The exitence of the MLE for the Poisson likelihood is determined by how and if the data set contains zero-counts. If there are no zero-counts, the MLE exists.
If there are zero-counts, the existence of the MLE is characterized as follows.
The \( n \) predictors \( \tilde{X}_1, \ldots, \tilde{X}_n \) span a convex cone in \( \mathbb{R}^{p} \) with
vertex in 0. The MLE exists if and only if
\[ \tilde{t} = \sum_{i=1}^n Y_i \tilde{X}_i \]
is in the interior of the cone, where the \( Y_i \)'s are the responses. If there is
an intercept in the model (which there usually is) and \( \tilde{X}_n = (1, X_n)^T \in \mathbb{R}^{p+1} \)
for \( X_n \in \mathbb{R}^p \) it can be shown that \( \tilde{t} \) is in the interior of
the cone if and only if
\[ t = \frac{1}{\sum_{i=1}^n Y_i} \sum_{i=1}^n Y_i X_i \]
is in the interior of the convex hull of \( X_1, \ldots, X_n \) (the smallest
convex set spanned by the predictors). This is easy to visualize for \( p = 1, 2 \).
We generate a small toy example with \( p = 2 \) and \( n = 4 \). First we construct the predictors and visualize their convex hull.
library(ggplot2)
X <- data.frame(x1 = c(-2, -1, 2, 0), x2 = c(1, -1, 0, 2))
p <- qplot(x1, x2, data = X, geom = "polygon", alpha = I(0.5)) + geom_point(size = 5,
alpha = 1)
p
Then we consider an example where the average \( t \) ends up in the interior of the convex hull,
and we fit the Poisson model using glm.
y <- c(1, 2, 1, 0)
Xy <- cbind(y, X)
t <- c(y %*% X$x1, y %*% X$x2)/sum(y)
p + geom_point(aes(t[1], t[2]), size = 5, color = "blue")
testGlm <- glm(y ~ x1 + x2, family = poisson, data = Xy)
summary(testGlm)
##
## Call:
## glm(formula = y ~ x1 + x2, family = poisson, data = Xy)
##
## Deviance Residuals:
## 1 2 3 4
## 0.358 -0.162 0.213 -0.745
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.0267 0.5461 0.05 0.96
## x1 -0.1237 0.3742 -0.33 0.74
## x2 -0.6550 0.5157 -1.27 0.20
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 2.77259 on 3 degrees of freedom
## Residual deviance: 0.75402 on 1 degrees of freedom
## AIC: 13.37
##
## Number of Fisher Scoring iterations: 5
Next we consider an example where the average \( t \) ends up on the boundary of the convex hull. We can fit
a Poisson model using glm, and we won't get any warnings that the MLE does not exist.
y <- c(1, 2, 0, 0)
Xy <- cbind(y, X)
t <- c(y %*% X$x1, y %*% X$x2)/sum(y)
p + geom_point(aes(t[1], t[2]), size = 5, color = "blue")
testGlm <- glm(y ~ x1 + x2, family = poisson, data = Xy)
summary(testGlm)
##
## Call:
## glm(formula = y ~ x1 + x2, family = poisson, data = Xy)
##
## Deviance Residuals:
## 1 2 3 4
## 0.00e+00 0.00e+00 -3.52e-07 -2.01e-05
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -12.84 25534.58 0 1
## x1 -8.79 17023.05 0 1
## x2 -4.74 8511.53 0 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 4.4987e+00 on 3 degrees of freedom
## Residual deviance: 4.0610e-10 on 1 degrees of freedom
## AIC: 10.61
##
## Number of Fisher Scoring iterations: 21
There are several signs in the summary output that the MLE does not exist. First, the residual deviance is almost driven to 0, which is a warning sign. Second, the estimated standard errors are huge. Moreover, the 21 iterations is actually a lot. Typically, the IWLS algorithm will converge in very few iterations.