Package 'logspline' reference manual

Title:	Routines for Logspline Density Estimation
Description:	Contains routines for logspline density estimation. The function oldlogspline() uses the same algorithm as the logspline package version 1.0.x; i.e. the Kooperberg and Stone (1992) algorithm (with an improved interface). The recommended routine logspline() uses an algorithm from Stone et al (1997) <DOI:10.1214/aos/1031594728>.
Authors:	Charles Kooperberg [aut, cre], Cleve Moler [ctb] (LINPACK routines in src), Jack Dongarra [ctb] (LINPACK routines in src)
Maintainer:	Charles Kooperberg <[email protected]>
License:	Apache License 2.0
Version:	2.1.22
Built:	2024-09-08 06:13:10 UTC
Source:	CRAN

Logspline Density Estimation

Description

Density (dlogspline), cumulative probability (plogspline), quantiles (qlogspline), and random samples (rlogspline) from a logspline density that was fitted using the 1997 knot addition and deletion algorithm (logspline). The 1992 algorithm is available using the oldlogspline function.

Usage

dlogspline(q, fit, log = FALSE) 
plogspline(q, fit) 
qlogspline(p, fit) 
rlogspline(n, fit) dlogspline(q, fit, log = FALSE) 
plogspline(q, fit) 
qlogspline(p, fit) 
rlogspline(n, fit)

Arguments

`q`	vector of quantiles. Missing values (NAs) are allowed.
`p`	vector of probabilities. Missing values (NAs) are allowed.
`n`	sample size. If `length(n)` is larger than 1, then `length(n)` random values are returned.
`fit`	`logspline` object, typically the result of `logspline`.
`log`	should dlogspline return densities (TRUE) or log-densities (FALSE)

Details

Elements of q or p that are missing will cause the corresponding elements of the result to be missing.

Value

Densities (dlogspline), probabilities (plogspline), quantiles (qlogspline), or a random sample (rlogspline) from a logspline density that was fitted using knot addition and deletion.

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Charles J. Stone, Mark Hansen, Charles Kooperberg, and Young K. Truong. The use of polynomial splines and their tensor products in extended linear modeling (with discussion) (1997). Annals of Statistics, 25, 1371–1470.

Examples

x <- rnorm(100)
fit <- logspline(x)
qq <- qlogspline((1:99)/100, fit)
plot(qnorm((1:99)/100), qq)                  # qq plot of the fitted density
pp <- plogspline((-250:250)/100, fit)
plot((-250:250)/100, pp, type = "l")
lines((-250:250)/100,pnorm((-250:250)/100))  # asses the fit of the distribution
dd <- dlogspline((-250:250)/100, fit)
plot((-250:250)/100, dd, type = "l")
lines((-250:250)/100, dnorm((-250:250)/100)) # asses the fit of the density
rr <- rlogspline(100, fit)                   # random sample from fit
x <- rnorm(100)
fit <- logspline(x)
qq <- qlogspline((1:99)/100, fit)
plot(qnorm((1:99)/100), qq)                  # qq plot of the fitted density
pp <- plogspline((-250:250)/100, fit)
plot((-250:250)/100, pp, type = "l")
lines((-250:250)/100,pnorm((-250:250)/100))  # asses the fit of the distribution
dd <- dlogspline((-250:250)/100, fit)
plot((-250:250)/100, dd, type = "l")
lines((-250:250)/100, dnorm((-250:250)/100)) # asses the fit of the density
rr <- rlogspline(100, fit)                   # random sample from fit

Logspline Density Estimation - 1992 version

Description

Probability density function (doldlogspline), distribution function (poldlogspline), quantiles (qoldlogspline), and random samples (roldlogspline) from a logspline density that was fitted using the 1992 knot deletion algorithm (oldlogspline). The 1997 algorithm using knot deletion and addition is available using the logspline function.

Usage

doldlogspline(q, fit) 
poldlogspline(q, fit) 
qoldlogspline(p, fit) 
roldlogspline(n, fit) doldlogspline(q, fit) 
poldlogspline(q, fit) 
qoldlogspline(p, fit) 
roldlogspline(n, fit)

Arguments

`q`	vector of quantiles. Missing values (NAs) are allowed.
`p`	vector of probabilities. Missing values (NAs) are allowed.
`n`	sample size. If `length(n)` is larger than 1, then `length(n)` random values are returned.
`fit`	`oldlogspline` object, typically the result of `oldlogspline`.

Details

Elements of q or p that are missing will cause the corresponding elements of the result to be missing.

Value

Densities (doldlogspline), probabilities (poldlogspline), quantiles (qoldlogspline), or a random sample (roldlogspline) from an oldlogspline density that was fitted using knot deletion.

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

x <- rnorm(100)
fit <- oldlogspline(x)
qq <- qoldlogspline((1:99)/100, fit)
plot(qnorm((1:99)/100), qq)                  # qq plot of the fitted density
pp <- poldlogspline((-250:250)/100, fit)
plot((-250:250)/100, pp, type = "l")
lines((-250:250)/100, pnorm((-250:250)/100)) # asses the fit of the distribution
dd <- doldlogspline((-250:250)/100, fit)
plot((-250:250)/100, dd, type = "l")
lines((-250:250)/100, dnorm((-250:250)/100)) # asses the fit of the density
rr <- roldlogspline(100, fit)                # random sample from fit
x <- rnorm(100)
fit <- oldlogspline(x)
qq <- qoldlogspline((1:99)/100, fit)
plot(qnorm((1:99)/100), qq)                  # qq plot of the fitted density
pp <- poldlogspline((-250:250)/100, fit)
plot((-250:250)/100, pp, type = "l")
lines((-250:250)/100, pnorm((-250:250)/100)) # asses the fit of the distribution
dd <- doldlogspline((-250:250)/100, fit)
plot((-250:250)/100, dd, type = "l")
lines((-250:250)/100, dnorm((-250:250)/100)) # asses the fit of the density
rr <- roldlogspline(100, fit)                # random sample from fit

Logspline Density Estimation

Description

Fits a logspline density using splines to approximate the log-density using the 1997 knot addition and deletion algorithm (logspline). The 1992 algorithm is available using the oldlogspline function.

Usage

logspline(x, lbound, ubound, maxknots = 0, knots, nknots = 0, penalty,
silent = TRUE, mind = -1, error.action = 2) logspline(x, lbound, ubound, maxknots = 0, knots, nknots = 0, penalty,
silent = TRUE, mind = -1, error.action = 2)

Arguments

`x`	data vector. The data needs to be uncensored. `oldlogspline` can deal with right- left- and interval-censored data.
`lbound`, `ubound`	lower/upper bound for the support of the density. For example, if there is a priori knowledge that the density equals zero to the left of 0, and has a discontinuity at 0, the user could specify `lbound = 0`. However, if the density is essentially zero near 0, one does not need to specify `lbound`.
`maxknots`	the maximum number of knots. The routine stops adding knots when this number of knots is reached. The method has an automatic rule for selecting maxknots if this parameter is not specified.
`knots`	ordered vector of values (that should cover the complete range of the observations), which forces the method to start with these knots. Overrules knots. If `knots` is not specified, a default knot-placement rule is employed.
`nknots`	forces the method to start with `nknots` knots. The method has an automatic rule for selecting `nknots` if this parameter is not specified.
`penalty`	the parameter to be used in the AIC criterion. The method chooses the number of knots that minimizes `-2 * loglikelihood + penalty * (number of knots - 1)`. The default is to use a penalty parameter of `penalty = log(samplesize)` as in BIC. The effect of this parameter is summarized in `summary.logspline`.
`silent`	should diagnostic output be printed?
`mind`	minimum distance, in order statistics, between knots.
`error.action`	how should `logspline` deal with non-convergence problems? Very-very rarely in some extreme situations `logspline` has convergence problems. The only two situations that I am aware of are when there is effectively a sharp bound, but this bound was not specified, or when the data is severly rounded. `logspline` can deal with this in three ways. If `error.action` is 2, the same data is rerun with the slightly more stable, but less flexible `oldlogspline`. The object is translated in a `logspline` object using `oldlogspline.to.logspline`, so this is almost invisible to the user. It is particularly useful when you run simulation studies, as he code can seemlessly continue. Only the `lbound` and `ubound` options are passed on to `oldlogspline`, other options revert to the default. If `error.action` is 1, a warning is printed, and `logspline` returns nothing (but does not crash). This is useful if you run a simulation, but do not like to revert to `oldlogspline`. If `error.action` is 0, the code crashes using the `stop` function.

Value

Object of the class logspline, that is intended as input for plot.logspline (summary plots), summary.logspline (fitting summary), dlogspline (densities), plogspline (probabilities), qlogspline (quantiles), rlogspline (random numbers from the fitted distribution).

The object has the following members:

`call`	the command that was executed.
`nknots`	the number of knots in the model that was selected.
`coef.pol`	coefficients of the polynomial part of the spline. The first coefficient is the constant term and the second is the linear term.
`coef.kts`	coefficients of the knots part of the spline. The `k`-th element is the coefficient of $(x-t(k))^3_+$ (where $x^3_+$ means the positive part of the third power of $x$ , and $t(k)$ means knot `k`).
`knots`	vector of the locations of the knots in the `logspline` model.
`maxknots`	the largest number of knots minus one considered during fitting (i.e. with `maxknots = 6` the maximum number of knots is 5).
`penalty`	the penalty that was used.
`bound`	first element: 0 - `lbound` was $-\inf$ 1 it was something else; second element: `lbound`, if specified; third element: 0 - `ubound` was $\inf$ , 1 it was something else; fourth element: `ubound`, if specified.
`samples`	the sample size.
`logl`	matrix with 3 columns. Column one: number of knots; column two: model fitted during addition (1) or deletion (2); column 3: log-likelihood.
`range`	range of the input data.
`mind`	minimum distance in order statistics between knots required during fitting (the actual minimum distance may be much larger).

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

y <- rnorm(100)
fit <- logspline(y)       
plot(fit)
#
# as (4 == length(-2, -1, 0, 1, 2) -1), this forces these initial knots,
# and does no knot selection
fit <- logspline(y, knots = c(-2, -1, 0, 1, 2), maxknots = 4, penalty = 0)  
#
# the following example give one of the rare examples where logspline
# crashes, and this shows the use of error.action = 2.
#
set.seed(118)
zz <- rnorm(300)
zz[151:300] <- zz[151:300]+5
zz <- round(zz)
fit <- logspline(zz)
#
# you could rerun this with 
# fit <- logspline(zz, error.action=0)
# or
# fit <- logspline(zz, error.action=1)
y <- rnorm(100)
fit <- logspline(y)       
plot(fit)
#
# as (4 == length(-2, -1, 0, 1, 2) -1), this forces these initial knots,
# and does no knot selection
fit <- logspline(y, knots = c(-2, -1, 0, 1, 2), maxknots = 4, penalty = 0)  
#
# the following example give one of the rare examples where logspline
# crashes, and this shows the use of error.action = 2.
#
set.seed(118)
zz <- rnorm(300)
zz[151:300] <- zz[151:300]+5
zz <- round(zz)
fit <- logspline(zz)
#
# you could rerun this with 
# fit <- logspline(zz, error.action=0)
# or
# fit <- logspline(zz, error.action=1)

Logspline Density Estimation - 1992 version

Description

Fits a logspline density using splines to approximate the log-density using the 1992 knot deletion algorithm (oldlogspline). The 1997 algorithm using knot deletion and addition is available using the logspline function.

Usage

oldlogspline(uncensored, right, left, interval, lbound,
ubound, nknots, knots, penalty, delete = TRUE) oldlogspline(uncensored, right, left, interval, lbound,
ubound, nknots, knots, penalty, delete = TRUE)

Arguments

`uncensored`	vector of uncensored observations from the distribution whose density is to be estimated. If there are no uncensored observations, this argument can be omitted. However, either `uncensored` or `interval` must be specified.
`right`	vector of right censored observations from the distribution whose density is to be estimated. If there are no right censored observations, this argument can be omitted.
`left`	vector of left censored observations from the distribution whose density is to be estimated. If there are no left censored observations, this argument can be omitted.
`interval`	two column matrix of lower and upper bounds of observations that are interval censored from the distribution whose density is to be estimated. If there are no interval censored observations, this argument can be omitted.
`lbound`, `ubound`	lower/upper bound for the support of the density. For example, if there is a priori knowledge that the density equals zero to the left of 0, and has a discontinuity at 0, the user could specify `lbound = 0`. However, if the density is essentially zero near 0, one does not need to specify `lbound`. The default for `lbound` is `-inf` and the default for `ubound` is `inf`.
`nknots`	forces the method to start with nknots knots (`delete = TRUE`) or to fit a density with nknots knots (`delete = FALSE`). The method has an automatic rule for selecting nknots if this parameter is not specified.
`knots`	ordered vector of values (that should cover the complete range of the observations), which forces the method to start with these knots (`delete = TRUE`) or to fit a density with these knots `delete = FALSE`). Overrules `nknots`. If `knots` is not specified, a default knot-placement rule is employed.
`penalty`	the parameter to be used in the AIC criterion. The method chooses the number of knots that minimizes `-2 * loglikelihood + penalty * (number of knots - 1)`. The default is to use a penalty parameter of `penalty = log(samplesize)` as in BIC. The effect of this parameter is summarized in `summary.oldlogspline`.
`delete`	should stepwise knot deletion be employed?

Value

Object of the class oldlogspline, that is intended as input for plot.oldlogspline, summary.oldlogspline, doldlogspline (densities), poldlogspline (probabilities),
qoldlogspline (quantiles), roldlogspline (random numbers from the fitted distribution). The function oldlogspline.to.logspline can translate an object of the class oldlogspline to an object of the class logspline.

The object has the following members:

`call`	the command that was executed.
`knots`	vector of the locations of the knots in the `oldlogspline` model. old
`coef`	coefficients of the spline. The first coefficient is the constant term, the second is the linear term and the k-th $(k>2)$ is the coefficient of $(x-t(k-2))^3_+$ (where $x^3_+$ means the positive part of the third power of $x$ , and $t(k-2)$ means knot $k-2$ ). If a coefficient is zero the corresponding knot was deleted from the model.
`bound`	first element: 0 - `lbound` was $-\inf$ 1 it was something else; second element: `lbound`, if specified; third element: 0 - `ubound` was $\inf$ , 1 it was something else; fourth element: `ubound`, if specified.
`logl`	the `k`-th element is the log-likelihood of the fit with `k+2` knots.
`penalty`	the penalty that was used.
`sample`	the sample size that was used.
`delete`	was stepwise knot deletion employed?

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

# A simple example
y <- rnorm(100)
fit <- oldlogspline(y)       
plot(fit)
# An example involving censoring and a lower bound
y <- rlnorm(1000)
censoring <- rexp(1000) * 4
delta <- 1 * (y <= censoring)
y[delta == 0] <- censoring[delta == 0]
fit <- oldlogspline(y[delta == 1], y[delta == 0], lbound = 0)
# A simple example
y <- rnorm(100)
fit <- oldlogspline(y)       
plot(fit)
# An example involving censoring and a lower bound
y <- rlnorm(1000)
censoring <- rexp(1000) * 4
delta <- 1 * (y <= censoring)
y[delta == 0] <- censoring[delta == 0]
fit <- oldlogspline(y[delta == 1], y[delta == 0], lbound = 0)

Logspline Density Estimation - 1992 to 1997 version

Description

Translates an oldlogspline object in an logspline object. This routine is mostly used in logspline, as it allows the routine to use oldlogspline for some situations where logspline crashes. The other use is when you have censored data, and thus have to use oldlogspline to fit, but wish to use the auxiliary routines from logspline.

Usage

oldlogspline.to.logspline(obj, data) oldlogspline.to.logspline(obj, data)

Arguments

`obj`	object of class `logspline`
`data`	the original data. Used to compute the `range` component of the new object. if `data` is not available, the 1/(n+1) and n/(n+1) quantiles of the fitted distribution are used for `range`.

Value

object of the class logspline. The call component of the new object is not useful. The delete component of the old object is ignored.

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

x <- rnorm(100)
fit.old <- oldlogspline(x)
fit.translate <- oldlogspline.to.logspline(fit.old,x)
fit.new <- logspline(x)
plot(fit.new)
plot(fit.old,add=TRUE,col=2)
#
# should look almost the same, the differences are the
# different fitting routines
#
x <- rnorm(100)
fit.old <- oldlogspline(x)
fit.translate <- oldlogspline.to.logspline(fit.old,x)
fit.new <- logspline(x)
plot(fit.new)
plot(fit.old,add=TRUE,col=2)
#
# should look almost the same, the differences are the
# different fitting routines
#

Logspline Density Estimation

Description

Plots a logspline density, distribution function, hazard function or survival function from a logspline density that was fitted using the 1997 knot addition and deletion algorithm (logspline). The 1992 algorithm is available using the oldlogspline function.

Usage

## S3 method for class 'logspline'
plot(x, n = 100, what = "d", add = FALSE, xlim, xlab = "",
ylab = "", type = "l", ...) 
## S3 method for class 'logspline'
plot(x, n = 100, what = "d", add = FALSE, xlim, xlab = "",
ylab = "", type = "l", ...)

Arguments

`x`	`logspline` object, typically the result of `logspline`.
`n`	the number of equally spaced points at which to plot the density.
`what`	what should be plotted: `"d"` (density), `"p"` (distribution function), `"s"` (survival function) or `"h"` (hazard function).
`add`	should the plot be added to an existing plot.
`xlim`	range of data on which to plot. Default is from the 1th to the 99th percentile of the density, extended by 10% on each end.
`xlab`, `ylab`	labels plotted on the axes.
`type`	type of plot.
`...`	other plotting options, as desired

Details

This function produces a plot of a logspline fit at n equally spaced points roughly covering the support of the density. (Use xlim = c(from, to) to change the range of these points.)

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

y <- rnorm(100)
fit <- logspline(y)       
plot(fit) 
y <- rnorm(100)
fit <- logspline(y)       
plot(fit)

Logspline Density Estimation - 1992 version

Description

Plots an oldlogspline density, distribution function, hazard function or survival function from a logspline density that was fitted using the 1992 knot deletion algorithm. The 1997 algorithm using knot deletion and addition is available using the logspline function.

Usage

## S3 method for class 'oldlogspline'
plot(x, n = 100, what = "d", xlim, xlab = "", ylab = "",
type = "l", add = FALSE, ...) ## S3 method for class 'oldlogspline'
plot(x, n = 100, what = "d", xlim, xlab = "", ylab = "",
type = "l", add = FALSE, ...)

Arguments

`x`	`logspline` object, typically the result of `logspline`.
`n`	the number of equally spaced points at which to plot the density.
`what`	what should be plotted: `"d"` (density), `"p"` (distribution function), `"s"` (survival function) or `"h"` (hazard function).
`xlim`	range of data on which to plot. Default is from the 1th to the 99th percentile of the density, extended by 10% on each end.
`xlab`, `ylab`	labels plotted on the axes.
`type`	type of plot.
`add`	should the plot be added to an existing plot.
`...`	other plotting options, as desired

Details

This function produces a plot of a oldlogspline fit at n equally spaced points roughly covering the support of the density. (Use xlim=c(from,to) to change the range of these points.)

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

y <- rnorm(100)
fit <- oldlogspline(y)       
plot(fit) 
y <- rnorm(100)
fit <- oldlogspline(y)       
plot(fit)

Logspline Density Estimation

Description

This function summarizes both the stepwise selection process of the model fitting by logspline, as well as the final model that was selected using AIC/BIC. A logspline object was fit using the 1997 knot addition and deletion algorithm. The 1992 algorithm is available using the oldlogspline function.

Usage

## S3 method for class 'logspline'
summary(object, ...) 
## S3 method for class 'logspline'
print(x, ...) ## S3 method for class 'logspline'
summary(object, ...) 
## S3 method for class 'logspline'
print(x, ...)

Arguments

`object`, `x`	`logspline` object, typically the result of `logspline`
`...`	other arguments are ignored.

Details

These function produce identical printed output. The main body is a table with five columns: the first column is a possible number of knots for the fitted model;

the second column is the log-likelihood for the fit;

the third column is -2 * loglikelihood + penalty * (number of knots - 1), which is the AIC criterion; logspline selected the model with the smallest value of AIC;

the fourth and fifth columns give the endpoints of the interval of values of penalty that would yield the model with the indicated number of knots. (NAs imply that the model is not optimal for any choice of penalty.) At the bottom of the table the number of knots corresponding to the selected model is reported, as is the value of penalty that was used.

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

y <- rnorm(100)
fit <- logspline(y)       
summary(fit) 
y <- rnorm(100)
fit <- logspline(y)       
summary(fit)

Logspline Density Estimation - 1992 version

Description

This function summarizes both the stepwise selection process of the model fitting by oldlogspline, as well as the final model that was selected using AIC/BIC. A logspline object was fit using the 1992 knot deletion algorithm (oldlogspline). The 1997 algorithm using knot deletion and addition is available using the logspline function.

Usage

## S3 method for class 'oldlogspline'
summary(object, ...) 
## S3 method for class 'oldlogspline'
print(x, ...)## S3 method for class 'oldlogspline'
summary(object, ...) 
## S3 method for class 'oldlogspline'
print(x, ...)

Arguments

`object`, `x`	`oldlogspline` object, typically the result of `oldlogspline`
`...`	other arguments are ignored.

Details

These function produces the same printed output. The main body is a table with five columns: the first column is a possible number of knots for the fitted model;

the second column is the log-likelihood for the fit;

the third column is -2 * loglikelihood + penalty * (number of knots - 1), which is the AIC criterion; logspline selected the model with the smallest value of AIC;

Author(s)

Charles Kooperberg [email protected].

References

Charles Kooperberg and Charles J. Stone. Logspline density estimation for censored data (1992). Journal of Computational and Graphical Statistics, 1, 301–328.

Examples

y <- rnorm(100)
fit <- oldlogspline(y)       
summary(fit) 
y <- rnorm(100)
fit <- oldlogspline(y)       
summary(fit)

Reformat data as vector or matrix

Description

This function tries to convert a date.frame or a matrix to a no-frills matrix without labels, and a vector or time-series to a no-frills vector without labels.

Usage

unstrip(x) unstrip(x)

Arguments

`x`	one- or two-dimensional object.

Details

Many of the functions for logspline, oldlogspline, were written in the “before data.frame” era; unstrip attempts to keep all these functions useful with more advanced input objects. In particular, many of these functions call unstrip before doing anything else.

Value

If x is two-dimensional a matrix without names, if x is one-dimensional a numerical vector

Author(s)

Charles Kooperberg [email protected].

Examples

data(co2)
unstrip(co2)
data(iris)
unstrip(iris)
data(co2)
unstrip(co2)
data(iris)
unstrip(iris)

Package 'logspline'

Help Index

Logspline Density Estimation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Logspline Density Estimation - 1992 version

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Logspline Density Estimation

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Logspline Density Estimation - 1992 version

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Logspline Density Estimation - 1992 to 1997 version

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Logspline Density Estimation

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

Logspline Density Estimation - 1992 version

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

Logspline Density Estimation

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

Logspline Density Estimation - 1992 version

Description

Usage

Arguments