Exercise 5
Question 1.
Let $Y_1,\cdots,Y_n \sim f_{\theta}(y)$ with an unknown parameter $\theta$.
- Show that the sample $\alpha$-quantile $y_{(\alpha)}$ is the M-estimator corresponding to $\rho(y,\theta)=\alpha (y-\theta)_++(1-\alpha)(\theta-y)_+$.
- What is the asymptotic distribution of $y_{(\alpha)}$? Find an (asymptotic) $(1-\delta)100\%$ confidence interval for the $\alpha$-quantile $F^{-1}_\theta(\alpha)$ of the distribution $f_\theta$.
Question 2.
The file
Hornets.dat provides the results of the research on hornets' cells building. The file contains
the numbers of hornets in the
i-th box and the numbers of cells per capita,
CPC, (hornet) that were built in the
i-th box.
- Fit a linear model of CPC as a function of log(Hornets). What can you say about the adequacy of the model? Try to find an appropriate transformation of the
response and re-fit the model. Comment the results.
- Fit robust regression using several M-estimators: Hampel, Huber, Tukey's bisquare, etc. Compare the results and
compare them with the OLS fit from the previous paragraph.
Question 3.
The file
Puromycin.dat contains the data on the substrate concentration of Puromycin, x (parts per
million, ppm) and the initial rate, or "velocity", y, of the enzymatic reaction (counts/min
2) in the presence of Puromycin. The velocity is assumed to depend on the substrate concentration according to
the Michaelis-Menten equation: y=θ
1 x/(θ
2+x).
- What is the physical/mathematical meaning of the parameters θ1 and θ2?
- Using transformations of x and y transform the original nonlinear model to a linear one. Fit the corresponding linear
model. Does it seem to be adequate?
- Find the gradient matrix D for the Michaelis-Menten original nonlinear model. Fit the nonlinear model using the results of the previous paragraph
for obtaining initial values for the parameters, and check the adequacy of the nonlinear model.
- Test the hypothesis θ1=200 applying F- and t-tests, comment the results.
- Predict velocity of the enzyme reaction when the substrate concentration of Puromycin is at level 0.5. Give the corresponding confidence and prediction intervals.
Computational Notes for R users:
- To fit various robust regression models use
rlm and
lqs functions (see help for details).
- To fit nonlinear regression use the function
nls (see help for details).