Bandwidth Selection for Location-Scale Kernel Quantile Regression

nplsqregbw selects bandwidths and the scalar location-scale shift parameter delta for nplsqreg by minimizing leave-one-out check loss.

Usage

nplsqregbw(bws, ...)

# S3 method for class 'formula'
nplsqregbw(bws, data = NULL, tau = 0.5,
       subset, na.action, ...)

# S3 method for class 'lsqregressionbandwidth'
nplsqregbw(bws, tau = bws$tau, ...)

# Default S3 method
nplsqregbw(bws,
       xdat = stop("invoked without data 'xdat'"),
       ydat = stop("invoked without data 'ydat'"),
       tau = 0.5,
       tau.search = c("full", "refined"),
       delta = NULL,
       scale = NULL,
       regtype = c("lc", "ll", "lp"),
       regtype.pilot = c("auto", "ll", "lc", "lp"),
       nomad = FALSE,
       nomad.pilot = FALSE,
       pilot.args = list(),
       bandwidth.compute = TRUE,
       delta.bounds = c(1e-4, 1 - 1e-4),
       optim.control = list(maxit = 50L),
       ...)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the bandwidth specification, formula/data interface, and training data.

bws: a formula, an lsqregressionbandwidth object, an rbandwidth object, a numeric bandwidth vector, or omitted for automatic selection. Ordinary npregbw rbandwidth objects may be used as starting values. Exact nplsqreg reuse is through an lsqregressionbandwidth object, typically fit$bws; fit$reg.bws is internal regression state.
data: an optional data frame, list or environment containing the variables in the model. If not found in data, the variables are taken from environment(bws).
subset: an optional vector specifying a subset of observations to be used by the formula method.
na.action: a function specifying the action to take when missing values are found by the formula method.
xdat: a $p$-variate data frame of explanatory data used as training data.
ydat: a one dimensional numeric vector of dependent data.

Quantile Index And Vector-Tau Search

These arguments identify the quantile probabilities and how vector-tau fits are orchestrated.

tau: a numeric scalar or vector specifying the quantile probability or probabilities $\tau$. Values must lie strictly in $(0,1)$.
tau.search: either "full" or "refined". The default "full" route performs a full bandwidth/delta search for each tau, sharing only the pilot scale. The explicit "refined" route fits the central tau first and uses the resulting bandwidths, delta, and when applicable selected degree vector as warm starts for the remaining quantiles.

Scale Pilot

These arguments control or supply the conditional standard deviation used in the location-scale transformation.

scale: an optional strictly positive scale vector interpreted as the conditional standard deviation at the training observations. If omitted, a pilot scale is computed automatically.
regtype.pilot: regression type used for the pilot mean and residual-variance smooths. The default "auto" uses local linear smoothing when continuous predictors are present and local constant smoothing for categorical-only designs.
nomad.pilot: logical value indicating whether the pilot regressions should use the automatic NOMAD local-polynomial route when compatible with regtype.pilot. Defaults to FALSE.
pilot.args: optional named list of additional arguments supplied to the pilot npreg calls.

Bandwidth, Delta And Degree Search

These arguments control the main check-loss bandwidth and delta optimization.

regtype: regression type for the main transformed-response fit and check-loss bandwidth search. This is independent of regtype.pilot and follows npregbw semantics.
delta: optional starting value for the shift parameter. If omitted, the first start uses 0.5; subsequent multistarts draw starting values from $(0,1)$.
delta.bounds: two numeric values giving the lower and upper bounds for delta. The interval must lie strictly inside $(0,1)$.
nomad: logical value indicating whether to use the local-polynomial NOMAD route for joint bandwidth, delta, and continuous degree-vector search. Defaults to FALSE.
bandwidth.compute: logical value indicating whether automatic bandwidth selection should be performed. Defaults to TRUE.
optim.control: a list of controls passed to the Powell optimizer used for fixed degree searches and NOMAD hot-start refinement.
...: additional bandwidth, kernel, local-polynomial, and search controls forwarded to the package bandwidth machinery. Common examples include regtype, bwtype, bwmethod, nmulti, degree, basis, bernstein.basis, search.engine, degree.min, degree.max, degree.start, degree.restarts, powell.remin, nomad.remin, and nomad.nmulti. The explanatory-variable kernel aliases used by npqreg and npcdistbw, such as cxkertype, cxkerorder, uxkertype, and oxkertype, are accepted and mapped to the corresponding regression-kernel controls. Response-side conditional-distribution kernel controls such as cykertype are not meaningful for the location-scale transformed-response estimator and fail clearly rather than being ignored.

Details

For a requested quantile probability $\tau$, nplsqregbw chooses bandwidths and delta for the transformed response $$ Y_i^\delta = Y_i + \hat\sigma(X_i)\Phi^{-1}(\delta) $$ by minimizing the leave-one-out check loss. The selected bandwidths are then used by nplsqreg for the final mixed-data npreg fit of the transformed response.

The default pilot scale is a residual scale: first npreg estimates the conditional mean, then npreg smooths squared residuals using the same bandwidth object, and the square root of the floored fitted variance is used as the scale. The local-linear pilot option follows Fan and Yao (1998). A user-supplied scale must be strictly positive and is interpreted as a conditional standard deviation, not a variance.

When nomad=TRUE, the search can include the continuous local-polynomial degree vector in addition to bandwidths and delta. The search.engine="nomad+powell" route uses NOMAD for degree search and Powell for hot-start refinement at the selected degree.

Value

nplsqregbw returns an object of class lsqregressionbandwidth. For vector tau, the object stores the per-tau bandwidth objects, selected delta values, objectives, fit order, warm-start provenance, and shared pilot scale.

References

Fan, J. and Q. Yao (1998), “Efficient Estimation of Conditional Variance Functions in Stochastic Regression,” Biometrika, 85, 645-660. doi:10.1093/biomet/85.3.645

Racine, J.S. and K. Li (2017), “Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach,” Journal of Econometrics, 201, 72-94. doi:10.1016/j.jeconom.2017.06.020

Racine, J.S. and I. Van Keilegom (2020), “A smooth nonparametric, multivariate, mixed-data location-scale test,” Journal of Business & Economic Statistics, 38, 784-795. doi:10.1080/07350015.2019.1574227

Author

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca