R code for Least-squares density estimation

# LEAST-SQUARES DENSITY ESTIMATION WITH SHAPE RESTRICTIONS

Smooth Decreasing Density Estimation

Given a sample x1,...,xn from a decreasing density f, this code provides a smooth decreasing estimate using regression splines.

The knot choices can be provide by the user or the "default" or automatic choices may be used.

If chosen by the user: The estimate will be over the interval [a,b] where the user-defined knots are at a=t0< t1 < ...< tk< tk+1=b.

The inputs to the code are the sample values and the knots. If knots=0, the default knot values will be used with the mode at the origin.

The outputs are the knots, the value of the criterion function, the values of the density estimate at the observed x-values, and values of the density estimate at a tightly-spaced grid of points for plotting.

### R code for smooth decreasing density

Here is some example code where the default knots are chosen.

### Example implementation of R code for estimating a smooth decreasing density

Smooth Decreasing Convex Density Estimation

Given a sample x1,...,xn from a decreasing convex density f on [a,b] and knots a=t0< t1 < ...< tk< tk+1, f is estimated using cubic regression splines

The knot choices can be provide by the user or the "default" or automatic choices may be used.

If chosen by the user: The estimate will be over the interval [a,b] where the user-defined knots are at a=t0< t1 < ...< tk< tk+1=b.

The inputs to the code are the sample values and the knots. If knots=0, the default knot values will be used with the mode at the origin.

The outputs are the knots, the value of the criterion function, the values of the density estimate at the observed x-values, and values of the density estimate at a tightly-spaced grid of points for plotting.

### R code for smooth decreasing convex density

Here is some example code where the default knots are chosen.

### Example implementation of R code for estimating a smooth decreasing convex density

Smooth Unimodal Density Estimation

Given a sample x1,...,xn from a unimodal density f on [a,b] and mode m, f is estimated using quadratic regression splines. Knots are placed evenly over [a,b], then extra knots are added where the data are "thickest."

### R code for estimating a smooth unimodal density

Here is some example code.

### Example implementation of R code for estimating a smooth unimodal density

meyer@stat.colostate.edu