@@ -166,50 +166,57 @@ These estimates can be found by maximizing the likelihood function given the
166166data.
167167168168The pdf of a lognormally distributed random variable $X$ is given by:
169+169170$$
170171f(x) = \frac{1}{x}\frac{1}{\sigma \sqrt{2\pi}} exp\left(\frac{-1}{2}\left(\frac{\ln x-\mu}{\sigma}\right)\right)^2
171172$$
172173173174Since $\ln X$ is normally distributed this is the same as
175+174176$$
175177f(x) = \frac{1}{x} \phi(x)
176178$$
179+177180where $\phi$ is the pdf of $\ln X$ which is normally distibuted with mean $\mu$ and variance $\sigma ^2$.
178181179182For a sample $x = (x_1, x_2, \cdots, x_n)$ the _likelihood function_ is given by:
183+180184$$
181185\begin{aligned}
182-L(\mu, \sigma | x_i) = \prod_{i=1}^{n} f(\mu, \sigma | x_i) \\
183-L(\mu, \sigma | x_i) = \prod_{i=1}^{n} \frac{1}{x_i} \phi(\ln x_i)
186+L(\mu, \sigma | x_i) &= \prod_{i=1}^{n} f(\mu, \sigma | x_i) \\
187+&= \prod_{i=1}^{n} \frac{1}{x_i} \phi(\ln x_i)
184188\end{aligned}
185189$$
186190191+187192Taking $\log$ on both sides gives us the _log likelihood function_ which is:
193+188194$$
189195\begin{aligned}
190-l(\mu, \sigma | x_i) = -\sum_{i=1}^{n} \ln x_i + \sum_{i=1}^n \phi(\ln x_i) \\
191-l(\mu, \sigma | x_i) = -\sum_{i=1}^{n} \ln x_i - \frac{n}{2} \ln(2\pi) - \frac{n}{2} \ln \sigma^2 - \frac{1}{2\sigma^2}
192-\sum_{i=1}^n (\ln x_i - \mu)^2
196+\ell(\mu, \sigma | x_i) &= -\sum_{i=1}^{n} \ln x_i + \sum_{i=1}^n \phi(\ln x_i) \\
197+&= -\sum_{i=1}^{n} \ln x_i - \frac{n}{2} \ln(2\pi) - \frac{n}{2} \ln \sigma^2 - \frac{1}{2\sigma^2} \sum_{i=1}^n (\ln x_i - \mu)^2
193198\end{aligned}
194199$$
195200196201To find where this function is maximised we find its partial derivatives wrt $\mu$ and $\sigma ^2$ and equate them to $0$.
197202198203Let's first find the MLE of $\mu$,
204+199205$$
200206\begin{aligned}
201-\frac{\delta l}{\delta \mu} = - \frac{1}{2\sigma^2} \times 2 \sum_{i=1}^n (\ln x_i - \mu) = 0 \\
202-\Rightarrow \sum_{i=1}^n \ln x_i - n \mu = 0 \\
203-\Rightarrow \hat{\mu} = \frac{\sum_{i=1}^n \ln x_i}{n}
207+\frac{\delta l}{\delta \mu} = - \frac{1}{2\sigma^2} \times 2 \sum_{i=1}^n (\ln x_i - \mu) &= 0 \\
208+\sum_{i=1}^n (\ln x_i - n \mu) &= 0 \\
209+\hat{\mu} &= \frac{\sum_{i=1}^n \ln x_i}{n}
204210\end{aligned}
205211$$
206212207213Now let's find the MLE of $\sigma$,
214+208215$$
209216\begin{aligned}
210-\frac{\delta l}{\delta \sigma^2} = - \frac{n}{2\sigma^2} + \frac{1}{2\sigma^4} \sum_{i=1}^n (\ln x_i - \mu)^2 = 0 \\
211-\Rightarrow \frac{n}{2\sigma^2} = \frac{1}{2\sigma^4} \sum_{i=1}^n (\ln x_i - \mu)^2 \\
212-\Rightarrow \hat{\sigma} = \left( \frac{\sum_{i=1}^{n}(\ln x_i - \hat{\mu})^2}{n} \right)^{1/2}
217+\frac{\delta l}{\delta \sigma^2} = - \frac{n}{2\sigma^2} + \frac{1}{2\sigma^4} \sum_{i=1}^n (\ln x_i - \mu)^2 &= 0 \\
218+\frac{n}{2\sigma^2} &= \frac{1}{2\sigma^4} \sum_{i=1}^n (\ln x_i - \mu)^2 \\
219+\hat{\sigma} &= \left( \frac{\sum_{i=1}^{n}(\ln x_i - \hat{\mu})^2}{n} \right)^{1/2}
213220\end{aligned}
214221$$
215222@@ -266,7 +273,6 @@ tr_lognorm
266273times as large.)
267274268275269-270276## Pareto distribution
271277272278We mentioned above that using maximum likelihood estimation requires us to make