# Profile likelihood

I am considering a normal distribution with mean $\beta_1 + \beta_2\exp(-\phi x)$ and variance $\sigma^2$, i.e. $y \sim N(\beta_1 + \beta_2\exp(-\phi x), \sigma^2)$.

My aim is to calculate the profile log likelihood $L_\ast(\sigma)$ for $\sigma$.

I have calculated the log likelihood to be of the form:

$-\frac{n}{2}\log(2 \pi) - n \log(\sigma) - \frac{1}{2 \sigma^2} \sum\limits_{i=1}^n \{(y_i - \beta_1 - \beta_2\exp(-\phi x))^2\}$

where $n$ is the number of data points

I am trying to show that $L_\ast(\sigma)$ is of the form $-n \log(\sigma) - \frac{n \hat{\sigma}^2}{2 \sigma^2}$ and then find an expression for $\hat{\sigma}^2$.

I know that the idea of profile likelihood here is to fix $\sigma$ and maximise with respect to the other parameters, i.e. $\beta_1, \beta_2$ and $\phi$. However I have been suggested not to do this by differentiating the likelihood function with respect to these parameters.

I was wondering if anyone had ideas of the best way to go about doing this?

• You have a sample size $n$, where $y_i$ corresponds to regressor $x_i$, or all you $x$'s are really equal, as notation suggests? In that case, the model will not be identifiable! – kjetil b halvorsen May 15 '15 at 14:27
• Why do you want this? In this model, the ml estimators of the regression parameters$\beta_1, \beta_2, \phi$ do not depend on $\sigma^2$, so the profile likelihood finction for $\sigma^2$ is only a constant, so profile likelihood reduces to the normal likelihood theory. What do you want to do, really? A profile likelihood for one of the regression parameters will be useful, but not for the variance. Profile likelihood here is usually used for eliminating $\sigma^2$, not for eliminating the regression parameters. – kjetil b halvorsen May 15 '15 at 15:28

Why do you want this? In this model, the maximum likelihood estimators of the regression parameters $\beta_1, \beta_2, \phi$ do not depend on $\sigma^2$, so the profile likelihood function for $\sigma^2$ is only a constant, so profile likelihood reduces to the normal likelihood theory. What do you want to do, really? A profile likelihood for one of the regression parameters will be useful, but not for the variance. Profile likelihood here is usually used for eliminating $\sigma^2$, not for eliminating the regression parameters.