Fitting Geometric Parameter via MLE

The log-likelihood function for the Geometric distribution for the sample {x1, …, xn} is

LL geometric distribution

The MLE value is achieved when

MLE fit geometric parameter

which is the same value as from the method of moments (see Method of Moments).

References

Forbes, C., Evans, M., Hastings, N., Peacock, B. (2011) Statistical distribution. Wiley
https://www.academia.edu/49056503/Statistical_distributions

Siegrist, K. (2022) Maximum-Likelihood
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.03%3A_Maximum_Likelihood

Millard, S. P. (2023) Estimate probability parameter of a geometric distribution
https://search.r-project.org/CRAN/refmans/EnvStats/html/egeom.html

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