical model

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1. Feb 7, 2017

the_dane

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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?

2. Feb 7, 2017

Stephen Tashi

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. Feb 8, 2017

Ray Vickson

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. Feb 8, 2017

the_dane

I use $L(Y_1,...Y_n;μ,φ) =Y_1*Y_2*...*Y_n$

5. Feb 8, 2017

Stephen Tashi

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. Feb 8, 2017

Ray Vickson

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. Feb 8, 2017

the_dane

Can I get the prob. distribution from the formula of $Y_i$.

8. Feb 8, 2017

Ray Vickson

Yes, that is exactly what you need to do.