Geometric distribution
Maximum likelihood estimation

The maximum likelihood estimator of p {\displaystyle p} is the value that maximizes the likelihood function given a sample. By finding the zero of the derivative of the log-likelihood function when the distribution is defined over N {\displaystyle \mathbb {N} }, the maximum likelihood estimator can be found to be p ^ = 1 x ¯ {\displaystyle {\hat {p}}={\frac {1}{\bar {x}}}}, where x ¯ {\displaystyle {\bar {x}}} is the sample mean. If the domain is N 0 {\displaystyle \mathbb {N} _{0}}, then the estimator shifts to p ^ = 1 x ¯ + 1 {\displaystyle {\hat {p}}={\frac {1}{{\bar {x}}+1}}}. As previously discussed in § Method of moments, these estimators are biased.

Regardless of the domain, the bias is equal to

which yields the bias-corrected maximum likelihood estimator,
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