the MLE question. what is the MLE if the variance is heteroskedasticity

#1
The normal distribution of MLE is start {x1,,,,,xn} random sample from N(u,σ^2)

This family of distributions has two parameters: θ = (μ, σ), so we maximize the likelihood, , over both parameters simultaneously using logarithm...

I know this. but what if the normal distribution's {x1,,,,,xn} random sample from N(u,σi^2)

I mean the variance is not homoscedasticity. now the variance is heteroskedasticity...

I atteched the image. My questions is that What is the MLE if variance change σ^2 to σi^2.
σ^2 - homoscedasticity,
σi^2- variance is heteroskedasticity.
So

f(x/u, σi^2) = 1/sqrt(2pi)σi exp((x-u)^2/2σi^2
I don't know.... How can I solved...
 
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Dason

Ambassador to the humans
#2
Your notation and phrasing make it difficult to understand what you're asking. Are you saying each observation has a different unknown variance?
 
#3
I atteched the image. My questions that What is the MLE if the variance is change σ^2 to σi^2.
σ^2 - homoscedasticity,
σi^2- variance is heteroskedasticity.
So f(x/u, σi^2) = 1/sqrt(2pi)σi exp((x-u)^2/2σi^2
 

Dason

Ambassador to the humans
#4
Your terminology doesn't make sense. Once again I am asking are you saying that every observation [/math]x_i[/math] has their own variance term [math]\sigma_i^2[/math]? Heteroskedasticity means that the variance isn't constant but typically if we model that we either have repeated observations (multiple x's take on the same value) or we model the variance as a function of the mean or something...

If you want to estimate the overall mean and estimate a completely different variance for each observation... then you're out of luck because you have one more parameter than you do observations and there isn't a unique solution.
 

Dason

Ambassador to the humans
#5
Your terminology doesn't make sense. Once again I am asking are you saying that every observation [/math]x_i[/math] has their own variance term [math]\sigma_i^2[/math]? Heteroskedasticity means that the variance isn't constant but typically if we model that we either have repeated observations (multiple x's take on the same value) or we model the variance as a function of the mean or something...

If you want to estimate the overall mean and estimate a completely different variance for each observation... then you're out of luck because you have one more parameter than you do observations and there isn't a unique solution.