Testing Poisson Models Using Chi-Square Goodness-of-Fit

Apply the chi-square goodness-of-fit test to count data to assess whether the underlying distribution is Poisson. Covers computing expected cell counts from the Poisson PMF, computing the chi-square statistic, determining the correct degrees of freedom (with or without estimating $\lambda$ ), combining cells with small expected counts, and reaching a test decision.

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Tutorial

Setting Up the Poisson Goodness-of-Fit Test

We have a sample of $n$ counts $X_1, X_2, \ldots, X_n$ and want to test whether they come from a Poisson distribution:

H_0: X \sim \text{Poisson}(\lambda) \qquad \text{vs.} \qquad H_1: X \text{ is not Poisson}(\lambda).

Group the data into $K$ cells by the value of $X$ : one cell for each of $X = 0, 1, \ldots, K-2$ and a tail cell for $X \ge K-1.$ Under $H_0,$ the probability of cell $k$ is the Poisson PMF

p_k = P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!},

and the expected count in cell $k$ is

E_k = n \cdot p_k.

The tail probability is computed as the complement:

P(X \ge K-1) = 1 - \sum_{k=0}^{K-2} p_k.

For example, with $n = 100$ and $\lambda = 1$ (so $e^{-1} \approx 0.3679$ ),

p_0 \approx 0.3679,\quad p_1 \approx 0.3679,\quad p_2 \approx 0.1839,\quad p_3 \approx 0.0613,\quad p_{\ge 4} \approx 0.0190,

so the expected counts are

E_0 \approx 36.79,\quad E_1 \approx 36.79,\quad E_2 \approx 18.39,\quad E_3 \approx 6.13,\quad E_{\ge 4} \approx 1.90.