The log-likelihood function for a sample {x1, …, xn} from a lognormal distribution with parameters μ and σ is
The log-likelihood function for a normal distribution is
Thus, the log-likelihood function for a sample {x1, …, xn} from a lognormal distribution is equal to the log-likelihood function from {ln x1, …, ln xn} minus the constant term ∑lnxi. Since the constant term doesn’t affect which parameter values produce the maximum value of LL, we conclude that the maximum is achieved for the same values of μ and σ on the sample {ln x1, …, ln xn} taken from a normal distribution, namely
A less biased value of σ2 is obtained by replacing n with n–1. Note too that the above values are identical to those obtained by the method of moments.
Examples Workbook
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References
Genos, B. F. (2009) Parameter estimation for the Lognormal distribution
https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=2927&context=etd
Wikipedia (2020) Log-normal distribution
https://en.wikipedia.org/wiki/Log-normal_distribution
the transformation y = exp(x) where x~N(m,s) has jacobian determinant 1/y, which does not depend on parameters and therefore is just a constant in the loglikelihood. In other words: to estimate m and s from y1…yn~lognormal(m,s):
1. take x_i = log(y_i) for all i =1,…,n
2. fit the normal model on all x_i
Hello, how can I estimate suning solver the parameters of Log normal and poisson distribution?
Hello,
I show how this is done for the Weibull distribution at
https://www.real-statistics.com/distribution-fitting/distribution-fitting-via-maximum-likelihood/fitting-weibull-parameters-mle/
The approach is similar for other distributions. For the Log-Normal distribution, you can find the formula for the Log-Likelihood function at
https://www.real-statistics.com/distribution-fitting/distribution-fitting-via-maximum-likelihood/fitting-lognormal-distribution-via-mle/
Charles