Question
Consider the lognormal distribution with the density function given in Section $6.9 .$ Suppose we have a random sample $x_{1}, x_{2}, \ldots, x_{n}$ from a lognormal distribution.(a) Write out the likelihood function.(b) Develop the maximum likelihood estimators of $\mu$ and $\sigma^{2}$.
Step 1
The pdf of a lognormal distribution is given by: \[f(x_i;\mu,\sigma) = \frac{1}{x_i\sigma\sqrt{2\pi}}e^{-\frac{(\ln x_i - \mu)^2}{2\sigma^2}}\] So, the likelihood function is: \[L(\mu,\sigma^2;x_1,...,x_n) = \prod_{i=1}^{n} Show more…
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