monvarresid provides a strictly monotone estimator of the variance function based on the nonparametric regression model.
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")
vector containing the x-values (design points) of a sample
vector containing the y-values (response) of a sample
lower bound of the support of the design points density function, or smallest fixed design point
upper bound of the support of the design points density function, or largest fixed design point
number or vector of evaluation points of the unconstrained nonparametric variance estimator (e.g. Nadaraya-Watson estimator)
number or vector of points where the monotone estimation is computed
bandwith of kernel \(K\) of the regression estimation step
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
bandwith of kernel \(K_d\) of the density estimation step
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
bandwith of kernel \(K_r\) of the variance estimation step
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.
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
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
For '0' the original variance function is estimated, for '1' the inverse of the variance function is estimated.
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.
monvarresid
returns a list of values
the input values x, standardized on the interval \([0,1]\)
input variable y
the points, for which the unconstrained function is estimated
the points, for which the monotone variance function will be estimated
length of the vector x
length of the vector z
length of the vector t
bandwidth used with the Kernel \(K\)
bandwidth used with the Kernel \(K_d\)
bandwidth used with the Kernel \(K_r\)
kernel used for the regression estimation step
kernel used for the monotonization step
kernel used for the unconstrained variance estimate
method, which was used for the unconstrained regression estimate
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
method, which was used for the unconstrained variance estimate
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
indicates, if the origin variance function or its inverse has been estimated
the monotone estimate for the variance function at the design points \(t\)
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 \(\hat s_I^{-1}\).
monreg
for monotone regression function estimation and monvardiff
for monotone variance function estimation by differences.