R: Massart Inequality for Distributions

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(sup_x|F_n(x)-F(x)|> t) <= 2exp(-2nt^2)

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.

### 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 `t`.
`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)

[Package DistributionUtils version 0.6-0 Index]