Class 1: Poisson model

1 Poisson distribution

We will use the Poisson distribution to model count data and (re)-introduce key concepts as probability density function, likelihood, maximum likelihood estimator (MLE), and estimator properties, such as unbiasedness, sufficiency, consistency and efficiency (Cramér-Rao lower bound).

1.1 Probability density function

A random variable \(y \in \{0,1,2,\ldots\}\) follows a Poisson distribution with rate \(\lambda>0\), if \[ Pr(y|\lambda) = \frac{\lambda^ye^{-\lambda}}{y!}, \ \ \ \mbox{for} \ y=0,1,2,\ldots. \] Below we illustrate the distributions of \(y\) following a Poisson(2):

• \(Pr(y=2|\lambda=2)=0.271\)

• \(Pr(y \leq 3|\lambda=2)=0.857\).

cbind(0:3,dpois(0:3,2))

##      [,1]      [,2]
## [1,]    0 0.1353353
## [2,]    1 0.2706706
## [3,]    2 0.2706706
## [4,]    3 0.1804470

sum(dpois(0:3,2))

## [1] 0.8571235

1.2 A few different rates \(\lambda\)

plot(0:10,dpois(0:10,1),type="h",xlab="Count (y)",ylab="Probability",lwd=2)
lines(0:10+0.1,dpois(0:10,2),col=2,lwd=2,type="h")
lines(0:10+0.2,dpois(0:10,3),col=3,lwd=2,type="h")
legend("topright",legend=c("Y~Poisson(1)","Y~Poisson(2)","Y~Poisson(3)"),col=1:3,lwd=2,bty="n")

2 Example: Coal mining disaster

Counts of coal mining disaster in Great Britain by year from 1851 to 1891

y = c(4,5,4,1,0,4,3,4,0,6,3,3,4,0,2,6,3,3,5,4,5,3,1,4,4,1,5,5,3,4,2,5,2,2,3,4,2,1,3,2,2)

c(mean(y),var(y))

## [1] 3.097561 2.540244

n = length(y)
freq = rep(0,7)
for (i in 1:7)
  freq[i] = sum(y==(i-1))

tab = cbind(0:6,freq,round(freq/n,2),round(dpois(0:6,2),2),round(dpois(0:6,3),2))
colnames(tab) = c("y","Freq","Rel.Freq","Poi(2)","Poi(3)")
tab

##      y Freq Rel.Freq Poi(2) Poi(3)
## [1,] 0    3     0.07   0.14   0.05
## [2,] 1    4     0.10   0.27   0.15
## [3,] 2    7     0.17   0.27   0.22
## [4,] 3    9     0.22   0.18   0.22
## [5,] 4   10     0.24   0.09   0.17
## [6,] 5    6     0.15   0.04   0.10
## [7,] 6    2     0.05   0.01   0.05

plot(0:6,freq/n,type="h",xlab="Counts",ylab="Relative frequency",lwd=4,ylim=c(0,0.3))
title("Coal mining disaster")
lines(0:6+0.05,dpois(0:6,2),col=2,lwd=2,type="h")
lines(0:6+0.1,dpois(0:6,3),col=3,lwd=2,type="h")
lines(0:6+0.15,dpois(0:6,mean(y)),col=4,lwd=2,type="h")
legend("topright",legend=c("Observed frequency","Y~Poisson(2)","Y~Poisson(3)","Y~Poisson(3.098)"),
       col=1:5,lwd=2,bty="n")