Editing Geometric distribution (section)

=== Maximum likelihood estimation ===
The [[maximum likelihood estimator]] of <math>p</math> is the value that maximizes the [[likelihood function]] given a sample.<ref name=":5" />{{Rp|page=308}} By finding the [[Zero of a function|zero]] of the [[derivative]] of the [[Log-likelihood|log-likelihood function]] when the distribution is defined over <math>\mathbb{N}</math>, the maximum likelihood estimator can be found to be <math>\hat{p} = \frac{1}{\bar{x}}</math>, where <math>\bar{x}</math> is the sample mean.<ref>{{Cite web |last=Siegrist |first=Kyle |date=2020-05-05 |title=7.3: Maximum Likelihood |url=https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.03%3A_Maximum_Likelihood |access-date=2024-06-20 |website=Statistics LibreTexts |language=en}}</ref> If the domain is <math>\mathbb{N}_0</math>, then the estimator shifts to <math>\hat{p} = \frac{1}{\bar{x}+1}</math>. As previously discussed in [[Geometric distribution#Method of moments|§ Method of moments]], these estimators are biased.

Regardless of the domain, the bias is equal to

: <math>
    b \equiv \operatorname{E}\bigg[\;(\hat p_\mathrm{mle} - p)\;\bigg]
        = \frac{p\,(1-p)}{n} 
  </math>

which yields the [[Maximum likelihood estimation#Higher-order properties|bias-corrected maximum likelihood estimator]],{{Cn|date=July 2024}}

: <math>
    \hat{p\,}^*_\text{mle} = \hat{p\,}_\text{mle} - \hat{b\,}
  </math>