Univariate local-polynomial likelihood kernel density estimation

The estimators can handle data with bounded, unbounded, and discrete support, see Details.

kde1d(
  x,
  xmin = NaN,
  xmax = NaN,
  mult = 1,
  bw = NA,
  deg = 2,
  weights = numeric(0)
)

Arguments

x

vector (or one-column matrix/data frame) of observations; can be numeric or ordered.

xmin

lower bound for the support of the density (only for continuous data); NaN means no boundary.

xmax

upper bound for the support of the density (only for continuous data); NaN means no boundary.

mult

positive bandwidth multiplier; the actual bandwidth used is \(bw*mult\).

bw

bandwidth parameter; has to be a positive number or NA; the latter uses the plug-in methodology of Sheather and Jones (1991) with appropriate modifications for deg > 0.

deg

degree of the polynomial; either 0, 1, or 2 for log-constant, log-linear, and log-quadratic fitting, respectively.

weights

optional vector of weights for individual observations.

Value

An object of class kde1d.

Details

A gaussian kernel is used in all cases. If xmin or xmax are finite, the density estimate will be 0 outside of \([xmin, xmax]\). A log-transform is used if there is only one boundary (see, Geenens and Wang, 2018); a probit transform is used if there are two (see, Geenens, 2014).

Discrete variables are handled via jittering (see, Nagler, 2018a, 2018b). A specific form of deterministic jittering is used, see equi_jitter().

References

Geenens, G. (2014). Probit transformation for kernel density estimation on the unit interval. Journal of the American Statistical Association, 109:505, 346-358, arXiv:1303.4121

Geenens, G., Wang, C. (2018). Local-likelihood transformation kernel density estimation for positive random variables. Journal of Computational and Graphical Statistics, to appear, arXiv:1602.04862

Nagler, T. (2018a). A generic approach to nonparametric function estimation with mixed data. Statistics & Probability Letters, 137:326–330, arXiv:1704.07457

Nagler, T. (2018b). Asymptotic analysis of the jittering kernel density estimator. Mathematical Methods of Statistics, in press, arXiv:1705.05431

Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53, 683–690.

See also

dkde1d(), pkde1d(), qkde1d(), rkde1d(), plot.kde1d(), lines.kde1d()

Examples

## unbounded data
x <- rnorm(500) # simulate data
fit <- kde1d(x) # estimate density
dkde1d(0, fit) # evaluate density estimate
#> [1] 0.3957527
summary(fit) # information about the estimate
#> kernel density estimate ('kde1d'), log-quadratic
#> -----------------------------------------------------------------
#> nobs = 500, bw = 0.71, loglik = -712.05, d.f. = 4.65
plot(fit) # plot the density estimate
curve(dnorm(x),
  add = TRUE, # add true density
  col = "red"
)


## bounded data, log-linear
x <- rgamma(500, shape = 1) # simulate data
fit <- kde1d(x, xmin = 0, deg = 1) # estimate density
dkde1d(seq(0, 5, by = 1), fit) # evaluate density estimate
#> [1] 2.409557e+02 3.409745e-01 1.155638e-01 3.951467e-02 8.084613e-03
#> [6] 1.126153e-03
summary(fit) # information about the estimate
#> kernel density estimate ('kde1d'), log-linear with bounded support (xmin = 0)
#> -----------------------------------------------------------------
#> nobs = 500, bw = 0.3, loglik = -450.93, d.f. = 12.89
plot(fit) # plot the density estimate
curve(dgamma(x, shape = 1), # add true density
  add = TRUE, col = "red",
  from = 1e-3
)


## discrete data
x <- rbinom(500, size = 5, prob = 0.5) # simulate data
x <- ordered(x, levels = 0:5) # declare as ordered
fit <- kde1d(x) # estimate density
dkde1d(sort(unique(x)), fit) # evaluate density estimate
#> [1] 0.04736923 0.16091189 0.29017110 0.30269678 0.16344180 0.03540920
summary(fit) # information about the estimate
#> (jittered) kernel density estimate ('kde1d'), log-quadratic
#> -----------------------------------------------------------------
#> nobs = 500, bw = 1.12, loglik = -789.98, d.f. = 10.08
plot(fit) # plot the density estimate
points(ordered(0:5, 0:5), # add true density
  dbinom(0:5, 5, 0.5),
  col = "red"
)


## weighted estimate
x <- rnorm(100) # simulate data
weights <- rexp(100) # weights as in Bayesian bootstrap
fit <- kde1d(x, weights = weights) # weighted fit
plot(fit) # compare with unweighted fit
lines(kde1d(x), col = 2)