Unknown parameter
In practical applications, we may be able to assume that the conditions for a Poisson process hold at least approximately, but the rate of events, \(\lambda\), is usually an unknown value that must be estimated from data.
We now give an example of one such data set — a rather unusual one!
Example
The following table shows the numbers of deaths by horse kicks in the Prussian army each year from 1875 to 1894, in each of 14 army corps.
Corps | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
G | 0 | 2 | 2 | 1 | 0 | 0 | 1 | 1 | 0 | 3 | 0 | 2 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
I | 0 | 0 | 0 | 2 | 0 | 3 | 0 | 2 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 2 | 0 | 3 | 1 | 0 |
II | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 1 | 1 | 0 | 0 | 2 | 1 | 1 | 0 | 0 | 2 | 0 | 0 |
III | 0 | 0 | 0 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 0 | 0 | 1 | 0 | 1 | 2 | 1 | 0 | 0 | 0 |
IV | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
V | 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
VI | 0 | 0 | 1 | 0 | 2 | 0 | 0 | 1 | 2 | 0 | 1 | 1 | 3 | 1 | 1 | 1 | 0 | 3 | 0 | 0 |
VII | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 2 | 0 | 0 | 2 | 1 | 0 | 2 | 0 |
VIII | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
IX | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | 1 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 0 | 1 | 0 | 0 |
X | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 1 | 3 | 0 | 1 | 1 |
XI | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 1 | 3 | 0 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 1 | 3 | 1 |
XIV | 1 | 1 | 2 | 1 | 1 | 3 | 0 | 4 | 0 | 1 | 0 | 3 | 2 | 1 | 0 | 2 | 1 | 1 | 0 | 0 |
XV | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 |
It is reasonable to assume that events (i.e. horse-kick deaths) occur randomly over the 20 years at a constant rate — a homogeneous Poisson process within each corps. If we make a further assumption that the rate is the same for all 14 corps, the counts in the above table will be a random sample from a Poisson distribution.
However the rate of deaths per year in each corps, \(\lambda\), is unknown and must be estimated from the data.
Maximum likelihood
Given a random sample, \(\{x_1, x_2, \dots, x_n\}\), from a \(\PoissonDistn(\lambda)\) distribution, we will use maximum likelihood to estimate \(\lambda\). The logarithm of the Poisson probability function is
\[ \log(p(x | \lambda)) \;=\; x \log(\lambda) - \lambda - \log(x!) \]so the log-likelihood function is
\[ \ell( \lambda) \;=\; \sum_{i=1}^n {x_i} \log(\lambda) - n\lambda + K \]where \(K\) is a constant that does not depend on \(\lambda\). To find the maximum likelihood estimate, we set the derivative of the log-likelihood to zero,
\[ \ell'( \lambda) \;=\; \frac {\sum {x_i}} {\lambda} - n \;=\; 0 \]so
\[ \hat{\lambda} \;=\; \frac {\sum {x_i}} n \;=\; \overline{x} \]