Numerically estimate the copula density for a sequence of $u$ and $v$ probabilities for which each sequence has equal steps that are equal to $\Delta(uv)$. The density $c(u,v)$ of a copula $\mathbf{C}(u,v)$ is numerically estimated by
$$c(u,v) = \bigl[\mathbf{C}(u_2,v_2) - \mathbf{C}(u_2,v_1) - \mathbf{C}(u_1,v_2) + \mathbf{C}(u_1,v_1)\bigr]\, /\, \bigl[\Delta(uv)\times\Delta(uv)\bigr]\mbox{,}$$
where $c(u,v) \ge 0$ (see Nelsen, 2006, p. 10; <code>densityCOPplot</code>). The joint density can be defined by the coupla density for continuous variables and is the ratio of the joint density funcion $f(x,y)$ for random variables $X$ and $Y$ to the product of the marginal densities ($f_x(x)$ and $f_y(y)$):
$$c\bigl(F_x(x), F_y(y)\bigr) = \frac{f(x,y)}{f_x(x)f_y(y)}\mbox{,}$$
where $F_x(x)$ and $F_y(y)$ are the respective cumulative distribution functions of $X$ and $Y$, and lastly $u = F_x(x)$ and $v = F_y(y)$.

densityCOP

Extensive functions for bivariate copula (bicopula) computations and related operations
for bicopula theory. The lower, upper, product, and select other bicopula are implemented along
with operations including the diagonal, survival copula, dual of a copula, co-copula, and
numerical bicopula density. Level sets, horizontal and vertical sections are supported. Numerical
derivatives and inverses of a bicopula are provided through which simulation is implemented.
Bicopula composition, convex combination, asymmetry extension, and products also are provided.
Support extends to the Kendall Function as well as the Lmoments thereof. Kendall Tau,
Spearman Rho and Footrule, Gini Gamma, Blomqvist Beta, Hoeffding Phi, Schweizer-
Wolff Sigma, tail dependency, tail order, skewness, and bivariate Lmoments are implemented, and
positive/negative quadrant dependency, left (right) increasing (decreasing) are available.
Other features include Kullback-Leibler Divergence, Vuong Procedure, spectral measure, and
Lcomoments for inference, maximum likelihood, and AIC, BIC, and RMSE for goodness-of-fit.

William Asquith

copBasic

General Bivariate Copula Theory and Many Utility Functions

densityCOP function

Density of a Copula — densityCOP

<dl>

 <dt>u</dt>
<dd>Nonexceedance probability $u$ in the $X$ direction;</dd>

 <dt>v</dt>
<dd>Nonexceedance probability $v$ in the $Y$ direction;</dd>

 <dt>cop</dt>
<dd>A copula function;</dd>

 <dt>para</dt>
<dd>Vector of parameters or other data structure, if needed, to pass to the copula;</dd>

 <dt>deluv</dt>
<dd>The change in the sequences $\{u, v\} \mapsto \delta, \ldots, 1-\delta; \delta = \Delta(uv)$ probabilities;</dd>

 <dt>truncate.at.zero</dt>
<dd>A density must be $c(u,v) \ge 0$, but because this computation is based on a rectangular approximation and not analytical, there exists a possibility that very small rectangles could result in numerical values in R that are less than zero. This usually can be blamed on rounding. This logical if <code>TRUE</code> truncates computed densities to zero, and the default assumes that the user is providing a proper copula. A <code>FALSE</code> value is used by the function <code>isfuncCOP</code>;</dd>

 <dt>the.zero</dt>
<dd>The value for &#8220;the zero&#8221; where a small number might be useful for pseudo-maximum likelihood estimation using <code>sumlogs</code>;</dd>

 <dt>sumlogs</dt>
<dd>Return the $\sum{\log c(u,v; \Theta)}$ where $\Theta$ are the parameters in <code>para</code> and this feature is provided for <code>mleCOP</code>; and</dd>

 <dt>...</dt>
<dd>Additional arguments to pass to the copula function.</dd>

</dl>

densityCOP: Density of a Copula

Description

Usage

Arguments

Value

References

See Also