BiCopChiPlot: Chi-plot for bivariate copula data
In CDVine: Statistical Inference of C- And D-Vine Copulas

Description Usage Arguments Details Value Author(s) References See Also Examples

This function creates a chi-plot of given bivariate copula data.

1	BiCopChiPlot(u1, u2, PLOT=TRUE, mode="NULL", ...)

`u1,u2`	Data vectors of equal length with values in [0,1].
`PLOT`	Logical; whether the results are plotted. If `PLOT = FALSE`, the values `lambda`, `chi` and `control.bounds` are returned (see below; default: `PLOT = TRUE`).
`mode`	Character; whether a general, lower or upper chi-plot is calculated. Possible values are `mode = "NULL"`, `"upper"` and `"lower"`. `"NULL"` = general chi-plot (default) `"upper"` = upper chi-plot `"lower"` = lower chi-plot
`...`	Additional plot arguments.

For observations u_{i,j}, i=1,...,N, j=1,2, the chi-plot is based on the following two quantities: the chi-statistics

χ_i=F_{U_1,U_2}(u_{i,1},u_{i,2})-F_{U_1}(u_{i,1})F_{U_2}(u_{i,2}) / (F_{U_1}(u_{i,1}) (1-F_{U_1}(u_{i,1}))F_{U_2}(u_{i,2})(1-F_{U_2}(u_{i,2}))^0.5,

and the lambda-statistics

λ_i=4 sgn( tildeF_{U_1}(u_{i,1}),tildeF_{U_2}(u_{i,2}) ) * max( tildeF_{U_1}(u_{i,1})^2,tildeF_{U_2}(u_{i,2})^2 ),

where F_{U_1}, F_{U_2} and F_{U_1U_2} are the empirical distribution functions of the uniform random variables U_1 and U_2 and of (U_1,U_2), respectively. Further, tildeF_{U_1}=F_{U_1}-0.5 and tildeF_{U_2}=F_{U_2}-0.5.

These quantities only depend on the ranks of the data and are scaled to the interval [0,1]. λ_i measures a distance of a data point (u_{i,1},u_{i,2}) to the center of the bivariate data set, while χ_i corresponds to a correlation coefficient between dichotomized values of U_1 and U_2. Under independence it holds that χ_i~N(0,1/N) and λ_i~U[0,1] asymptotically, i.e., values of χ_i close to zero indicate independence—corresponding to F_{U_1U_2}=F_{U_1}F_{U_2}.

When plotting these quantities, the pairs of (λ_i,χ_i) will tend to be located above zero for positively dependent margins and vice versa for negatively dependent margins. Control bounds around zero indicate whether there is significant dependence present.

If mode = "lower" or "upper", the above quantities are calculated only for those u_{i,1}'s and u_{i,2}'s which are smaller/larger than the respective means of u1=(u_{1,1},...,u_{N,1}) and u2=(u_{1,2},...,u_{N,2}).

`lambda`	Lambda-statistics (x-axis).
`chi`	Chi-statistics (y-axis).
`control.bounds`	A 2-dimensional vector of bounds ((1.54/√{n},-1.54/√{n}), where n is the length of `u1` and where the chosen values correspond to an approximate significance level of 10%.

Natalia Belgorodski, Ulf Schepsmeier

Abberger, K. (2004). A simple graphical method to explore tail-dependence in stock-return pairs. Discussion Paper, University of Konstanz, Germany.

Genest, C. and A. C. Favre (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12 (4), 347-368.

BiCopMetaContour, BiCopKPlot, BiCopLambda

## Not run: 
# chi-plots for bivariate Gaussian copula data
n = 500
tau = 0.5

# simulate copula data
fam = 1	
theta = BiCopTau2Par(fam,tau)
dat = BiCopSim(n,fam,theta)	

# create chi-plots
dev.new(width=16,height=5)
par(mfrow=c(1,3))
BiCopChiPlot(dat[,1],dat[,2],xlim=c(-1,1),ylim=c(-1,1),
             main="General chi-plot")
BiCopChiPlot(dat[,1],dat[,2],mode="lower",xlim=c(-1,1),
             ylim=c(-1,1),main="Lower chi-plot")
BiCopChiPlot(dat[,1],dat[,2],mode="upper",xlim=c(-1,1),
             ylim=c(-1,1),main="Upper chi-plot")

## End(Not run)