GKSPvalue | R Documentation |
Compute the pvalue of generalized Kolmogorov-Smirnov test statistics, see details on how to use the function.
GKSPvalue( stat = NULL, n = NULL, index = NULL, indexL = NULL, indexU = NULL, x = NULL, statName = NULL )
stat |
a |
n |
Integer, the sample size of the data. |
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. |
x |
Numeric, the sample data. |
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 numeric value representing the pvalue
## Generate samples x <- rbeta(10, 1, 2) ## Perform KS test ks_res <- GKSStat(x = x, statName = "KS") ## Compute the pvalue for the KS test GKSPvalue(stat = ks_res) ## For any observed statistic GKSPvalue(stat = 0.2, n = 10, statName = "KS") ## Change the detection range of the KS test ## to test only the first 3 ordered samples ## All gives the same result GKSPvalue(stat = 0.2, n = 10, alpha0 = 0.3, statName = "KS") GKSPvalue(stat = 0.2, n = 10, index = 1:3, statName = "KS") GKSPvalue(stat = 0.2, n = 10, indexL = 1:3, indexU = 1:3, statName = "KS")
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