abvnonpar {evd} | R Documentation

## Non-parametric Estimates for Dependence Functions of the Bivariate Extreme Value Distribution

### Description

Calculate or plot non-parametric estimates for the dependence function A of the bivariate extreme value distribution.

### Usage

    abvnonpar(x = 0.5, data, nsloc1 = NULL, nsloc2 = NULL,
        method = c("cfg", "pickands", "deheuvels", "hall", "tdo"),
        convex = FALSE, wf = function(t) t, plot = FALSE,
        add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, xlim = c(0, 1),
        ylim = c(0.5, 1), xlab = "", ylab = "", ...)

### Arguments

`x` | A vector of values at which the dependence function is evaluated (ignored if plot or add is `TRUE`). A(1/2) is returned by default since it is often a useful summary of dependence.
`data` | A matrix or data frame with two columns, which may contain missing values.
`nsloc1, nsloc2` | A data frame with the same number of rows as `data`, for linear modelling of the location parameter on the first/second margin. The data frames are treated as covariate matrices, excluding the intercept. A numeric vector can be given as an alternative to a single column data frame.
`method` | The estimation method (see Details). The `"cfg"` method is used by default.
`convex` | Logical; take the convex minorant?
`wf` | The weight function used in the `"cfg"` method (see Details). The function must be vectorized.
`plot` | Logical; if `TRUE` the function is plotted. The x and y values used to create the plot are returned invisibly. If `plot` and `add` are `FALSE` (the default), the arguments following `add` are ignored.
`add` | Logical; add to an existing plot? The existing plot should have been created using either `abvnonpar` or `abvpar`, the latter of which plots (or calculates) the dependence function for a number of parametric models.
`lty, blty` | Function and border line types. Set `blty` to zero to omit the border.
`lwd` | Line width.
`col` | Line colour.
`xlim, ylim` | x and y-axis limits.
`xlab, ylab` | x and y-axis labels.
`...` | Other high-level graphics parameters to be passed to `plot`.

### Details

The dependence function A() of the bivariate extreme value distribution is defined in `abvpar`. Non-parametric estimates are constructed as follows. Suppose (z_{i1},z_{i2}) for i=1,...,n are n bivariate observations that are passed using the `data` argument. The marginal parameters are estimated (under the assumption of independence) and the data is transformed using

y_{i1} = {1 + s'_1(z_{i1}-a'_1)/b'_1}^(-1/s'_1)

and

y_{i2} = {1 + s'_2(z_{i2}-a'_2)/b'_2}^(-1/s'_2)

for i = 1,...,n, where (a'_1,b'_1,s'_1) and (a'_2,b'_2,s'_2) are the maximum likelihood estimates for the location, scale and shape parameters on the first and second margins. If `nsloc1` or `nsloc2` are given, the location parameters may depend on i (see `fgev`).

Five different estimators of the dependence function can be implemented. They are defined (on 0 <= w <= 1) as follows.

`method = "cfg"` (Caperaa, Fougeres and Genest, 1997)

A_c(w) = exp{ [1-p(w)] integral_0^w (H(x) - x)/[x(1-x)] dx - p(w) integral_w^1 (H(x) - x)/[x(1-x)] dx }

`method = "pickands"` (Pickands, 1981)

A_p(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}/(1-w)]}

`method = "deheuvels"` (Deheuvels, 1991)

A_d(w) = n / {sum_{i=1}^n min[y_{i1}/w, y_{i2}(1-w)] - w sum_{i=1}^n y_{i1} - (1-w) sum_{i=1}^n y_{i2} + n}

`method = "hall"` (Hall and Tajvidi, 2000)

A_h(w) = n (sum_{i=1}^n min[y_{i1}/(by_1 w), y_{i2}/(by_2 (1-w))])^{-1}

`method = "tdo"` (Tiago de Oliveira, 1997)

A_t(w) = 1 - 1/(1 + log n) sum_{i=1}^n min[w/(1 + n y_{i1}), (1 - w)/(1 + n y_{i2})]

In the estimator A_h(), by_j = (sum_{i=1}^n y_{ij})/n for j = 1,2. In the estimator A_c(), H(x) is the empirical distribution function of x_1,...,x_n, where x_i = y_{i1} / (y_{i1} + y_{i2}) for i = 1,...,n, and p(w) is any bounded function on [0,1], which can be specified using the argument `wf`. By default `wf` is the identity function.

Let A_n() be any estimator of A(). The constraint A_n(0) = A_n(1) = 1 is satisfied by A_d(), A_t() and A_h(), and by A_c() when p(0) = 0 and p(1) = 1. None of the estimators satisfy max(w,1-w) <= A_n(w) <= 1 for all 0 <= w <= 1. An obvious modification is

A'_n(w) = min(1, max{A_n(w), w, 1-w}).

This modification is always implemented.

A_t(w) is the only estimator that is convex. Convex estimators can be derived from other methods by taking the convex minorant, which can be achieved by setting `convex` to `TRUE`.

### Value

`abvnonpar` calculates or plots a non-parametric estimate of the dependence function of the bivariate extreme value distribution.

### Note

Appendix A of the User's Guide contains a short simulation study that compares the estimators defined above. The estimators A_p(), A_d() and A_h() are very similar, and may not be distinguishable when plotted.

### References

Caperaa, P. Fougeres, A.-L. and Genest, C. (1997) A non-parametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.

Deheuvels, P. (1991) On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Letters, 12, 429–439.

Hall, P. and Tajvidi, N. (2000) Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli, 6, 835–844.

Pickands, J. (1981) Multivariate extreme value distributions. Proc. 43rd Sess. Int. Statist. Inst., 49, 859–878.

Tiago de Oliveira, J. (1997) Statistical Analysis of Extremes. Pendor.

### See Also

`abvpar`, `atvnonpar`, `fgev`

### Examples

    bvdata <- rbvevd(100, dep = 0.7, model = "log")
    abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE)
    abvnonpar(data = bvdata, method = "d", plot = TRUE)

    M1 <- fitted(fbvevd(bvdata, model = "log"))
    abvpar(dep = M1["dep"], model = "log", plot = TRUE)
    abvnonpar(data = bvdata, add = TRUE, lty = 2)

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