# qqbounds

##### Computation of confidence intervals for qqplot

We compute confidence intervals for QQ plots. These can be simultaneous (to check whether the whole data set is compatible) or pointwise (to check whether each (single) data point is compatible);

- Keywords
- hplot, distribution

##### Usage

```
qqbounds(x,D,alpha,n,withConf.pw, withConf.sim,
exact.sCI=(n
```

##### Arguments

- x
data to be checked for compatibility with distribution

`D`

.- D
object of class

`"UnivariateDistribution"`

, the assumed data distribution.- alpha
confidence level

- n
sample size

- withConf.pw
logical; shall pointwise confidence lines be computed?

- withConf.sim
logical; shall simultaneous confidence lines be computed?

- exact.pCI
logical; shall pointwise CIs be determined with exact Binomial distribution?

- exact.sCI
logical; shall simultaneous CIs be determined with exact kolmogorov distribution?

- nosym.pCI
logical; shall we use (shortest) asymmetric CIs?

- debug
logical; if

`TRUE`

additional output to debug confidence bounds.

##### Details

Both simultaneous and pointwise confidence intervals come in a
finite-sample and an asymptotic version;
the finite sample versions will get quite slow
for large data sets `x`

, so in these cases the asymptotic version
will be preferrable.
For simultaneous intervals,
the finite sample version is based on C function `"pkolmogorov2x"`

from package stats, while the asymptotic one uses
R function `pkstwo`

again from package stats, both taken
from the code to `ks.test`

.

Both finite sample and asymptotic versions use the fact, that the distribution of the supremal distance between the empirical distribution \(\hat F_n\) and the corresponding theoretical one \(F\) (assuming data from \(F\)) does not depend on \(F\) for continuous distribution \(F\) and leads to the Kolmogorov distribution (compare, e.g. Durbin(1973)). In case of \(F\) with jumps, the corresponding Kolmogorov distribution is used to produce conservative intervals.

For pointwise intervals, the finite sample version is based on corresponding binomial distributions, (compare e.g., Fisz(1963)), while the asymptotic one uses a CLT approximation for this binomial distribution. In fact, this approximation is only valid for distributions with strictly positive density at the evaluation quantiles.

In the finite sample version, the binomial distributions will in general not
be symmetric, so that, by setting `nosym.pCI`

to `TRUE`

we may
produce shortest asymmetric confidence intervals (albeit with a considerable
computational effort).

The symmetric intervals returned by default will be conservative (which also applies to distributions with jumps in this case).

For distributions with jumps or with density (nearly) equal to 0 at the
corresponding quantile, we use the approximation of `(D-E(D))/sd(D)`

by the standard normal at these points; this latter approximation is only
available if package distrEx is installed; otherwise the corresponding
columns will be filled with `NA`

.

##### Value

A list with components `crit`

--- a matrix with the lower and upper confidence
bounds, and `err`

a logical vector of length 2.

Component `crit`

is a matrix with `length(x)`

rows
and four columns `c("sim.left","sim.right","pw.left","pw.right")`

.
Entries will be set to `NA`

if the corresponding `x`

component
is not in `support(D)`

or if the computation method returned an error
or if the corresponding parts have not been required (if `withConf.pw`

or `withConf.sim`

is `FALSE`

).

`err`

has components `pw`

---do we have a non-error return value for the computation of pointwise CI's
(`FALSE`

if `withConf.pw`

is `FALSE`

)--- and `sim`

---do we have a non-error return value for the computation of simultaneous CI's
(`FALSE`

if `withConf.sim`

is `FALSE`

).

##### References

Durbin, J. (1973)
*Distribution theory for tests based on the sample distribution
function*. SIAM.

Fisz, M. (1963). *Probability Theory and Mathematical Statistics*.
3rd ed. Wiley, New York.

##### See Also

`qqplot`

from package stats -- the standard QQ plot
function, `ks.test`

again from package stats
for the implementation of the Kolmogorov distributions;
`qqplot`

from package distr for
comparisons of distributions, and
`qqplot`

from package distrMod for comparisons
of data with models, as well as `qqplot`

for
checking of corresponding robust esimators.

##### Examples

```
# NOT RUN {
qqplot(Norm(15,sqrt(30)), Chisq(df=15))
## uses:
qqbounds(x = rnorm(30),Norm(),alpha=0.95,n=30,
withConf.pw = TRUE, withConf.sim = TRUE,
exact.sCI=TRUE ,exact.pCI= TRUE,
nosym.pCI = FALSE)
qqbounds(x = rchisq(30,df=4),Chisq(df=4),alpha=0.95,n=30,
withConf.pw = TRUE, withConf.sim = TRUE,
exact.sCI=FALSE ,exact.pCI= FALSE,
nosym.pCI = FALSE)
qqbounds(x = rchisq(30,df=4),Chisq(df=4),alpha=0.95,n=30,
withConf.pw = TRUE, withConf.sim = TRUE,
exact.sCI=TRUE ,exact.pCI= TRUE,
nosym.pCI = TRUE)
# }
```

*Documentation reproduced from package distr, version 2.8.0, License: LGPL-3*