Solution Found!
The data of this exercise were gathered as part of a study
Chapter , Problem 46(choose chapter or problem)
The data of this exercise were gathered as part of a study to estimate the population size of the bowhead whale (Raftery and Zeh 1993). The statistical procedures for estimating the population size along with an assessment of the variability of the estimate were quite involved, and this problem deals with only one aspect of the problem - a study of the distribution of whale swimming speeds. Pairs of sightings and corresponding locations that could be reliably attributed to the same whale were collected, thus providing an estimate of velocity for each whale. The velocities, \(v_{1}, v_{2}, \ldots, v_{210}\) (km/h), were converted into times \(t_{1}, t_{2} \ldots . t_{210}\) to swim \(1 \mathrm{~km}-t_{i}=1 / v_{i}\). The distribution of the ti was then fit by a gamma distribution. The times are contained in the file whales.
a. Make a histogram of the 210 values of \(t_i\). Does it appear that a gamma distribution would be a plausible model to fit?
b. Fit the parameters of the gamma distribution by the method of moments.
c. Fit the parameters of the gamma distribution by maximum likelihood. How do these values compare to those found before?
d. Plot the two gamma densities on top of the histogram. Do the fits look reasonable?
e. Estimate the sampling distributions and the standard errors of the parameters fit by the method of moments by using the bootstrap.
f. Estimate the sampling distributions and the standard errors of the parameters fit by maximum likelihood by using the bootstrap. How do they compare to the results found previously?
g. Find approximate confidence intervals for the parameters estimated by maximum likelihood.
Questions & Answers
QUESTION:
The data of this exercise were gathered as part of a study to estimate the population size of the bowhead whale (Raftery and Zeh 1993). The statistical procedures for estimating the population size along with an assessment of the variability of the estimate were quite involved, and this problem deals with only one aspect of the problem - a study of the distribution of whale swimming speeds. Pairs of sightings and corresponding locations that could be reliably attributed to the same whale were collected, thus providing an estimate of velocity for each whale. The velocities, \(v_{1}, v_{2}, \ldots, v_{210}\) (km/h), were converted into times \(t_{1}, t_{2} \ldots . t_{210}\) to swim \(1 \mathrm{~km}-t_{i}=1 / v_{i}\). The distribution of the ti was then fit by a gamma distribution. The times are contained in the file whales.
a. Make a histogram of the 210 values of \(t_i\). Does it appear that a gamma distribution would be a plausible model to fit?
b. Fit the parameters of the gamma distribution by the method of moments.
c. Fit the parameters of the gamma distribution by maximum likelihood. How do these values compare to those found before?
d. Plot the two gamma densities on top of the histogram. Do the fits look reasonable?
e. Estimate the sampling distributions and the standard errors of the parameters fit by the method of moments by using the bootstrap.
f. Estimate the sampling distributions and the standard errors of the parameters fit by maximum likelihood by using the bootstrap. How do they compare to the results found previously?
g. Find approximate confidence intervals for the parameters estimated by maximum likelihood.
ANSWER:Step 1 of 8
a)
We decided to use 9 equally spaced intervals in drawing the histogram.
The length of each interval is the difference between the maximum and minimum value, divided by the number of intervals (and then rounded off to one decimal place less than the data are).