如何判断30个样本的数据集是否符合泊松分布？

Alright, let's break down how to check if your 30-point dataset follows a Poisson distribution—since you've already ruled out normality and suspect Poisson is a fit, here's a practical, step-by-step approach that includes GLM and other key checks:

Poisson distributions have specific traits, so start here to rule out obvious mismatches:

• Check data type: Poisson applies to non-negative integers (count data). If your dataset has decimals or negative values, Poisson is immediately out of the question.
• Compare mean and variance: A defining feature of Poisson is that the population mean equals the population variance. For small samples like yours, they won't be identical, but if your sample variance is drastically larger (hint: overdispersion, try negative binomial) or smaller than the sample mean, Poisson is unlikely.
• Visualize the data: Plot a bar chart (since it's discrete) or histogram. Poisson distributions are typically right-skewed, especially when the mean is small—if your plot looks nothing like that, you might want to reconsider your hypothesis.

For small samples, choose tests that work well with limited data:

• Chi-Squared Goodness-of-Fit Test: This is standard for discrete distributions, but you need to adjust for small expected counts:
  1. Group your data into bins, merging any bins where the expected count would be less than 5 (this keeps the test valid).
  2. Use your sample mean as the estimate for the Poisson parameter λ.
  3. Calculate the test statistic: χ² = Σ[(Observed - Expected)² / Expected]
  4. The degrees of freedom are (number of bins) - 1 - 1 (subtract an extra 1 because you estimated λ from the data). If the p-value is > 0.05, you can't reject the hypothesis that your data follows Poisson.
• Kolmogorov-Smirnov (KS) Test: Note that KS is designed for continuous distributions, so it's less ideal here, but you can use it as a secondary check. It tends to be conservative for discrete data, so don't rely on it alone.

You mentioned GLM—this is a great way to tie the distribution check to model fit, which is often more useful than a standalone test:

• Fit a Poisson GLM with only an intercept: If you don't have predictor variables, this model just estimates the Poisson parameter λ (the intercept exponentiated gives you λ, since Poisson GLM uses the log link: log(λ) = intercept).
• Check model diagnostics:
  • Deviance: The model's deviance should follow a chi-squared distribution with n - p degrees of freedom (here, n=30, p=1, so df=29). If the deviance is close to the degrees of freedom, and the p-value for the deviance test is > 0.05, that's strong evidence your data fits Poisson.
  • Pearson Residuals: Calculate the Pearson chi-squared statistic: Σ[(Observed - μ)² / μ] where μ is the predicted mean from the model. If this statistic is roughly equal to the degrees of freedom (29), that supports the Poisson assumption.
  • Residual plots: Plot the deviance or Pearson residuals against predicted values. If residuals are randomly scattered with no clear pattern (no upward/downward trends, no outliers pulling the plot), that's a good sign the Poisson fit is appropriate.

• With only 30 data points, formal tests might not have enough power to detect small deviations from Poisson. So combine test results with your EDA and GLM diagnostics—don't rely on a single p-value.
• If you notice overdispersion (variance >> mean), even if the tests are borderline, Poisson isn't the right fit. Switch to a negative binomial GLM instead, which accounts for extra variance.

内容的提问来源于stack exchange，提问作者CS3420