np.smoothcoef.bw: Smooth Coefficient Kernel Regression Bandwidth Selection

In np: Nonparametric Kernel Smoothing Methods for Mixed Data Types

Smooth Coefficient Kernel Regression Bandwidth Selection

Description

npscoefbw computes a bandwidth object for a smooth coefficient kernel regression estimate of a one (1) dimensional dependent variable on p+q-variate explanatory data, using the model Y_i = W_{i}^{\prime} \gamma (Z_i) + u_i where W_i'=(1,X_i') given training points (consisting of explanatory data and dependent data), and a bandwidth specification, which can be a rbandwidth object, or a bandwidth vector, bandwidth type and kernel type.

Usage

npscoefbw(...)

## S3 method for class 'formula'
npscoefbw(formula, 
          data, 
          subset, 
          na.action, 
          call, 
          ...)

## Default S3 method:
npscoefbw(xdat = stop("invoked without data 'xdat'"),
          ydat = stop("invoked without data 'ydat'"),
          zdat = NULL,
          bws,
          backfit.iterate,
          backfit.maxiter,
          backfit.tol,
          bandwidth.compute = TRUE,
          basis,
          bernstein.basis,
          bwmethod,
          bwscaling,
          bwtype,
          ckerbound,
          ckerlb,
          ckerorder,
          ckertype,
          ckerub,
          cv.iterate,
          cv.num.iterations,
          degree,
          degree.select = c("manual", "coordinate", "exhaustive"),
          search.engine = c("nomad+powell", "cell", "nomad"),
          nomad = FALSE,
          nomad.nmulti = 0L,
          degree.min = NULL,
          degree.max = NULL,
          degree.start = NULL,
          degree.restarts = 0L,
          degree.max.cycles = 20L,
          degree.verify = FALSE,
          nmulti,
          okertype,
          optim.abstol,
          optim.maxattempts,
          optim.maxit,
          optim.method,
          optim.reltol,
          random.seed,
          regtype,
          ukertype,
          scale.factor.init.lower = 0.1,
          scale.factor.init.upper = 2.0,
          scale.factor.init = 0.5,
          lbd.init = 0.5,
          hbd.init = 1.5,
          dfac.init = 1.0,
          scale.factor.search.lower = NULL,
          ...)

## S3 method for class 'scbandwidth'
npscoefbw(xdat = stop("invoked without data 'xdat'"),
          ydat = stop("invoked without data 'ydat'"),
          zdat = NULL,
          bws,
          backfit.iterate = FALSE,
          backfit.maxiter = 100,
          backfit.tol = .Machine$double.eps,
          bandwidth.compute = TRUE,
          cv.iterate = FALSE,
          cv.num.iterations = 1,
          nmulti,
          optim.abstol = .Machine$double.eps,
          optim.maxattempts = 10,
          optim.maxit = 500,
          optim.method = c("Nelder-Mead", "BFGS", "CG"),
          optim.reltol = sqrt(.Machine$double.eps),
          random.seed = 42,
          scale.factor.init.lower = 0.1,
          scale.factor.init.upper = 2.0,
          scale.factor.init = 0.5,
          lbd.init = 0.5,
          hbd.init = 1.5,
          dfac.init = 1.0,
          scale.factor.search.lower = NULL,
          ...)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the smooth-coefficient data, formula interface, and whether bandwidths are supplied or computed.

bandwidth.compute	a logical value which specifies whether to do a numerical search for bandwidths or not. If set to FALSE, a scbandwidth object will be returned with bandwidths set to those specified in bws. Defaults to TRUE.
bws	a bandwidth specification. This can be set as a scbandwidth object returned from a previous invocation, or as a vector of bandwidths, with each element i corresponding to the bandwidth for column i in xdat. In either case, the bandwidth supplied will serve as a starting point in the numerical search for optimal bandwidths. If specified as a vector, then additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, selection methods, and so on. This can be left unset.
call	the original function call. This is passed internally by np when a bandwidth search has been implied by a call to another function. It is not recommended that the user set this.
data	an optional data frame, list or environment (or object coercible to a data frame by as.data.frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which the function is called.
formula	a symbolic description of variables on which bandwidth selection is to be performed. The details of constructing a formula are described below.
na.action	a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The (recommended) default is na.omit.
subset	an optional vector specifying a subset of observations to be used in the fitting process.
xdat	a p-variate data frame of explanatory data (training data), which, by default, populates the columns 2 through p+1 of W in the model equation, and in the absence of zdat, will also correspond to Z from the model equation.
ydat	a one (1) dimensional numeric or integer vector of dependent data, each element i corresponding to each observation (row) i of xdat.
zdat	an optionally specified q-variate data frame of explanatory data (training data), which corresponds to Z in the model equation. Defaults to be the same as xdat.

Automatic Degree Search Controls

These arguments control automatic local-polynomial degree search.

degree.max	optional scalar or integer vector giving upper bounds for automatic degree search when degree.select != "manual".
degree.max.cycles	positive integer giving the maximum number of coordinate-search sweeps over the degree vector. Ignored for "manual" and "exhaustive" degree selection.
degree.min	optional scalar or integer vector giving lower bounds for automatic degree search when degree.select != "manual".
degree.restarts	non-negative integer giving the number of additional deterministic coordinate-search restarts. Ignored for "manual" and "exhaustive" degree selection.
degree.select	character string controlling local-polynomial degree handling when regtype="lp". "manual" (default) treats degree as fixed. "coordinate" performs cached coordinate-wise search over admissible degree vectors for the continuous z variables. "exhaustive" evaluates the full admissible degree grid when search.engine="cell". For NOMAD-based search engines, any non-"manual" value requests direct joint search over degree and bandwidth coordinates.
degree.start	optional starting degree vector for automatic coordinate search. If omitted, the search starts from the degree-zero local-constant baseline on the continuous z variables.
degree.verify	logical value indicating whether a coordinate-search solution should be exhaustively verified over the admissible degree grid after the heuristic phase completes. Available only for search.engine="cell".

Backfitting Controls

These controls tune the optional smooth-coefficient backfitting iterations.

backfit.iterate	boolean value specifying whether or not to iterate evaluations of the smooth coefficient estimator, for extra accuracy, during the cross-validated backfitting procedure. Defaults to FALSE.
backfit.maxiter	integer specifying the maximum number of times to iterate the evaluation of the smooth coefficient estimator in the attempt to obtain the desired accuracy. Defaults to 100.
backfit.tol	tolerance to determine convergence of iterated evaluations of the smooth coefficient estimator. Defaults to .Machine$double.eps.

Bandwidth Criterion And Representation

These arguments choose the selection criterion and the way continuous bandwidths are represented.

bwmethod	which method was used to select bandwidths. cv.ls specifies least-squares cross-validation, which is all that is currently supported. Defaults to cv.ls.
bwscaling	a logical value that when set to TRUE the supplied bandwidths are interpreted as ‘scale factors’ (c_j), otherwise when the value is FALSE they are interpreted as ‘raw bandwidths’ (h_j for continuous data types, \lambda_j for discrete data types). For continuous data types, c_j and h_j are related by the formula h_j = c_j \sigma_j n^{-1/(2P+l)}, where \sigma_j is an adaptive measure of spread of continuous variable j defined as min(standard deviation, mean absolute deviation, interquartile range/1.349), n the number of observations, P the order of the kernel, and l the number of continuous variables. For discrete data types, c_j and h_j are related by the formula h_j = c_jn^{-2/(2P+l)}, where here j denotes discrete variable j. Defaults to FALSE.
bwtype	character string used for the continuous variable bandwidth type, specifying the type of bandwidth provided. Defaults to fixed. Option summary: fixed: fixed bandwidths or scale factors generalized_nn: generalized nearest neighbors adaptive_nn: adaptive nearest neighbors

Categorical Search Initialization

These controls set categorical search starts.

dfac.init	deterministic fixed-bandwidth start factor for ordered and unordered categorical coordinates. Used only when bwtype="fixed". Defaults to 1.0. Values must not exceed 2.
hbd.init	upper bound for random fixed-bandwidth start factors for ordered and unordered categorical coordinates. Used only when bwtype="fixed". Defaults to 1.5. Must be greater than or equal to lbd.init and must not exceed 2.
lbd.init	lower bound for random fixed-bandwidth start factors for ordered and unordered categorical coordinates. Used only when bwtype="fixed". Defaults to 0.5. Values must not exceed 2 so that categorical fixed starts remain within lawful bandwidth bounds.

Continuous Kernel Support Controls

These controls choose and parameterize bounded support for continuous kernels.

ckerbound	character string controlling continuous-kernel support handling. Can be set as none (default kernel on full support), range (use sample min/max), or fixed (use ckerlb/ckerub). The bounded-kernel route reuses the selected continuous kernel and renormalizes it on the chosen support; see np.kernels.
ckerlb	numeric scalar/vector of lower bounds for continuous variables used when ckerbound="fixed". Must satisfy lower-bound validity for each continuous variable (e.g., <= min(variable)). Use -Inf for unbounded below. See np.kernels for bounded-kernel normalization details.
ckerub	numeric scalar/vector of upper bounds for continuous variables used when ckerbound="fixed". Must satisfy upper-bound validity for each continuous variable (e.g., >= max(variable)). Use Inf for unbounded above. See np.kernels for bounded-kernel normalization details.

Continuous Scale-Factor Search Initialization

These controls define deterministic and random continuous scale-factor starts and the lower admissibility floor for fixed-bandwidth search.

scale.factor.init	deterministic initial scale factor for continuous fixed-bandwidth search. Defaults to 0.5. The value supplied by the user is not rewritten, but the effective first start passed to the optimizer is max(scale.factor.init, scale.factor.search.lower). See Details.
scale.factor.init.lower	lower endpoint for random continuous scale-factor starts. Defaults to 0.1. The value supplied by the user is not rewritten, but the effective random-start lower endpoint is max(scale.factor.init.lower, scale.factor.search.lower). See Details.
scale.factor.init.upper	upper endpoint for random continuous scale-factor starts. Defaults to 2.0. It must be greater than or equal to the effective lower endpoint, max(scale.factor.init.lower, scale.factor.search.lower); otherwise bandwidth search errors rather than silently expanding the interval. See Details.
scale.factor.search.lower	optional nonnegative scalar giving the hard lower admissibility bound for continuous fixed-bandwidth search candidates. Defaults to NULL. If NULL, an existing bandwidth object's stored value is inherited when available; otherwise the package default 0.1 is used. This floor applies to computed/search bandwidth candidates and to effective search starts only. It does not rewrite explicit bandwidths supplied for storage with bandwidth.compute = FALSE. Final fixed-bandwidth search candidates must also have a finite valid raw objective value.

Cross-Validation Iteration Controls

These controls tune iterative cross-validation behavior.

cv.iterate	boolean value specifying whether or not to perform iterative, cross-validated backfitting on the data. See details for limitations of the backfitting procedure. Defaults to FALSE.
cv.num.iterations	integer specifying the number of times to iterate the backfitting process over all covariates. Defaults to 1.

Kernel Type Controls

These controls choose continuous, unordered, and ordered kernels.

ckerorder	numeric value specifying kernel order (one of (2,4,6,8)). Kernel order specified along with a uniform continuous kernel type will be ignored. Defaults to 2.
ckertype	character string used to specify the continuous kernel type. Can be set as gaussian, epanechnikov, or uniform. Defaults to gaussian.
okertype	character string used to specify the ordered categorical kernel type. Can be set as wangvanryzin, liracine, or racineliyan. Defaults to liracine.
ukertype	character string used to specify the unordered categorical kernel type. Can be set as aitchisonaitken or liracine. Defaults to aitchisonaitken.

Local-Polynomial Model Specification

These arguments control the local-polynomial estimator, basis, and fixed degree specification.

basis	for regtype="lp", the polynomial basis family used for local polynomial fitting. Options are "glp" (default), "additive", and "tensor".
bernstein.basis	for regtype="lp", logical flag selecting Bernstein-basis representation where supported. When automatic degree search is requested and bernstein.basis is not explicitly supplied, the search route defaults to TRUE for numerical stability. Explicit bernstein.basis=FALSE is honored, but raw-polynomial search can be poorly conditioned at higher degrees.
degree	for regtype="lp", polynomial degree specification for each continuous z variable.
regtype	a character string specifying local smoothing type for the z surface. Options are "lc" (default), "ll", and "lp".

NOMAD Search Controls

These arguments control the optional NOMAD direct-search route for local-polynomial degree and bandwidth search.

nomad	logical shortcut for the recommended automatic local-polynomial NOMAD route. When TRUE, any missing values among regtype, search.engine, degree.select, bernstein.basis, degree.min, degree.max, degree.verify, and bwtype are filled with regtype="lp", search.engine="nomad+powell", degree.select="coordinate", bernstein.basis=TRUE, degree.min=0L, degree.max=10L, degree.verify=FALSE, and bwtype="fixed". Explicit incompatible settings error immediately; in particular, nomad=TRUE currently requires regtype="lp", bwtype="fixed", automatic degree search, bernstein.basis=TRUE, no explicit degree, and search.engine %in% c("nomad", "nomad+powell"). This shortcut does not change the meaning of nmulti or nomad.nmulti: nmulti remains the outer restart count, while nomad.nmulti controls inner crs::snomadr() multistarts within each outer restart. Returned bandwidth objects retain this normalized preset metadata in bw$nomad.shortcut for a returned object bw; when available, nomad.time and powell.time record the direct-search and Powell-polish timing components.
nomad.nmulti	non-negative integer controlling the inner crs::snomadr() multistart count used within each outer NOMAD restart when regtype="lp" and automatic degree search uses search.engine="nomad" or "nomad+powell". Defaults to 0L, which preserves the current one-start-per- restart behavior. This does not replace nmulti: nmulti controls outer restarts, while nomad.nmulti controls inner NOMAD multistarts within each outer restart.
search.engine	character string controlling the automatic local-polynomial search backend when regtype="lp" and degree.select != "manual". "nomad+powell" (default) performs direct joint search over the zdat-side continuous bandwidth coordinates and degree vector using crs::snomadr(), then applies one Powell hot start from the NOMAD solution. "nomad" omits the Powell refinement. "cell" profiles the criterion over the admissible degree grid using repeated fixed-degree bandwidth solves. NOMAD-based search currently requires bwtype="fixed", degree.verify=FALSE, and the suggested package crs to be installed.

Numerical Search Controls

These controls set search restart behavior.

nmulti

integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points. Defaults to min(2,ncol(xdat)).

Optimization Controls

These arguments control outer optimization behavior for the semiparametric search.

optim.abstol	the absolute convergence tolerance used by optim. Only useful for non-negative functions, as a tolerance for reaching zero. Defaults to .Machine$double.eps.
optim.maxattempts	maximum number of attempts taken trying to achieve successful convergence in optim. Defaults to 100.
optim.maxit	maximum number of iterations used by optim. Defaults to 500.
optim.method	method used by optim for minimization of the objective function. See ?optim for references. Defaults to "Nelder-Mead". the default method is an implementation of that of Nelder and Mead (1965), that uses only function values and is robust but relatively slow. It will work reasonably well for non-differentiable functions. method "BFGS" is a quasi-Newton method (also known as a variable metric algorithm), specifically that published simultaneously in 1970 by Broyden, Fletcher, Goldfarb and Shanno. This uses function values and gradients to build up a picture of the surface to be optimized. method "CG" is a conjugate gradients method based on that by Fletcher and Reeves (1964) (but with the option of Polak-Ribiere or Beale-Sorenson updates). Conjugate gradient methods will generally be more fragile than the BFGS method, but as they do not store a matrix they may be successful in much larger optimization problems.
optim.reltol	relative convergence tolerance used by optim. The algorithm stops if it is unable to reduce the value by a factor of 'reltol * (abs(val) + reltol)' at a step. Defaults to sqrt(.Machine$double.eps), typically about 1e-8.
random.seed	an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.

Additional Arguments

These arguments collect remaining controls passed through S3 methods.

...	additional arguments supplied to specify the regression type, bandwidth type, kernel types, selection methods, and so on, detailed below.

Details

The scale.factor.* controls are dimensionless search controls. The package converts scale factors to bandwidths using the estimator-specific scaling encoded in the bandwidth object, including kernel order and the number of continuous variables relevant for the estimator. Users should not pre-multiply these controls by sample-size or standard-deviation factors.

scale.factor.init controls the deterministic first search start when that control is exposed. scale.factor.init.lower and scale.factor.init.upper define the random multistart interval when exposed. scale.factor.search.lower is the lower admissibility bound for continuous fixed-bandwidth search candidates. The effective first start is max(scale.factor.init, scale.factor.search.lower) when both controls are present, and the effective random-start lower endpoint is max(scale.factor.init.lower, scale.factor.search.lower). scale.factor.init.upper must be at least that effective lower endpoint; the package errors rather than silently expanding the user's interval.

When scale.factor.search.lower is NULL, an existing bandwidth object's stored floor is inherited when available; otherwise the package default 0.1 is used. Explicit bandwidths supplied for storage with bandwidth.compute = FALSE are not rewritten by the search floor.

Categorical search-start controls such as dfac.init, lbd.init, and hbd.init have separate semantics and are not affected by scale.factor.search.lower.

Documentation guide: see np.kernels for kernels, np.options for global options, and plot for plotting options.

For S3 plotting help, use methods("plot") and query class-specific help topics such as ?plot.npregression and ?plot.rbandwidth. You can inspect implementations with getS3method("plot","npregression").

npscoefbw implements a variety of methods for semiparametric regression on multivariate (p+q-variate) explanatory data defined over a set of possibly continuous data. The approach is based on Li and Racine (2003) who employ ‘generalized product kernels’ that admit a mix of continuous and discrete data types.

Three classes of kernel estimators for the continuous data types are available: fixed, adaptive nearest-neighbor, and generalized nearest-neighbor. Adaptive nearest-neighbor bandwidths change with each sample realization in the set, x_i, when estimating the density at the point x. Generalized nearest-neighbor bandwidths change with the point at which the density is estimated, x. Fixed bandwidths are constant over the support of x.

npscoefbw may be invoked either with a formula-like symbolic description of variables on which bandwidth selection is to be performed or through a simpler interface whereby data is passed directly to the function via the xdat, ydat, and zdat parameters. Use of these two interfaces is mutually exclusive.

Data contained in the data frame xdat may be continuous and in zdat may be of mixed type. Data can be entered in an arbitrary order and data types will be detected automatically by the routine (see np for details).

Data for which bandwidths are to be estimated may be specified symbolically. A typical description has the form dependent data ~ parametric explanatory data | nonparametric explanatory data, where dependent data is a univariate response, and parametric explanatory data and nonparametric explanatory data are both series of variables specified by name, separated by the separation character '+'. For example, y1 ~ x1 + x2 | z1 specifies that the bandwidth object for the smooth coefficient model with response y1, linear parametric regressors x1 and x2, and nonparametric regressor (that is, the slope-changing variable) z1 is to be estimated. See below for further examples. In the case where the nonparametric (slope-changing) variable is not specified, it is assumed to be the same as the parametric variable.

A variety of kernels may be specified by the user. Kernels implemented for continuous data types include the second, fourth, sixth, and eighth order Gaussian and Epanechnikov kernels, and the uniform kernel. Unordered discrete data types use a variation on Aitchison and Aitken's (1976) kernel, while ordered data types use a variation of the Wang and van Ryzin (1981) kernel.

Setting nomad=TRUE is a convenience preset for this automatic LP route, not a generic optimizer alias. For smooth coefficient regression it expands any missing values to the equivalent long-form call

   npscoefbw(...,
             regtype = "lp",
             search.engine = "nomad+powell",
             degree.select = "coordinate",
             bernstein.basis = TRUE,
             degree.min = 0L,
             degree.max = 10L,
             degree.verify = FALSE,
             bwtype = "fixed")
 

Compatible explicit tuning arguments are respected. Incompatible explicit settings fail fast so the shortcut never silently changes user-selected semantics.

When regtype="lp" and degree.select != "manual", npscoefbw can jointly determine the zdat-side local polynomial degree vector together with the associated bandwidth coordinates. With search.engine="cell", the criterion is profiled over the admissible degree grid using cached coordinate-wise or exhaustive search together with repeated fixed-degree bandwidth solves. With search.engine="nomad" or "nomad+powell", the criterion is optimized directly over the joint degree/bandwidth space using crs::snomadr(); "nomad+powell" then performs one Powell hot start from the NOMAD solution and keeps the better of the direct NOMAD and polished answers. This polynomial-adaptive joint-search route is motivated by Hall and Racine (2015). When bernstein.basis is not explicitly supplied, the automatic search route defaults to bernstein.basis=TRUE for numerical stability.

Value

if bwtype is set to fixed, an object containing bandwidths (or scale factors if bwscaling = TRUE) is returned. If it is set to generalized_nn or adaptive_nn, then instead the kth nearest neighbors are returned for the continuous variables while the discrete kernel bandwidths are returned for the discrete variables. Bandwidths are stored in a vector under the component name bw. Backfitted bandwidths are stored under the component name bw.fitted.

The functions predict, summary, and plot support objects of this class.

Usage Issues

If you are using data of mixed types, then it is advisable to use the data.frame function to construct your input data and not cbind, since cbind will typically not work as intended on mixed data types and will coerce the data to the same type.

Caution: multivariate data-driven bandwidth selection methods are, by their nature, computationally intensive. Virtually all methods require dropping the ith observation from the data set, computing an object, repeating this for all observations in the sample, then averaging each of these leave-one-out estimates for a given value of the bandwidth vector, and only then repeating this a large number of times in order to conduct multivariate numerical minimization/maximization. Furthermore, due to the potential for local minima/maxima, restarting this procedure a large number of times may often be necessary. This can be frustrating for users possessing large datasets. For exploratory purposes, you may wish to override the default search tolerances, say, setting optim.reltol=.1 and conduct multistarting (the default is to restart min(2,ncol(zdat)) times). Once the procedure terminates, you can restart search with default tolerances using those bandwidths obtained from the less rigorous search (i.e., set bws=bw on subsequent calls to this routine where bw is the initial bandwidth object). A version of this package using the Rmpi wrapper is under development that allows one to deploy this software in a clustered computing environment to facilitate computation involving large datasets.

Support for backfitted bandwidths is experimental and is limited in functionality. The code does not support asymptotic standard errors or out of sample estimates with backfitting.

Author(s)

Tristen Hayfield [e-mail], Jeffrey S. Racine [e-mail]

References

Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.

Cai Z. (2007), “Trending time-varying coefficient time series models with serially correlated errors,” Journal of Econometrics, 136, 163-188.

Hastie, T. and R. Tibshirani (1993), “Varying-coefficient models,” Journal of the Royal Statistical Society, B 55, 757-796.

Hall, P. and J.S. Racine (2015), “Infinite Order Cross-Validated Local Polynomial Regression,” Journal of Econometrics, 185, 510-525.

Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.

Li, Q. and J.S. Racine (2010), “Smooth varying-coefficient estimation and inference for qualitative and quantitative data,” Econometric Theory, 26, 1-31.

Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.

Li, Q. and D. Ouyang and J.S. Racine (2013), “Categorical semiparametric varying-coefficient models,” Journal of Applied Econometrics, 28, 551-589.

Li, A. and Q. Li and J.S. Racine (under revision), “Boundary Adjusted, Polynomial Adaptive, Nonparametric Kernel Conditional Density Estimation,” Econometric Reviews.

Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301-309.

See Also

np.kernels, np.options, plot npregbw, npreg

Examples

## Not run: 
# EXAMPLE 1 (INTERFACE=FORMULA):
set.seed(42)

n <- 100
x <- runif(n)
z <- runif(n, min=-2, max=2)
y <- x*exp(z)*(1.0+rnorm(n,sd = 0.2))
bw <- npscoefbw(formula=y~x|z)
summary(bw)

# EXAMPLE 1 (INTERFACE=DATA FRAME): 

n <- 100
x <- runif(n)
z <- runif(n, min=-2, max=2)
y <- x*exp(z)*(1.0+rnorm(n,sd = 0.2))
bw <- npscoefbw(xdat=x, ydat=y, zdat=z)
summary(bw)

## End(Not run) 

np documentation built on May 3, 2026, 1:07 a.m.