spacings {Renext}

Methods computing spacings between Largest Order Statistics

Description

Methods computing the random spacings for the Largest Order Statistics of the marks in a POT renewal.

Usage

spacings(object, ...)

## S3 method for class 'numeric'
spacings(object, wExp = TRUE, ...)

## S3 method for class 'data.frame'
spacings(object, varName, wExp = TRUE, ...)

## S3 method for class 'Rendata'
spacings(object, type = c("MAX", "OTS", "OT"), wExp = TRUE, ...)

Arguments

object

A object containing the marks X_i. This is can be a vector or a data frame for data aggregated by blocks. For an object of class data.frame one of the three data.frame slots: OTdata, MAXdata or OTSdata can be used using the suitable value of type.

varName

Character vector of length 1 giving the name of the variable when object is a data.frame.

wExp

Logical. If TRUE, the spacings are weighted as explained in Details.

type

Character specifying the data.frame to be used when object has class "Rendata".

...

Not used yet.

Details

The spacings are the differences between the Largest Order Statistics. They are useful for some estimation tasks or diagnostics. Given random variables X_i, the i-th spacing Y_i is the difference X_{(i)}-X_{(i+1)} between the i-th largest order statistic X_{(i)} and the next in the decreasing order i.e. X_{(i+1)}.

When the r.vs X_i form a sample of an exponential or Gumbel distribution, the spacings associated with the largest order statistics are or tend to be independent and exponentially distributed. More precisely, the weighted spacings i \times Y_i have or tend to have the same exponential distribution. This can be used to estimate the shape parameter of the underlying distribution using only the largest order statistics. Moreover the r-1 spacings Y_i built from the r largest order statistics i \le r are or tend to be independent from the r-th largest order statistic X_{(r)}.

When wExp is TRUE, the returned values are the weighted spacings i \times Y_i.

Value

A list or vector containing the spacings. When the data is structured in blocks as is the MAXdata slot of an object of class "Rendata", the spacings are computed form the order statistics within each block, to maintain independence with the next order statistic in decreasing order.

Caution

By default, the spacings are scaled as explained above, thus assuming that the marks are exponentially distributed.

Author(s)

Yves Deville

References

Embrechts P., Klüppelberg C. and Mikosch T. (1997) Modelling Extremal Events for Insurance and Finance. Springer.

Examples

sp <- spacings(rgumbel(200, loc = 0, scale = 1))
expplot(sp)
sp1 <- spacings(rgev(200, loc = 0, scale = 1, shape = 0.3))
expplot(sp1)
## spacings are computed by block
sp2 <- spacings(object = Garonne$MAXdata,
                varName = Garonne$info$varName)
expplot(unlist(sp2))
sp3 <- spacings(object = Garonne, type = "OT")
expplot(sp3)