# How do we find the maximum likelihood estimate of $\mu$ from $\mathcal{N}(\mu,\sigma^{2})$ through the Newton-Raphson method if $\sigma^{2}$ is known?

Here it is the problem: I am supposed to obtain the maximum likelihood estimate of the mean for some normal distribution $$\mathcal{N}(\mu,\sigma^{2})$$ where $$\sigma^{2}$$ is known (let it be $$\sigma = 1$$ for convenience). As far as I have understood, I am asked to generate some data from the corresponding normal distribution (where $$\mu$$ is also known), apply the Newton-Raphson method to the score function, which gives the sought estimate, and verify how accurate it is.

My question is: am I on the right track? That is to say, should I generate data and compare the estimate with the real value?

By the way, I am new to statistics and I am still learning R language.

Any help is appreciated. Thanks in advance!

• The MLE in this case has a well-known and simple form. It is unclear why you would use iterative methods such as Newton-Raphson to try to get it. – Reinstate Monica Apr 29 '19 at 1:01
• I know how to obtain the MLE here. I think the Professor wants us to compare the closed form to its numerical counterpart. – user1337 Apr 29 '19 at 1:32

To generate the data in R, you can use the rnorm function: rnorm(n, mean = 0, sd = 1) where n is the number of observations you would like to generate.

To find the mean and variance of a Gaussian distribution by MLE, you do not need to use Newton-Raphson. By calculating the derivatives of the log likelihood function with respect to $$\sigma$$ and $$\mu$$, you can find the mean and variance.

Gaussian distribution density: $$N(\mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(y - \mu)^2 / 2\sigma^2},$$

Log-Likelihood of the Gaussian distribution:

$$\ell = \frac{-n}{2} log(2\pi\sigma^2) - \frac{1}{2\sigma^2} \sum_i (x_i - \mu)^2 .$$

Taking the derivative with respect to $$\mu$$ will let you find the mean: $$\widehat{\mu} = \bar{x} = \sum_i \frac{x_i}{n}$$

Taking the derivative with respect to $$\sigma$$ will let you find the variance: $$\widehat{\sigma^2} = \frac{1}{n} \sum_i (x_i - \mu)^2 .$$