Kernel Consistent Univariate Density Equality Test with Mixed Data Types

npunitest implements the consistent metric entropy test of Maasoumi and Racine (2002) for two arbitrary, stationary univariate nonparametric densities on common support.

Usage

npunitest(data.x = NULL,
          data.y = NULL,
          method = c("integration","summation"),
          bootstrap = TRUE,
          boot.num = 399,
          bw.x = NULL,
          bw.y = NULL,
          random.seed = 42,
          ...)

Arguments

Data, Bandwidth Inputs And Formula Interface

These arguments identify the two samples, statistic variant, and any supplied bandwidths.

bw.x, bw.y: numeric (scalar) bandwidths. Defaults to plug-in (see details below).
data.x, data.y: common support univariate vectors containing the variables.
method: a character string used to specify whether to compute the integral version or the summation version of the statistic. Can be set as integration or summation. Defaults to integration. See ‘Details’ below for important information regarding the use of summation when data.x and data.y lack common support and/or are sparse.

Bootstrap Controls

These arguments control bootstrap execution and reproducibility settings.

boot.num: an integer value specifying the number of bootstrap replications to use. Defaults to 399.
bootstrap: a logical value which specifies whether to conduct the bootstrap test or not. If set to FALSE, only the statistic will be computed. Defaults to TRUE.
random.seed: an integer used to seed R's random number generator. This is to ensure replicability. Defaults to 42.

Additional Arguments

Further arguments are passed to the bandwidth-selection routines used by the test.

...: additional arguments supplied to specify the bandwidth type, kernel types, and so on. This is used since we specify bw as a numeric scalar and not a bandwidth object, and is of interest if you do not desire the default behaviours. To change the defaults, you may specify any of bwscaling, bwtype, ckertype, ckerorder, ukertype, okertype.

Value

npunitest returns an object of type unitest with the following components

Srho: the statistic Srho
Srho.bootstrap: contains the bootstrap replications of Srho
P: the P-value of the statistic
boot.num: number of bootstrap replications
bw.x, bw.y: scalar bandwidths for data.x, data.y

summary supports object of type unitest.

Book And Method Pointers

npunitest uses an entropy-distance measure to compare univariate density or probability functions. The statistic is based on the normalized Hellinger-type distance described in the entropy-test literature referenced below.

For book-length background, see Racine (2019), Chapter 2 Continuous Density and Cumulative Distribution Functions, especially the entropy and density-test material. For related density and testing background, see Li and Racine (2007), Chapters 1, 12, and 13.

References

Granger, C.W. and E. Maasoumi and J.S. Racine (2004), “A dependence metric for possibly nonlinear processes”, Journal of Time Series Analysis, 25, 649-669.

Maasoumi, E. and J.S. Racine (2002), “Entropy and predictability of stock market returns,” Journal of Econometrics, 107, 2, pp 291-312.

Author

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca

Details

Documentation guide: see np.kernels for kernels, np.options for global options, and plot for plotting options.

npunitest computes the nonparametric metric entropy (normalized Hellinger of Granger, Maasoumi and Racine (2004)) for testing equality of two univariate density/probability functions, \(D[f(x), f(y)]\). See Maasoumi and Racine (2002) for details. Default bandwidths are of the plug-in variety (bw.SJ for continuous variables and direct plug-in for discrete variables). The bootstrap is conducted via simple resampling with replacement from the pooled data.x and data.y (data.x only for summation).

The summation version of this statistic can be numerically unstable when data.x and data.y lack common support or when the overlap is sparse (the summation version involves division of densities while the integration version involves differences, and the statistic in such cases can be reported as exactly 0.5 or 0). Warning messages are produced when this occurs (‘integration recommended’) and should be heeded.

Numerical integration can occasionally fail when the data.x and data.y distributions lack common support and/or lie an extremely large distance from one another (the statistic in such cases will be reported as exactly 0.5 or 0). However, in these extreme cases, simple tests will reveal the obvious differences in the distributions and entropy-based tests for equality will be clearly unnecessary.

Usage Issues

See the example below for proper usage.

Examples

if (FALSE) { # \dontrun{
set.seed(1234)
n <- 1000

## Compute the statistic only for data drawn from same distribution

x <- rnorm(n)
y <- rnorm(n)

npunitest(x,y,bootstrap=FALSE)

if (interactive()) Sys.sleep(5)

## Conduct the test for this data

npunitest(x,y,boot.num=99)

if (interactive()) Sys.sleep(5)

## Conduct the test for data drawn from different distributions having
## the same mean and variance

x <- rchisq(n,df=5)
y <- rnorm(n,mean=5,sd=sqrt(10))
mean(x)
mean(y)
sd(x)
sd(y)

npunitest(x,y,boot.num=99)

if (interactive()) Sys.sleep(5)

## Two sample t-test for equality of means
t.test(x,y)
## F test for equality of variances and asymptotic
## critical values
Fstat <- var(x)/var(y)
qf(c(0.025,0.975),df1=n-1,df2=n-1)

## Plot the nonparametric density estimates on the same axes

fx <- density(x)
fy <- density(y)
xlim <- c(min(fx$x,fy$x),max(fx$x,fy$x))
ylim <- c(min(fx$y,fy$y),max(fx$y,fy$y))
if (interactive()) plot(fx,xlim=xlim,ylim=ylim,xlab="Data",main="f(x), f(y)")
lines(fy$x,fy$y,col="red")

if (interactive()) Sys.sleep(5)

## Test for equality of log(wage) distributions

data(wage1)
with(wage1, {
lwage.male <- lwage[female=="Male"]
lwage.female <- lwage[female=="Female"]

npunitest(lwage.male,lwage.female,boot.num=99)

if (interactive()) Sys.sleep(5)

## Plot the nonparametric density estimates on the same axes

f.m <- density(lwage.male)
f.f <- density(lwage.female)
xlim <- c(min(f.m$x,f.f$x),max(f.m$x,f.f$x))
ylim <- c(min(f.m$y,f.f$y),max(f.m$y,f.f$y))
plot(f.m,xlim=xlim,ylim=ylim,
     xlab="log(wage)",
     main="Male/Female log(wage) Distributions")
lines(f.f$x,f.f$y,col="red",lty=2)
rug(lwage.male)
legend(-1,1.2,c("Male","Female"),lty=c(1,2),col=c("black","red"))

})

if (interactive()) Sys.sleep(5)

## Conduct the test for data drawn from different discrete probability
## distributions

x <- factor(rbinom(n,2,.5))
y <- factor(rbinom(n,2,.1))

npunitest(x,y,boot.num=99)
} # }