# BiCopEst: Parameter estimation for bivariate copula data using... In CDVine: Statistical Inference of C- And D-Vine Copulas

## Description

This function estimates the parameter(s) for a bivariate copula using either inversion of empirical Kendall's tau for single parameter copula families or maximum likelihood estimation for one and two parameter copula families supported in this package.

## Usage

 ```1 2``` ```BiCopEst(u1, u2, family, method="mle", se=FALSE, max.df=30, max.BB=list(BB1=c(5,6),BB6=c(6,6),BB7=c(5,6),BB8=c(6,1))) ```

## Arguments

 `u1,u2` Data vectors of equal length with values in [0,1]. `family` An integer defining the bivariate copula family: `0` = independence copula `1` = Gaussian copula `2` = Student t copula (t-copula) `3` = Clayton copula `4` = Gumbel copula `5` = Frank copula `6` = Joe copula `7` = BB1 copula `8` = BB6 copula `9` = BB7 copula `10` = BB8 copula `13` = rotated Clayton copula (180 degrees; “survival Clayton”) `14` = rotated Gumbel copula (180 degrees; “survival Gumbel”) `16` = rotated Joe copula (180 degrees; “survival Joe”) `17` = rotated BB1 copula (180 degrees; “survival BB1”) `18` = rotated BB6 copula (180 degrees; “survival BB6”) `19` = rotated BB7 copula (180 degrees; “survival BB7”) `20` = rotated BB8 copula (180 degrees; “survival BB8”) `23` = rotated Clayton copula (90 degrees) `24` = rotated Gumbel copula (90 degrees) `26` = rotated Joe copula (90 degrees) `27` = rotated BB1 copula (90 degrees) `28` = rotated BB6 copula (90 degrees) `29` = rotated BB7 copula (90 degrees) `30` = rotated BB8 copula (90 degrees) `33` = rotated Clayton copula (270 degrees) `34` = rotated Gumbel copula (270 degrees) `36` = rotated Joe copula (270 degrees) `37` = rotated BB1 copula (270 degrees) `38` = rotated BB6 copula (270 degrees) `39` = rotated BB7 copula (270 degrees) `40` = rotated BB8 copula (270 degrees) `method` Character indicating the estimation method: either maximum likelihood estimation (`method = "mle"`; default) or inversion of Kendall's tau (`method = "itau"`). For `method = "itau"` only one parameter bivariate copula families can be used (`family = 1,3,4,5,6,13,14,16,23,24,26,33,34` or `36`). `se` Logical; whether standard error(s) of parameter estimates is/are estimated (default: `se = FALSE`). `max.df` Numeric; upper bound for the estimation of the degrees of freedom parameter of the t-copula (default: `max.df = 30`). `max.BB` List; upper bounds for the estimation of the two parameters (in absolute values) of the BB1, BB6, BB7 and BB8 copulas (default: `max.BB = list(BB1=c(5,6),BB6=c(6,6),BB7=c(5,6),BB8=c(6,1))`).

## Details

If `method = "itau"`, the function computes the empirical Kendall's tau of the given copula data and exploits the one-to-one relationship of copula parameter and Kendall's tau which is available for many one parameter bivariate copula families (see `BiCopPar2Tau` and `BiCopTau2Par`). The inversion of Kendall's tau is however not available for all bivariate copula families (see above). If a two parameter copula family is chosen and `method = "itau"`, a warning message is returned and the MLE is calculated.

For `method = "mle"` copula parameters are estimated by maximum likelihood using starting values obtained by `method = "itau"`. If no starting values are available by inversion of Kendall's tau, starting values have to be provided given expert knowledge and the boundaries `max.df` and `max.BB` respectively.

A warning message is returned if the estimate of the degrees of freedom parameter of the t-copula is larger than `max.df`. For high degrees of freedom the t-copula is almost indistinguishable from the Gaussian and it is advised to use the Gaussian copula in this case. As a rule of thumb `max.df = 30` typically is a good choice. Moreover, standard errors of the degrees of freedom parameter estimate cannot be estimated in this case.

## Value

 `par, par2` Estimated copula parameter(s). `se,se2` Standard error(s) of the parameter estimate(s) (if `se = TRUE`).

## Author(s)

Ulf Schepsmeier, Eike Brechmann, Jakob Stoeber, Carlos Almeida

## References

Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.

`BiCopPar2Tau`, `BiCopTau2Par`, `CDVineSeqEst`, `BiCopSelect`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41``` ```## Example 1: bivariate Gaussian copula dat = BiCopSim(500,1,0.7) u1 = dat[,1] v1 = dat[,2] # empirical Kendall's tau tau1 = cor(u1,v1,method="kendall") # inversion of empirical Kendall's tau BiCopTau2Par(1,tau1) BiCopEst(u1,v1,family=1,method="itau")\$par # maximum likelihood estimate for comparison BiCopEst(u1,v1,family=1,method="mle")\$par ## Example 2: bivariate Clayton and survival Gumbel copulas # simulate from a Clayton copula dat = BiCopSim(500,3,2.5) u2 = dat[,1] v2 = dat[,2] # empirical Kendall's tau tau2 = cor(u2,v2,method="kendall") # inversion of empirical Kendall's tau for the Clayton copula BiCopTau2Par(3,tau2) BiCopEst(u2,v2,family=3,method="itau",se=TRUE) # inversion of empirical Kendall's tau for the survival Gumbel copula BiCopTau2Par(14,tau2) BiCopEst(u2,v2,family=14,method="itau",se=TRUE) # maximum likelihood estimates for comparison BiCopEst(u2,v2,family=3,method="mle",se=TRUE) BiCopEst(u2,v2,family=14,method="mle",se=TRUE) ## Example 3: fit of a t-copula to standardized residuals of ## S&P 500 and DAX returns data(worldindices) BiCopEst(worldindices[,1],worldindices[,4],family=2,method="mle",se=TRUE) ```