Editing Exponential distribution (section)

===Parameter estimation===

The [[maximum likelihood]] estimator for λ is constructed as follows.

The [[likelihood function]] for λ, given an [[independent and identically distributed]] sample ''x'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) drawn from the variable, is:
<math display="block"> L(\lambda) = \prod_{i=1}^n\lambda\exp(-\lambda x_i) = \lambda^n\exp\left(-\lambda \sum_{i=1}^n x_i\right) = \lambda^n\exp\left(-\lambda n\overline{x}\right), </math>

where:
<math display="block">\overline{x} = \frac{1}{n}\sum_{i=1}^n x_i</math>
is the sample mean.

The derivative of the likelihood function's logarithm is:
<math display="block">
  \frac{d}{d\lambda} \ln L(\lambda) =
  \frac{d}{d\lambda} \left( n \ln\lambda - \lambda n\overline{x} \right) =
  \frac{n}{\lambda} - n\overline{x}\ \begin{cases}
    > 0, & 0 < \lambda < \frac{1}{\overline{x}}, \\[8pt]
    = 0, & \lambda = \frac{1}{\overline{x}}, \\[8pt]
    < 0, & \lambda > \frac{1}{\overline{x}}.
  \end{cases}
</math>

Consequently, the [[maximum likelihood]] estimate for the rate parameter is:
<math display="block">\widehat{\lambda}_\text{mle} = \frac{1}{\overline{x}} = \frac{n}{\sum_i x_i}</math>

This is {{em|not}} an [[unbiased estimator]] of <math>\lambda,</math> although <math>\overline{x}</math> {{em|is}} an unbiased<ref name="Dean W. Wichern-2007">{{cite book|author1=Richard Arnold Johnson|author2=Dean W. Wichern|title=Applied Multivariate Statistical Analysis|url=https://books.google.com/books?id=gFWcQgAACAAJ|access-date=10 August 2012|year=2007 |publisher=Pearson Prentice Hall|isbn=978-0-13-187715-3}}</ref> MLE<ref>''[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm NIST/SEMATECH e-Handbook of Statistical Methods]''</ref> estimator of <math>1/\lambda</math> and the distribution mean.

The bias of <math> \widehat{\lambda}_\text{mle} </math> is equal to
<math display="block">B \equiv \operatorname{E}\left[\left(\widehat{\lambda}_\text{mle} - \lambda\right)\right] = \frac{\lambda}{n - 1} </math>
which yields the [[Maximum likelihood estimation#Second-order efficiency after correction for bias|bias-corrected maximum likelihood estimator]]
<math display="block">\widehat{\lambda}^*_\text{mle} = \widehat{\lambda}_\text{mle} - B.</math>

An approximate minimizer of [[mean squared error]] (see also: [[bias–variance tradeoff]]) can be found, assuming a sample size greater than two, with a correction factor to the MLE:
<math display="block">\widehat{\lambda} = \left(\frac{n - 2}{n}\right) \left(\frac{1}{\bar{x}}\right) = \frac{n - 2}{\sum_i x_i}</math>
This is derived from the mean and variance of the [[inverse-gamma distribution]], <math display="inline">\mbox{Inv-Gamma}(n, \lambda)</math>.<ref>{{cite journal |first1=Abdulaziz |last1=Elfessi |first2=David M. |last2=Reineke |title=A Bayesian Look at Classical Estimation: The Exponential Distribution |journal=Journal of Statistics Education |volume=9 |issue=1 |year=2001 |doi=10.1080/10691898.2001.11910648|doi-access=free }}</ref>