Kernel Shape Constrained Bounded Univariate Density Estimation

npuniden.sc computes shape constrained kernel univariate unconditional density estimates given a vector of continuously distributed training data and a bandwidth. Lower and upper bounds [a,b] can be supplied (default is [0,1]) and if a is set to -Inf there is only one bound on the right, while if b is set to Inf there is only one bound on the left.

npuniden.sc(X = NULL,
            Y = NULL,
            h = NULL,
            a = 0,
            b = 1,
            lb = NULL,
            ub = NULL,
            extend.range = 0,
            num.grid = 0,
            function.distance = TRUE,
            integral.equal = FALSE,
            constraint = c("density",
                           "mono.incr",
                           "mono.decr",
                           "concave",
                           "convex",
                           "log-concave",
                           "log-convex"))

Arguments

X: a required numeric vector of training data lying in \([a,b]\)
Y: an optional numeric vector of evaluation data lying in \([a,b]\)
h: a bandwidth (\(>0\))
a: an optional lower bound on the support of X or Y (defaults to 0)
b: an optional upper bound on the support of X or Y (defaults to 1)
lb: a scalar lower bound (\(\ge 0\)) to be used in conjunction with constraint="density"
ub: a scalar upper bound (\(\ge 0\) and \(\ge\) lb) to be used in conjunction with constraint="density"
extend.range: number specifying the fraction by which the range of the training data should be extended for the additional grid points (passed to the function extendrange)
num.grid: number of additional grid points (in addition to X and Y) placed on an equi-spaced grid spanning extendrange(c(X,Y),f=extend.range) (if num.grid=0 no additional grid points will be used regardless of the value of extend.range)
function.distance: a logical value that, if TRUE, minimizes the squared deviation between the constrained and unconstrained estimates, otherwise, minimizes the squared deviation between the constrained and unconstrained weights
integral.equal: a logical value, that, if TRUE, adjusts the constrained estimate to have the same probability mass over the range X, Y, and the additional grid points
constraint: a character string indicating whether the estimate is to be constrained to be monotonically increasing (constraint="mono.incr"), decreasing (constraint="mono.decr"), convex (constraint="convex"), concave (constraint="concave"), log-convex (constraint="log-convex"), or log-concave (constraint="log-concave")

Details

Typical usages are (see below for a complete list of options and also the examples at the end of this help file)


    model <- npuniden.sc(X,a=-2,b=3)

npuniden.sc implements a methods for estimating a univariate density function defined over a continuous random variable in the presence of bounds subject to a variety of shape constraints. The bounded estimates use the truncated Gaussian kernel function.

Note that for the log-constrained estimates, the derivative estimate returned is that for the log-constrained estimate not the non-log value of the estimate returned by the function. See Example 5 below hat manually plots the log-density and returned derivative (no transformation is needed when plotting the density estimate itself).

If the quadratic program solver fails to find a solution, the unconstrained estimate is returned with an immediate warning. Possible causes to be investigated are undersmoothing, sparsity, and the presence of non-sample grid points. To investigate the possibility of undersmoothing try using a larger bandwidth, to investigate sparsity try decreasing extend.range, and to investigate non-sample grid points try setting num.grid to 0.

Mean square error performance seems to improve generally when using additional grid points in the empirical support of X and Y (i.e., in the observed range of the data sample) but appears to deteriorate when imposing constraints beyond the empirical support (i.e., when extend.range is positive). Increasing the number of additional points beyond a hundred or so appears to have a limited impact.

The option function.distance=TRUE appears to perform better for imposing convexity, concavity, log-convexity and log-concavity, while function.distance=FALSE appears to perform better for imposing monotonicity, whether increasing or decreasing (based on simulations for the Beta(s1,s2) distribution with sample size \(n=100\)).

Value

A list with the following elements:

f: unconstrained density estimate
f.sc: shape constrained density estimate
se.f: asymptotic standard error of the unconstrained density estimate
se.f.sc: asymptotic standard error of the shape constrained density estimate
f.deriv: unconstrained derivative estimate (of order 1 or 2 or log thereof)
f.sc.deriv: shape constrained derivative estimate (of order 1 or 2 or log thereof)
F: unconstrained distribution estimate
F.sc: shape constrained distribution estimate
integral.f: the integral of the unconstrained estimate over X, Y, and the additional grid points
integral.f.sc: the integral of the constrained estimate over X, Y, and the additional grid points
solve.QP: logical, if TRUE solve.QP succeeded, otherwise failed
attempts: number of attempts when solve.QP fails (max = 9)

References

Du, P. and C. Parmeter and J. Racine (2024), “Shape Constrained Kernel PDF and PMF Estimation”, Statistica Sinica, 34 (1), 257–289, https://doi.org/10.5705/ss.202021.0112

Author

Jeffrey S. Racine racinej@mcmaster.ca

Examples

if (FALSE) { # \dontrun{
n <- 100
set.seed(42)

## Example 1: N(0,1), constrain the density to lie within lb=.1 and ub=.2

X <- sort(rnorm(n))
h <- npuniden.boundary(X,a=-Inf,b=Inf)$h
foo <- npuniden.sc(X,h=h,constraint="density",a=-Inf,b=Inf,lb=.1,ub=.2)
ylim <- range(c(foo$f.sc,foo$f))
plot(X,foo$f.sc,type="l",ylim=ylim,xlab="X",ylab="Density")
lines(X,foo$f,col=2,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

## Example 2: Beta(5,1), DGP is monotone increasing, impose valid
## restriction

X <- sort(rbeta(n,5,1))
h <- npuniden.boundary(X)$h

foo <- npuniden.sc(X=X,h=h,constraint=c("mono.incr"))

par(mfrow=c(1,2))
ylim <- range(c(foo$f.sc,foo$f))
plot(X,foo$f.sc,type="l",ylim=ylim,xlab="X",ylab="Density")
lines(X,foo$f,col=2,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

ylim <- range(c(foo$f.sc.deriv,foo$f.deriv))
plot(X,foo$f.sc.deriv,type="l",ylim=ylim,xlab="X",ylab="First Derivative")
lines(X,foo$f.deriv,col=2,lty=2)
abline(h=0,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

## Example 3: Beta(1,5), DGP is monotone decreasing, impose valid
## restriction

X <- sort(rbeta(n,1,5))
h <- npuniden.boundary(X)$h

foo <- npuniden.sc(X=X,h=h,constraint=c("mono.decr"))

par(mfrow=c(1,2))
ylim <- range(c(foo$f.sc,foo$f))
plot(X,foo$f.sc,type="l",ylim=ylim,xlab="X",ylab="Density")
lines(X,foo$f,col=2,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

ylim <- range(c(foo$f.sc.deriv,foo$f.deriv))
plot(X,foo$f.sc.deriv,type="l",ylim=ylim,xlab="X",ylab="First Derivative")
lines(X,foo$f.deriv,col=2,lty=2)
abline(h=0,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")
 
## Example 4: N(0,1), DGP is log-concave, impose invalid concavity
## restriction

X <- sort(rnorm(n))
h <- npuniden.boundary(X,a=-Inf,b=Inf)$h

foo <- npuniden.sc(X=X,h=h,a=-Inf,b=Inf,constraint=c("concave"))

par(mfrow=c(1,2))
ylim <- range(c(foo$f.sc,foo$f))
plot(X,foo$f.sc,type="l",ylim=ylim,xlab="X",ylab="Density")
lines(X,foo$f,col=2,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")
ylim <- range(c(foo$f.sc.deriv,foo$f.deriv))

plot(X,foo$f.sc.deriv,type="l",ylim=ylim,xlab="X",ylab="Second Derivative")
lines(X,foo$f.deriv,col=2,lty=2)
abline(h=0,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

## Example 45: Beta(3/4,3/4), DGP is convex, impose valid restriction

X <- sort(rbeta(n,3/4,3/4))
h <- npuniden.boundary(X)$h

foo <- npuniden.sc(X=X,h=h,constraint=c("convex"))

par(mfrow=c(1,2))
ylim <- range(c(foo$f.sc,foo$f))
plot(X,foo$f.sc,type="l",ylim=ylim,xlab="X",ylab="Density")
lines(X,foo$f,col=2,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

ylim <- range(c(foo$f.sc.deriv,foo$f.deriv))
plot(X,foo$f.sc.deriv,type="l",ylim=ylim,xlab="X",ylab="Second Derivative")
lines(X,foo$f.deriv,col=2,lty=2)
abline(h=0,lty=2)
rug(X)
legend("topleft",c("Constrained","Unconstrained"),lty=1:2,col=1:2,bty="n")

## Example 6: N(0,1), DGP is log-concave, impose log-concavity
## restriction

X <- sort(rnorm(n))
h <- npuniden.boundary(X,a=-Inf,b=Inf)$h

foo <- npuniden.sc(X=X,h=h,a=-Inf,b=Inf,constraint=c("log-concave"))

par(mfrow=c(1,2))

ylim <- range(c(log(foo$f.sc),log(foo$f)))
plot(X,log(foo$f.sc),type="l",ylim=ylim,xlab="X",ylab="Log-Density")
lines(X,log(foo$f),col=2,lty=2)
rug(X)
legend("topleft",c("Constrained-log","Unconstrained-log"),lty=1:2,col=1:2,bty="n")

ylim <- range(c(foo$f.sc.deriv,foo$f.deriv))
plot(X,foo$f.sc.deriv,type="l",ylim=ylim,xlab="X",ylab="Second Derivative of Log-Density")
lines(X,foo$f.deriv,col=2,lty=2)
abline(h=0,lty=2)
rug(X)
legend("topleft",c("Constrained-log","Unconstrained-log"),lty=1:2,col=1:2,bty="n")
} # }