Hi everyone!
I'm trying to understand the profile log-likelihood, with some trouble. To explain my trouble, I will take as an example a theoretical exercise about the profile log-likelihood of the gaussian distribution:
So in the point a, I've to compute the profile log-likelihood function in the case of n iid normal random variables. So my idea about the procedure is the following:
1)Consider the parameters of the function, select the parameter in which you are interested (in the exercise in this post, the mean of a gaussian, [MATH]μ^2[/MATH])
2)Compute the maximum likelihood estimation of the other parameters (in this case [MATH]σ^2[/MATH])
3)Return to the likelihood function, replace [MATH]σ^2[/MATH] with the result obtained putting the derivative of the function with respect to [MATH]σ^2[/MATH] and maximizing it, that's the following:
[MATH]\frac{1}{n}*{\sum_{i=1}^{n}(x_{i}-\mu)^2}[/MATH]
So, in the case of the exercise, I should take the likelihood function of the gaussian [MATH](2\pi\sigma^2)^{-\frac{n}{2}}e^{-\frac{\sum_{i=1}^{n}(x_{i}-\mu)^2}{2\sigma^2}}[/MATH] and replacing sigma with [MATH]\frac{1}{n}*{\sum_{i=1}^{n}(x_{i}-\mu)^2}[/MATH] and replacing [MATH]σ^2[/MATH] with [MATH]\frac{1}{n}*{\sum_{i=1}^{n}(x_{i}-\mu)^2}[/MATH], and finally compute the logarithmic function of the result.
Is it correct? I'm not sure of this because my results are different from the ones of my classmates.
I'm trying to understand the profile log-likelihood, with some trouble. To explain my trouble, I will take as an example a theoretical exercise about the profile log-likelihood of the gaussian distribution:
So in the point a, I've to compute the profile log-likelihood function in the case of n iid normal random variables. So my idea about the procedure is the following:
1)Consider the parameters of the function, select the parameter in which you are interested (in the exercise in this post, the mean of a gaussian, [MATH]μ^2[/MATH])
2)Compute the maximum likelihood estimation of the other parameters (in this case [MATH]σ^2[/MATH])
3)Return to the likelihood function, replace [MATH]σ^2[/MATH] with the result obtained putting the derivative of the function with respect to [MATH]σ^2[/MATH] and maximizing it, that's the following:
[MATH]\frac{1}{n}*{\sum_{i=1}^{n}(x_{i}-\mu)^2}[/MATH]
So, in the case of the exercise, I should take the likelihood function of the gaussian [MATH](2\pi\sigma^2)^{-\frac{n}{2}}e^{-\frac{\sum_{i=1}^{n}(x_{i}-\mu)^2}{2\sigma^2}}[/MATH] and replacing sigma with [MATH]\frac{1}{n}*{\sum_{i=1}^{n}(x_{i}-\mu)^2}[/MATH] and replacing [MATH]σ^2[/MATH] with [MATH]\frac{1}{n}*{\sum_{i=1}^{n}(x_{i}-\mu)^2}[/MATH], and finally compute the logarithmic function of the result.
Is it correct? I'm not sure of this because my results are different from the ones of my classmates.