HW7 - homework PDF

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HW #7 due Friday 5/23/14, In Lecture. This HW covers through the bootstrapping material in Notes_7.ppt (up through slide #35). Read Chapter 13 on nonlinear regression, pp. 27—33 and 564—567 on MLE, and pp. 458—464 and 529—536 for bootstrapping. You can just skim over Section 13.2, which discusses numerical optimization algorithms for fitting nonlinear regression models. For now, you can skip Section 13.6, which discusses neural network models. We will cover Section 13.6 in the next HW assignment. 1) In class, we showed that the MLE of the mean of a random sample {y1, y2, . . ., yn} from an N(µ, σ 2) population is exactly the sample average (i.e., µˆ = y ). Show that the MLE for the standard deviation is:

σˆ =

1 n 2 ∑ (y − y ) n i =1 i

Hint: Find the partial derivative of the likelihood function with respect to σ, then set this equal to zero and solve for σ. The result will be a function of the unknown µ, but you can plug in the MLE of µ from class. Hence, we are really finding the joint MLEs of µ and σ. 2) Do Problem 13.10 from the text. For part (b), fit the nonlinear regression model two ways: (i) using Excel's Solver tool and (ii) using R's nlm() function. 3) Refer to the data from Problem 13.10. In this problem you will use R to calculate crude bootstrap confidence intervals for the estimated parameters γ0 and γ1. For a conceptual overview of bootstrapping using Excel (which is not a very practical way to implement bootstrapping), refer to Lab 5 and the Lab5_Data.xls and also the inclass example Learning_Curve_Nonlinear_Regression.xls. You can use the boot() command in R (requires the boot package to be loaded with the library(boot) command) with at least 20,000 bootstrap replicates. If you use any built-in R functions to calculate the following, make sure you know what they are doing and how you would calculate the same quantities yourself. (a) Calculate and plot bootstrapped histograms of ˆγ 0 and ˆγ 1, and calculate the corresponding bootstrapped standard errors. Interpret the standard errors as they relate to the uncertainty in the estimates from Problem 13.10. (b) Calculate “crude” two-sided 95% CIs on γ0 and γ1 using the normal approximation to their bootstrapped distributions. (c) Calculate the reflected two-sided 95% CIs on γ0 and γ1 (this corresponds to the type = “basic” option of the boot.ci() function). 4) Include your R code in your HW assignment hard copy. When you turn in your homework assignment, also post your Excel file for Problem 2 on Blackboard. 1...