Solved: The article “Feeding Ecology of the Red-Eyed Vireo

Chapter 14, Problem 14E

(choose chapter or problem)

The article “Feeding Ecology of the Red-Eyed Vireo and Associated Foliage-Gleaning Birds” (Ecological Monographs, 1971: 129–152) presents the accompanying data on the variable X = the number of hops before the first flight and preceded by a flight. The author then proposed and fit a geometric probability distribution [\(\left[p(x)=P(X=x)=p^{x-1} \cdot q \text { for } x=1,2, \ldots,\right.\) where q=1-p] to the data. The total sample size was n = 130.

a. The likelihood is \(\left(p^{x_{1}-1} \cdot q\right) \cdot \cdots \cdot\left(p^{x_{n}-1} \cdot q\right)\) = \(p^{\Sigma x_{i}-n} \cdot q^{n}\). Show that the mle of p is given by \(\hat{p}\) = \(\left(\sum x_{i}-n\right) / \sum x_{i}\) and compute \(\hat{p}\) for the given data.

b. Estimate the expected cell counts using pˆ of part (a) [expected cell counts\(=n \cdot(\hat{p})^{x-1} \cdot \hat{q} \text { for } x=1,2, \ldots\)], and test the fit of the model using a X2 test by combining the counts for x = 7, 8,…, and 12 into one cell (x \(\geq\) 7).