## 3.6 Exercises

This is the list of evaluable exercises for Chapter 3. The number of stars represents an estimate of their difficulty: easy (\(\star\)), medium (\(\star\star\)), and hard (\(\star\star\star\)).

**Exercise 3.6**(theoretical, \(\star\)) Show that the local polynomial estimator yields the Nadaraya–Watson when \(p=0\). Use (3.18) to obtain (3.12).

**Exercise 3.7**(theoretical, \(\star\star\)) Obtain the optimization problem for the local Poisson regression (for the first degree) and the local binomial regression (of first degree also).

**Exercise 3.8**(theoretical, \(\star\star\)) Show that the Nadaraya–Watson is unbiased (in conditional expectation with respect to \(X_1,\ldots,X_n\)) when the regression function is constant: \(m(x)=c\), \(c\in\mathbb{R}\). Show the same for the local linear estimator for a linear regression function \(m(x)=a+bx\), \(a,b\in\mathbb{R}\).

*Hint*: use (3.19).

**Exercise 3.9**(theoretical, \(\star\star\star\)) Obtain the weight expressions (3.19) of the local linear estimator.

*Hint:*use the matrix inversion formula for \(2\times2\) matrices.

**Exercise 3.10**(theoretical, \(\star\star\star\)) Prove the two implications of Proposition 3.1 for the Nadaraya–Watson estimator (\(p=0\)).

**Exercise 3.11**(practical, \(\star\star\), Example 4.6 in Wasserman (2006)) The dataset at http://www.stat.cmu.edu/~larry/all-of-nonpar/=data/bpd.dat (alternative link) contains information about the presence of bronchopulmonary dysplasia (binary response) and the birth weight in grams (predictor) of 223 babies. Use the function

`locfit`

of the `locfit`

library with the argument `family = "binomial"`

and plot its output. Explore and comment on the resulting estimates, providing insights about the data.
**Exercise 3.12**(practical, \(\star\star\)) The

`ChickWeight`

dataset in R contains 578 observations of `weight`

and `Times`

of chicks. Fit a local binomial or local Poisson regression of `weight`

on `Times`

. Use the function `locfit`

of the `locfit`

library with the argument `family = "binomial"`

or `family = "poisson"`

and explore the bandwidth effect. Explore and comment on the resulting estimates. What is the estimated expected time of a chick that weights 200 grams?
**Exercise 3.13**(practical, \(\star\star\star\)) Implement your own version of the local linear estimator. The function must take a sample

`X`

, a sample `Y`

, the points `x`

at which the estimate should be obtained, the bandwidth `h`

and the kernel `K`

. Test its correct behavior by estimating an artificial dataset that follows a linear model.
**Exercise 3.14 **(practical, \(\star\star\star\)) Implement your own version of the local likelihood estimator of first degree for exponential response. The function must take a sample `X`

, a sample `Y`

, the points `x`

at which the estimate should be obtained, the bandwidth `h`

and the kernel `K`

. Test its correct behavior by estimating an artificial dataset that follows a generalized linear model with exponential response, this is,

\[\begin{align*} Y|X=x \sim \mathrm{Exp}(\lambda(x)),\quad \lambda(x)=e^{\beta_0+\beta_1x}, \end{align*}\]

using a cross-validated bandwidth. *Hint*: use `optim`

or `nlm`

for optimizing a function in R.

### References

Wasserman, L. 2006. *All of Nonparametric Statistics*. Springer Texts in Statistics. New York: Springer. https://doi.org/10.1007/0-387-30623-4.