distIneqMassart {DistributionUtils} | R Documentation |
Massart Inequality for Distributions
Description
This function implements a test of the random number generator and distribution function based on an inequality due to Massart (1990).
Usage
distIneqMassart(densFn = "norm", n = 10000, probBound = 0.001, ...)
Arguments
densFn |
Character. The root name of the distribution to be tested. |
n |
Numeric. The size of the sample to be used. |
probBound |
Numeric. The value of the bound on the right hand side of the Massart inequality. See Details. |
... |
Additional arguments to allow specification of the parameters of the distribution. |
Details
Massart (1990) gave a version of the Dvoretsky-Kiefer-Wolfowitz inequality with the best possible constant:
P\left(\sup_{x}|\hat F_n(x)-F(x)|> t\right) \leq%
2\exp(-2nt^2)
where \hat F_n
is the empirical distribution function for
a sample of n
independent and identically distributed random
variables with distribution function F
. This inequality is true
for all distribution functions, for all n
and t
.
This test is used in base R to check the standard distribution
functions. The code may be found in the file p-r-random.tests.R
in the tests
directory.
Value
sup |
Numeric. The supremum of the absolute difference between the empirical distribution and the true distribution function. |
probBound |
Numeric. The value of the bound on the right hand side of the Massart inequality. |
t |
Numeric. The lower bound which the supremum of the absolute difference between the empirical distribution and the true distribution function must exceed. |
pVal |
Numeric. The probability that the absolute difference
between the empirical distribution and the true distribution function
exceeds |
check |
Logical. Indicates whether the inequality is satisfied or not. |
Author(s)
David Scott d.scott@auckland.ac.nz, Christine Yang Dong c.dong@auckland.ac.nz
References
Massart P. (1990) The tight constant in the Dvoretsky-Kiefer-Wolfovitz inequality. Ann. Probab., 18, 1269–1283.
Examples
## Normal distribution is the default
distIneqMassart()
## Specify parameter values
distIneqMassart(mean = 1, sd = 2)
## Gamma distribution has no default value for shape
distIneqMassart("gamma", shape = 1)