# 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?

##### 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 `pkstwo`

again from package `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 `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 `ks.test`

again from package `qqplot`

from package `qqplot`

from package `qqplot`

for
checking of corresponding robust esimators.

##### Examples

```
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 2.3.1, License: LGPL-3*