Description Usage Arguments Details Value Note Author(s) See Also Examples

Function for creating a table of x-errors of a numerical inversion method (i.e., it uses an approximate quantile function of the target distribution). Thus the domain of the inverse distribution function is partitioned into intervals for which maxima, minima and some other quantiles of the x-errors are computed.

Currently the function only works for generators for continuous univariate distribution.

1 2 |

`n` |
sample size for one repetition. |

`aqdist` |
approximate inverse distribution function (quantile function) for a continuous univariate distribution. |

`qdist` |
(Exact) quatile function of distribution. |

`... ` |
parameters to be passed to |

`trunc` |
boundaries of truncated domain. (optional) |

`udomain` |
domain of investigation for (approximate) quantile
function |

`res` |
resolution (number of intervals). |

`kind` |
kind of x-error. |

`tails` |
logical. If |

`plot` |
logical. If |

The absolute x-error of an approximate inverse distribution function
(quantile function) *G^[-1]* for some
*u in (0,1)* is given by

*e_x(u) = |F^[-1](u) - G^[-1](u)|*

where *F^[-1]* denotes the (exact) quantile function of
the distribution.
The relative x-error is then defined as

*e_x(u) / |F^[-1](u)|*

Computing, plotting and analyzing of such x-errors can be quite time consuming.

*e_x(u)*

is a very volatile function and requires the computation at a lot of points. For plotting we can condense the information by partitioning (0,1) into intervals of equal length. In each of these the x-error is computed at equidistributed points and some quantiles (see below) are estimated and stored. Thus we save memory and it is much faster to plot and compare x-errors for different methods or distributions.

If `trunc`

is given, then function `qdist`

is rescaled to
this given domain. Notice, however, that this has some influence on
the accuracy of the results of the “exact” quantile function
`qdist`

.

Using argument `udomain`

it is possible to restrict the domain of
the given (approximate) quantile function `aqdist`

, i.e., of its
argument *u*.

When `tails=TRUE`

we use additional `n`

points for the first
and last interval (which correspond to the tail regions of the
distribution).

An object of class `"rvgt.ierror"`

, see
`uerror`

for details.

It should be noted that `xerror`

computes the difference
between the approximate inversion function `aqdist(u)`

and the
given ‘exact’ quantile function `qdist`

.
Thus one needs a quantile function `qdist`

that is
numerically (much) more accurate than `aqdist`

.

The random variate generator `rdist`

can alternatively be a
generator object form the
Runuran package.

Josef Leydold [email protected]

See `plot.rvgt.ierror`

for the syntax of the plotting
method. See `uerror`

for computing u-errors.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
## Create a table of absolute x-errors for spline interpolation of
## the inverse CDF of the standard normal distribution.
aq <- splinefun(x=pnorm((-100:100)*0.05), y=(-100:100)*0.05,
method="monoH.FC")
## Use a sample of size of 10^5 random variates.
xerr <- xerror(n=1e5, aqdist=aq, qdist=qnorm, kind="abs")
## Plot x-errors
plot(xerr)
## Same for the relative error.
## But this time we use a resolution of 500, and
## we immediately plot the error.
xerr <- xerror(n=1e5, aqdist=aq, qdist=qnorm,
res=500, kind="rel", plot=TRUE)
## An inverse CDF for a truncated normal distribution
aqtn <- splinefun(x=(pnorm((0:100)*0.015) - pnorm(0))/(pnorm(1.5)-pnorm(0)),
y=(0:100)*0.015, method="monoH.FC")
xerrtn <- xerror(n=1e5, aqdist=aqtn, qdist=qnorm, trunc=c(0,1.5),
plot=TRUE)
``` |

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