distIneqMassart function

Massart Inequality for Distributions

This function implements a test of the random number generator and distribution function based on an inequality due to Massart (1990).

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:

where $F_n$ 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.

Returns

• 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 t.

• check: Logical. Indicates whether the inequality is satisfied or not.

References

Massart P. (1990) The tight constant in the Dvoretsky-Kiefer-Wolfovitz inequality. Ann. Probab., 18 , 1269--1283.

Author(s)

David Scott d.scott@auckland.ac.nz , Christine Yang Dong c.dong@auckland.ac.nz

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)