Maximum Likelihood Estimator for Log Normal Distribution
If $X_{1}, X_{2},......X_{n} ~ LogN(\mu, \sigma^2) \tag{1}$ The pdf of log normal distribution is given by `f(x; \mu, \sigma^2) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp \left( -\frac{1}{2} \left( \frac{\log(x) - \mu}{\sigma} \right)^2 \right), \quad x > 0 \tag{2}` The likelihood function of log Normal pdf is given by `L(\mu, \sigma^2) = \prod_{i=1}^{n} f_X (x_i) \tag{3}` `L(\mu, \sigma^2) = \prod_{i=1}^{n} \frac{1}{x_i \sigma \sqrt{2 \pi}} \exp \left[ -\frac{1}{2} \left( \frac{\log(x_i) - \mu}{\sigma} \right)^2 \right] \tag{4}` `L(\mu, \sigma^2) = \frac{1}{(\sigma \sqrt{2 \pi})^n \prod_{i=1}^{n} x_i} \exp \left( -\frac{1}{2} \sum_{i=1}^{n} \left( \frac{\log(x_i) - \mu}{\sigma} \right)^2 \right) \tag{5}` Now taking the log both sides `\log L(\mu, \sigma^2) = -n \log(\sigma \sqrt{2 \pi}) - \sum_{i=1}^{n} \log(x_i) - \frac{1}{2} \sum_{i=1}^{n} \left( \frac{\log(x_i) - \mu}{\sigma} \right)^2 \tag{6}` Now...
