opt_w: Optimal weights
In koohyun-kwon/HTEBand: Uniform confidence band for heterogenous treatment effects

Solve for optimal weigths by the quantile matching method

opt_w(
  method,
  C.vec,
  y,
  x,
  d = NULL,
  eval,
  T.grad.mat,
  level,
  deg,
  kern,
  M,
  seed = NULL,
  useloop = TRUE,
  root.robust = FALSE,
  ng = 10,
  resid.1 = NULL,
  resid.0 = NULL,
  var.reg = "npr",
  c.method = "supp2",
  c.supp = NULL
)

`method`	character string used to define the parameter of interest and the function space. Possible choices are "reg.Hol" Regression function values under Hölder space "reg.Lip" Regression function values under Lipschitz space "TE.Lip" CATE under Lipschitz space "TE.Lip.eqbw" CATE under Lipschitz space using the same bandwidths for the treatment and the control groups "TE.Lip" CATE under Hölder space "TE.Lip.eqbw" CATE under Hölder space using the same bandwidths for the treatment and the control groups
`C.vec`	smoothness parameter for the function space; possibly a vector.
`y`	dependent variable; possibly a matrix with `nrow(y)` equals the number of observations.
`x`	a vector of independent variable
`d`	a vector of treatment indicator; it can be arbitrarily specified when not used.
`eval`	a vector of indices
`T.grad.mat`	A `n.T` by `k` dimensional gradient matrix of `T_t f` for `t` = 1,..., `n.T`.
`level`	level of quantile
`deg`	degree of local polynomial regression to be used
`kern`	kernel used to calculate conditional variance function; supports `"tri"`, `"epa"`, `"uni"`, and `"gau"`. Default is `kern = "epa"`.
`M`	number of bootstrap simulations
`seed`	seed for the random number generation; default is `seed = NULL`.
`useloop`	If `TRUE`, the function is implemented by `for` loop over `t` = 1,..., `n.T`; currently only `useloop = TRUE` is implemented.
`root.robust`	if `TRUE`, the fuction conducts diagnostic test whether optimizaiton worked well; default is `root.robust = FALSE`. Currently, this feature is obsolete.
`ng`	the number of grids of quantile values over which the diagnostic test would be peformed if `root.robust = TRUE`; default is `ng = 10`. Currently, this feature is obsolete.
`resid.1`	residuals corresponding to the treated observations; it can be a vector or a matrix if `ncol(y) > 1`. For treatment effect models, its ith component corresponds to the ith component of `y[d == 1, ]`. If there are no treatment and control groups, `resid.1` corresponds to residuals for the entire observations. It can be specified to be `NULL`.
`resid.0`	residuals corresponding to the control observations; it can be a vector or a matrix if `ncol(y) > 1`. Its ith component corresponds to the ith component of `y[d == 0, ]`. It can be specified to be `NULL`.
`var.reg`	nonparametric regression method used to calculate residuals; either `"npr"` (standing for `nprobust` package) or `"locpol"` (standing for `locpol` package). Default is `var.reg = "npr"`.
`c.method`	method to calculate the optimal value of c_n. Currently, using the default value `c.method = "supp2"` is recommended and other options are obsolete.
`c.supp`	value of c_n to be used when `c.method = "supp"`; default is `c.supp = NULL`.

a list of the following components

w.1: a 3-dim array of weigths corresponding to the treated observations; the second and third dimension correponds to ncol(y) and length(eval)
w.0: a 3-dim array of weigths corresponding to the control observations; the second and third dimension correponds to ncol(y) and length(eval). If there is no control observations, returns array of 0.
c.root: the quantile value corresponding to the optimal weigths.
increasing: boolean value testing whether the objective function is increasing or not; provided only if root.robust = TRUE
opt.grid: a data.frame object containing evaluations of the objective function over grid values of quantiles; provided only if root.robust = TRUE