monvardiff {monreg}

Estimating Monotone Variance Functions Using Pseudo-Residuals

Description

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

Usage

monvardiff(x,y,a=min(x),b=max(x),N=length(x),t=length(x),r=2,hr,Kr="epanech",
           hd,Kd="epanech",degree=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

r

order of the difference scheme, i.e. weights d_0,...,d_r to calculate the pseudo-residuals

hr

bandwith of kernel Kr 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

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

degree

determines the method for the unconstrained variance estimation. '0' for the classical Nadaraya-Watson estimate, '1' for the local linear estimate. As well degree can be the vector of the unconditional estimator provided by the user for the design points given in 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. monvardiff performs a monotone estimate of the unknown variance function s=\sigma^2. monvardiff first estimates s 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). This estimation contains the usage of the Pseudo-Residuals. In a second step the inverse of the (monotone) variance function is calculated by monotonizing the unconstrained estimate from the first step. With the above notation and \hat s for the unconstrained estimate, the second 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

monvardiff 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

r

order of the difference scheme, i.e. number of weights to calculate the pseudo-residuals

hr

bandwidth used with the Kernel K_r

hd

bandwidth used with the Kernel K_d

Kr

kernel used for the unconstrained variance estimate

Kd

kernel used for the monotonization step

degree

method, which was used for the unconstrained variance estimate

ldeg.vektor

length of the vector degree. If ldeg.vektor is not equal to 1 the user provided the vector of the unconditional variance estimator for the design points given in the vector N

inverse

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

estimation

the monotone estimate 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 monvarresid for monotone variance function estimation by nonparametric residuals.