In this lecture, we explain how to derive the maximum likelihood estimator (MLE) of the parameter of a Poisson distribution.
Before reading this lecture, you might want to revise the pages on:
We observe independent draws from a Poisson distribution.
In other words, there are independent Poisson random variablesand we observe their realizations
The probability mass function of a single draw iswhere:
is the parameter of interest (for which we want to derive the MLE);
the support of the distribution is the set of non-negative integer numbers:
is the factorial of .
The likelihood function is
The observations are independent. As a consequence, the likelihood function is equal to the product of their probability mass functions:Furthermore, the observed values necessarily belong to the support . So, we have
The log-likelihood function is
By taking the natural logarithm of the likelihood function derived above, we get the log-likelihood:
The maximum likelihood estimator of is
The MLE is the solution of the following maximization problem The first order condition for a maximum is The first derivative of the log-likelihood with respect to the parameter isImpose that the first derivative be equal to zero, and get
Therefore, the estimator is just the sample mean of the observations in the sample.
This makes intuitive sense because the expected value of a Poisson random variable is equal to its parameter , and the sample mean is an unbiased estimator of the expected value.
The estimator is asymptotically normal with asymptotic mean equal to and asymptotic variance equal to
The score isThe Hessian isThe information equality implies thatwhere we have used the fact that the expected value of a Poisson random variable with parameter is equal to . Finally, the asymptotic variance is
Thus, the distribution of the maximum likelihood estimator can be approximated by a normal distribution with mean and variance .
On StatLect you can find detailed derivations of MLEs for numerous other distributions and statistical models.
Type | Solution | |
---|---|---|
Exponential distribution | Univariate distribution | Analytical |
Normal distribution | Univariate distribution | Analytical |
T distribution | Univariate distribution | Numerical |
Multivariate normal distribution | Multivariate distribution | Analytical |
Normal linear regression model | Regression model | Analytical |
Logistic classification model | Classification model | Numerical |
Probit classification model | Classification model | Numerical |
Gaussian mixture | Mixture of distributions | Numerical (EM) |
Please cite as:
Taboga, Marco (2021). "Poisson distribution - Maximum Likelihood Estimation", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/Poisson-distribution-maximum-likelihood.
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