GKSCritical | R Documentation |

Compute the critical value of generalized Kolmogorov-Smirnov test statistics, see details on how to use the function.

GKSCritical( n, alpha, index = NULL, indexL = NULL, indexU = NULL, statName = c("KS", "KS+", "KS-", "BJ", "BJ+", "BJ-", "HC", "HC+", "HC-", "Simes") )

`n` |
Integer, the sample size of the data. |

`alpha` |
numeric, the type I error rate for the critical value. Please do
not be confused with |

`index` |
Integer, controlling which ordered samples will be used in the statistics, see details. |

`indexL` |
Integer, controlling which ordered samples will be used in the statistics, see details. |

`indexU` |
Integer, controlling which ordered samples will be used in the statistics, see details. |

`statName` |
Character, the name of the statistic that will be computed. The default is "KS". |

**statistics definitions**

The function compute the test statistics which aggregate the significant signal
from the order statistics of the samples, that is, if `T`

is a statistic
and `X_1`

,`X_2`

,...,`X_n`

are the samples, the value of `T`

is purely
based on the value of `X_(1)`

,`X_(2)`

,...,`X_(n)`

,
where `X_(i)`

is the ith ascending sorted samples of `X1`

,`X2`

,...,`Xn`

.
Moreover, the rejection region of the statistic `T`

can be written as
a set of rejection regions of the ordered samples `X_(1)`

,`X_(2)`

,...,`X_(n)`

.
In other words, there exist two sequences `{l_i}`

and `{u_i}`

for `i=1,...,n`

and the statistic `T`

is rejected if and only if there exist
one `i`

such that `X_(i) < l_i`

or `X_(i) > u_i`

.

The most well-known statistic which takes this form is the Kolmogorov-Smirnov
statistic. Other statistics like Berk-Jones or the higher criticism also have
similar formulas but define different sets of `{l_i}`

and `{u_i}`

.

**alpha0, index, indexL and indexU**

As mentioned previouly, the rejection of a test can be determined by the
sequences of `{l_i}`

and `{u_i}`

. Therefore, the parameter `alpha0`

, `index`

`indexL`

and `indexU`

. provide a way to control which `l_i`

and `u_i`

will be considered in the test procedure. If no argument is provided, all `l_i`

s
and `u_i`

s will be compared with their corresponding sorted sample `X_(i)`

.
This yields the traditional test statistics. If `alpha0`

is used, only
the data `X_(1),...X_(k)`

will be used in the test where `k`

is the nearest
integer of `alpha0*n`

. If `index`

is provided, only `X_(i)`

for `i`

in `index`

will be considered in the test. If `indexL`

and/or `indexU`

is not `NULL`

,
only `l_i`

for `i`

in `indexL`

and `u_i`

for `i`

in `indexU`

will be used as the
rejection boundary for the test. These can be used to generate an one-sided version
of the test statistic. For example, if `indexL`

is from `1`

to the length of `x`

and
`indexU`

is `NULL`

, this will yield a test specifically sensitive to smaller samples.
The test statistics like `KS+`

, `HC+`

and `BJ+`

are implemented by calling
`GKSStat(..., indexU = NULL)`

, where `indexU`

is always `NULL`

.

A critical value

## Compute the critical value of the KS test ## of sample size 10 GKSCritical(alpha = 0.05, n = 10, statName = "KS") ## The critical value for the test that ## only considers the first 3 ordered samples ## All gives the same result. GKSCritical(alpha = 0.05, n = 10, alpha0 = 0.3, statName = "KS") GKSCritical(alpha = 0.05, n = 10, index = 1:3, statName = "KS") GKSCritical(alpha = 0.05, n = 10, indexL = 1:3, indexU = 1:3, statName = "KS")

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