monvarresid: Estimating Monotone Variance Functions Using Nonparametric...
In monreg: Nonparametric Monotone Regression

Description Usage Arguments Details Value Author(s) See Also

View source: R/monvarresid.r

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

1 2	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")

`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.

Nonparametric regression models are of the form Y_i = m(X_i) + σ(X_i) \cdot \varepsilon_i, where m is the regression funtion and σ the variance function. monvarresid performs a monotone estimate of the unknown variance function s=σ^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} ∑\limits_{i=1}^N \int\limits_{-∞}^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}.

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