Log-normal Distribution - Maximum Likelihood Estimation of Parameters

Maximum Likelihood Estimation of Parameters

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that

where by ƒL we denote the probability density function of the log-normal distribution and by ƒN that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

\begin{align} \ell_L (\mu,\sigma | x_1, x_2, \dots, x_n) & {} = - \sum _k \ln x_k + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n) \\ & {} = \operatorname {constant} + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n). \end{align}

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ℓL and ℓN, reach their maximum with the same μ and σ. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that

\widehat \mu = \frac {\sum_k \ln x_k} n, \widehat \sigma^2 = \frac {\sum_k \left( \ln x_k - \widehat \mu \right)^2} {n}.

Read more about this topic:  Log-normal Distribution

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