2. ESTIMATION AND NONPARAMETRIC FITS

As stated previous section, different ways to approximate nonparametric part will get the corresponding estimators of . This section will present several macros to explain how to calculate the estimates.

2.1. KERNEL REGRESSION

Let be a kernel function satisfying certain conditions, a bandwidth parameter. The weight function is defined as

Then least squares estimator of

is defined as

where

with

and

with

, and the nonparametric estimator of g(t):

The detailed discussions on asymptotic theories of the estimators and are referred to Gao, Hong and Liang (1995) and Speckman (1988).

plmk presents the estimates of the parameter and the nonparametric part g(t) by using kernel method to handle g(t).

res.hbeta: vector or scalar, the estimate of

res.hsigma: scalar, the estimate of the variance of

when it is homoscedastic

The next picture , "plmk-ex.xpl", gives an example of XploRe code to generate a sample from the PLM model, and then shows how to compute the PLM estimates by plmk. Saving the example as a file (plmk-ex.xpl) and then linking "Execute", the actual parameter estimates are shown in the second picture (XploRe_out).

An example by using plmk

The estimate result of plmk-ex.xpl

Nonparametric fit by

plmk

^{Green curve stands
for true value, blue for parametric fit and red for nonparametric
fit..}

2.2. LEAST SQUARES SPLINE

Suppose that g has m-1 absolutely continuous derivatives and m-th derivative that is square integrable and satisfies for a specified C>0. Via a Taylor expansion, the partially linear model can be rewitten as

where

. By using a quadrature rule, Rem(t) can be approximated by a sum of the form

for some set of coefficients

and points

. Let

and set

. The least squares spline estimator is to minimize

Conveniently with matrix notations, denote

with

for

and

. and

are found as the solution to the minimizing of

If the problem has an unique solution, its form is the same as

plmp. Otherwise, we may use ridge idea to modify the estimator.

plmls presents the estimates of parameter and non parametric function by fitting the nonparametric part with least squares spline.

2.3. PIECEWISE POLYNOMIAL

We assume that g are Hölder continuous smooth of order p(=m+r), that is, let r and M denote nonnegative real constants , m is nonnegative integer such that

Now we describe piecewise polynomial approximation for the function

, defined in [0, 1]. Given a positive

, devide [0, 1] in M_nintervals with equal length

. The estimator has the form of a piecewise polynomial of degree m based on

the intervals, where the

coefficients are chosen by the method of least squares on the basis of the data.

Let be the indicator function of the -th interval, and be the midpoint of the -th interval, so that or 0 according to for and or not. Let be the m-order Taylor expansion of g(t) on . Denote

Consider the piecewise polynomial approximation of g of degree m given by

Suppose we have n observing data

. Denote

and

Then

Hence we need to find

and

to minimize

Suppose the minimization problem has an unique solution. Then the estimators of b and

are

and

, where

and

. The estimate of g(t) can be described as

for a suitable z.

plmp presents the estimates and .

res.hbeta: vector or scalar, the estimate of

The next picture , "plmp-ex.xpl", gives an example of XploRe code to generate a sample from the PLM model, and then shows how to compute the PLM estimates by plmp. Saving the example as a file (plmp-ex.xpl) and then linking "Execute", the actual parameter estimates are shown in the second picture(XploRe_out).

An example by using plmp

The parameter estimate value is listed as follows

The result of nonparametric fitting with following picture

2.4. LOCAL POLYNOMIAL

Suppose that the derivative of g(t) at the point exists. We then approximate the unknown regression function g(t) locally by a polynomial of order p. A Taylor expansion given, for t in a neighborhood of ,

To estimate and g(t), we first estimate the's as functions of , denoted , by minimizing

where h is a bandwidth controlling the size of the local neighborhood, and

with K a kernel function assigning weights to each datum. Then minimize

Denote the solution of (7) by

. Fianlly let

be the estimates of

, and dnote by

. It is clear from the Taylor expansion in (5) that

is an estimator for

. To estimate the entire function

we solve the above weighted least squares problem for all points

in the domain of interest.

It is more convenient to work with matrix notation. Denote by the design matrix of T in problem (6). Denote

and put

Further, let

be the

diagonal matric of weights:

. Then the weighted least squres problems (6) and (7) can be rewritten as

and

with

. The solution vectors are provided by weighted least squares and are given by

plmlorg is designed to impliment the above arguments in XploRe. lpregest is used to estimate nonparametric regression functions. Interpolation idea is employed to calculate the estimators of beta and g(t).

Considering the same example in previous sections, we here approximate the nonlinear part with the 2-nd local polynomial approximation. The results for parametric and nonparametric parts are listed as follows.

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