When method of moments returns the impossible

Question

You fit a Binomial(n, p) by method of moments, using \hat p = 1 - \dfrac{	ext{sample variance}}{	ext{sample mean}}. Your data give a sample mean of 5 and a sample variance of 6. Compute \hat p, and explain what the result is telling you.

Accepted Answer

Plug in. \hat p = 1 - \frac{6}{5} = 1 - 1.2 = -0.2. A probability of -0.2 is impossible: the estimate has fallen outside the allowed range [0, 1]. Why it happened. A genuine binomial always has variance smaller than its mean, because \operatorname{Var}(X) = np(1-p) is the mean np multiplied by (1 - p) < 1. Your data have variance larger than the mean (they are overdispersed), which no binomial can produce. The method of moments, asked to match an impossible pair of moments, dutifully returns an out-of-bounds answer. What to do about it. The nonsensical estimate is a useful alarm, not a failur…

When method of moments returns the impossible

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