Consider the lognormal distribution with the density...

Consider the lognormal distribution with the density function given in Section $6.9 .$ Suppose we have a sample $x \mid, x_{2}, \ldots x_{n}$ from a lognormal distribution.
(a) Write out the likelihood function.
(b) Develop the maximum likelihood estimators of $p$ and $a^{2}$.

Key Concepts

Lognormal Distribution

The lognormal distribution is a probability distribution of a random variable whose logarithm is normally distributed. This distribution is used to model phenomena where the data points are positively skewed and multiplicative in nature. Key parameters typically include the mean and variance of the underlying normal distribution after the logarithmic transformation, which then translate into the scale and shape of the lognormal distribution.

Likelihood Function

The likelihood function is a function of the parameters of a statistical model, given observed data. It is constructed by considering the joint probability (or probability density) of the observed data under the model, treating the observed sample as fixed and the parameters as variable. This function forms the basis for many estimation procedures, including maximum likelihood estimation, by quantifying how plausible different parameter values are given the data.

Maximum Likelihood Estimation (MLE)

Maximum likelihood estimation is a method of estimating the parameters of a statistical model by maximizing the likelihood function. The MLE approach selects the parameter values that make the observed data most probable, effectively 'fitting' the model to the data. In practice, this usually involves taking the logarithm of the likelihood function to simplify differentiation, computing the derivatives with respect to the parameters, and then setting them to zero to solve for the estimates.

Parameter Transformation

For distributions like the lognormal, it is often useful to apply a transformation to the data to simplify the estimation process. By taking the natural logarithm of a lognormal random variable, the resulting variable follows a normal distribution, which has well-known properties and simpler estimation procedures. This transformation facilitates the derivation of estimators and the analysis of their properties in a more familiar normal framework.

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