# dip

##### Compute Hartigans' Dip Test Statistic for Unimodality

Computes Hartigans' dip test statistic for testing unimodality, and additionally the modal interval.

- Keywords
- distribution, htest

##### Usage

`dip(x, full.result = FALSE, min.is.0 = FALSE, debug = FALSE)`

##### Arguments

- x
- numeric; the data.
- full.result
- logical or string;
`dip(., full.result=TRUE)`

returns the full result list; if`"all"`

it additionally uses the`mn`

and`mj`

components to compute the initial GCM and LCM, see below. - min.is.0
- logical indicating if the
**min**imal value of the dip statistic $D_n$ can be zero or not. Arguably should be set to`TRUE`

for internal consistency reasons, but is false by default both for continuity and backwards compatibilit - debug
- logical; if true, some tracing information is printed (from the C routine).

##### Value

- depending on
`full.result`

either a number, the dip statistic, or an object of class`"dip"`

which is a`list`

with components x the sorted `unname()`

d data.n `length(x)`

.dip the dip statistic lo.hi indices into `x`

for lower and higher end of modal intervalxl, xu lower and upper end of modal interval gcm, lcm (last used) indices for **g**reatest**c**onvex**m**inorant and the**l**east**c**oncave**m**ajorant.mn, mj index vectors of length `n`

for the GC minorant and the LC majorant respectively.- For
full results of class`"dip"`

, there are`print`

and`plot`

methods, the latter with its own manual page.

##### Note

For $n \le 3$ where `n <- length(x)`

, the dip
statistic $D_n$ is always the same minimum value,
$1/(2n)$, i.e., there's no possible dip test.
Note that up to May 2011, from Hartigan's original Fortran code, `Dn`

was set to zero, when all `x`

values were identical. However,
this entailed discontinuous behavior, where for arbitrarily close
data $\tilde x$, $D_n(\tilde x) = \frac 1{2n}$.

Yong Lu `")"`
in the original Fortran code. This bug has been corrected for diptest
version 0.25-0.

Nick Cox (Durham Univ.) said (on March 20, 2008 on the Stata-list):
As it comes from a bimodal husband-wife collaboration, the name
perhaps should be * Hartigan-Hartigan dip test*, but that
does not seem to have caught on. Some of my less statistical
colleagues would sniff out the hegemony of patriarchy there, although
which Hartigan is being overlooked is not clear.

Martin Maechler, as a Swiss, and politician, would say:
Let's find a compromise, and call it * Hartigans' dip test*,
so we only have to adapt orthography (:-).

##### References

P. M. Hartigan (1985)
Computation of the Dip Statistic to Test for Unimodality;
*Applied Statistics (JRSS C)* **34**, 320--325.
Corresponding (buggy!) Fortran code of

J. A. Hartigan and P. M. Hartigan (1985)
The Dip Test of Unimodality;
*Annals of Statistics* **13**, 70--84.

##### See Also

`dip.test`

to compute the dip *and* perform the unimodality test,
based on P-values, interpolated from `qDiptab`

;
`isoreg`

for isotonic regression.

##### Examples

```
data(statfaculty)
plot(density(statfaculty))
rug(statfaculty, col="midnight blue"); abline(h=0, col="gray")
dip(statfaculty)
(dS <- dip(statfaculty, full = TRUE, debug = TRUE))
plot(dS)
## even more output -- + plot showing "global" GCM/LCM:
(dS2 <- dip(statfaculty, full = "all", debug = 3))
plot(dS2)
data(faithful)
fE <- faithful$eruptions
plot(density(fE))
rug(fE, col="midnight blue"); abline(h=0, col="gray")
dip(fE, debug = 2) ## showing internal work
(dE <- dip(fE, full = TRUE)) ## note the print method
plot(dE, do.points=FALSE)
data(precip)
plot(density(precip))
rug(precip, col="midnight blue"); abline(h=0, col="gray")
str(dip(precip, full = TRUE, debug = TRUE))
##----------------- The 'min.is.0' option : ---------------------
##' dip(.) continuity and 'min.is.0' exploration:
dd <- function(x, debug=FALSE) {
x_ <- x ; x_[1] <- 0.9999999999 * x[1]
rbind(dip(x , debug=debug),
dip(x_, debug=debug),
dip(x , min.is.0=TRUE, debug=debug),
dip(x_, min.is.0=TRUE, debug=debug), deparse.level=2)
}
dd( rep(1, 8) ) # the 3rd one differs ==> min.is.0=TRUE is *dis*continuous
dd( 1:7 ) # ditto
dd( 1:7, debug=TRUE)
## border-line case ..
dd( 1:2, debug=TRUE)
## Demonstrate that 'min.is.0 = TRUE' does not change the typical result:
B.sim <- 1000 # or larger
D5 <- {set.seed(1); replicate(B.sim, dip(runif(5)))}
D5. <- {set.seed(1); replicate(B.sim, dip(runif(5), min.is.0=TRUE))}
stopifnot(identical(D5, D5.), all.equal(min(D5), 1/(2*5)))
hist(D5, 64); rug(D5)
D8 <- {set.seed(7); replicate(B.sim, dip(runif(8)))}
D8. <- {set.seed(7); replicate(B.sim, dip(runif(8), min.is.0=TRUE))}
stopifnot(identical(D8, D8.))
```

*Documentation reproduced from package diptest, version 0.75-5, License: GPL (>= 2)*