monvarresid {monreg}

Estimating Monotone Variance Functions Using Nonparametric Residuals

Description

monvarresid provides a strictly monotone estimator of the variance function based on the nonparametric regression model.

Usage

monvarresid(x,y,a=min(x),b=max(x),N=length(x),t=length(x),h,K="epanech",hd,Kd="epanech",
            hr,Kr="epanech",mdegree=1,sdegree=1,inverse=0,monotonie="isoton")

Arguments

x

vector containing the x-values (design points) of a sample

y

vector containing the y-values (response) of a sample

a

lower bound of the support of the design points density function, or smallest fixed design point

b

upper bound of the support of the design points density function, or largest fixed design point

N

number or vector of evaluation points of the unconstrained nonparametric variance estimator (e.g. Nadaraya-Watson estimator)

t

number or vector of points where the monotone estimation is computed

h

bandwith of kernel K of the regression estimation step

K

Kernel for the regression estimation step. 'epanech' for Epanechnikov, 'rectangle' for rectangle, 'biweight' for biweight, 'triweight' for triweight, 'triangle' for triangle, 'cosine' for cosine kernel

hd

bandwith of kernel K_d of the density estimation step

Kd

Kernel for the density estimation step (monotonization step). 'epanech' for "Epanechnikov, 'rectangle' for rectangle, 'biweight' for biweight, 'triweight' for triweight, 'triangle' for triangle, 'cosine' for cosine kernel

hr

bandwith of kernel K_r of the variance estimation step

Kr

Kernel for the variance estimation step (unconstrained estimation). 'epanech' for "Epanechnikov, 'rectangle' for rectangle, 'biweight' for biweight, 'triweight' for triweight, 'triangle' for triangle, 'cosine' for cosine kernel.

mdegree

determines the method for the regression estimation. '0'for the classical Nadaraya-Watson estimate, '1' for the local linear estimate. As well mdegree can be the vector of the estimator provided by the user for the design points given by the vector x

sdegree

Determines the method for the unconstrained variance estimation. '0' for the classical Nadaraya-Watson estimate, '1' for the local linear estimate. As well sdegree can be the vector of the unconditional estimator provided by the user for the design points given by the vector N

inverse

For '0' the original variance function is estimated, for '1' the inverse of the variance function is estimated.

monotonie

Determines the type of monotonicity. 'isoton' if the variance function is assumed to be isotone, 'antinton' if the variance function is assumed to be antitone.

Details

Nonparametric regression models are of the form Y_i = m(X_i) + \sigma(X_i) \cdot \varepsilon_i, where m is the regression funtion and \sigma the variance function. monvarresid performs a monotone estimate of the unknown variance function s=\sigma^2. monvarresid first estimates m by an unconstrained nonparametric method, the classical Nadaraya-Watson estimate or the local-linear estimate (unless the user decides to pass his or her own estimate). In a second step an unconstrained estimation for s is performed, again by the classical Nadaraya-Watson method or the local-linear estimate (unless the user decides to pass his or her own estimate). In a third step the inverse of the (monotone) variance function is calculated, by monotonizing the unconstrained estimate from the second step. With the above notation and \hat s for the unconstrained estimate, the third step writes as follows,

\hat s_I^{-1} = \frac{1}{Nh_d} \sum\limits_{i=1}^N \int\limits_{-\infty}^t K_d \Bigl( \frac{\hat s (\frac{i}{N} ) - u}{h_d} \Bigr) \; du.

Finally, the monotone estimate is achieved by inversion of \hat s_I^{-1}.

Value

monvarresid returns a list of values

xs

the input values x, standardized on the interval [0,1]

y

input variable y

z

the points, for which the unconstrained function is estimated

t

the points, for which the monotone variance function will be estimated

length.x

length of the vector x

length.z

length of the vector z

length.t

length of the vector t

h

bandwidth used with the Kernel K

hd

bandwidth used with the Kernel K_d

hr

bandwidth used with the Kernel K_r

K

kernel used for the regression estimation step

Kd

kernel used for the monotonization step

Kr

kernel used for the unconstrained variance estimate

mdegree

method, which was used for the unconstrained regression estimate

lmdeg

length of the vector mdegree. If lmdeg is not equal to 1 the user provided the vector of the unconditional regression estimator for the design points given by the vector x

sdegree

method, which was used for the unconstrained variance estimate

lsdeg

length of the vector sdegree. If lsdeg is not equal to 1 the user provided the vector of the unconditional variance estimator for the design points given by the vector N

inverse

indicates, if the origin variance function or its inverse has been estimated

estimation

the monotone estimate for the variance function at the design points t

Author(s)

This R Package was developed by Kay Pilz and Stefanie Titoff. Earlier developements of the estimator were made by Holger Dette and Kay Pilz.

See Also

monreg for monotone regression function estimation and monvardiff for monotone variance function estimation by differences.