Lmoments (version 1.3-2)

cauchypoly: Cauchy-polynomial quantile mixture

Description

Density, distribution function, quantile function and random generation for the Cauchy-polynomial quantile mixture.

Usage

dcauchypoly(x,param)
pcauchypoly(x,param)
qcauchypoly(cp,param)
rcauchypoly(n,param)
cauchypoly_pdf(x,param)
cauchypoly_cdf(x,param)
cauchypoly_inv(cp,param)
cauchypoly_rnd(n,param)

Value

'dcauchypoly' gives the density, 'pcauchypoly' gives the cumulative distribution function, 'qcauchypoly' gives the quantile function, and 'rcauchypoly' generates random deviates.

Arguments

x

vector of quantiles

cp

vector of probabilities

n

number of observations

param

vector of parameters

Author

Juha Karvanen

Details

The length the parameter vector specifies the order of the polynomial in the quantile mixture. If k<-length(param) then param[1:(k-1)] contains the mixture coefficients of polynomials starting from the constant and param[k] is the mixture coefficient for Cauchy distribution. (Functions cauchypoly_pdf, cauchypoly_cdf, cauchypoly_inv and cauchypoly_rnd are aliases for compatibility with older versions of this package.)

References

Karvanen, J. 2006. Estimation of quantile mixtures via L-moments and trimmed L-moments, Computational Statistics & Data Analysis 51, (2), 947--959. https://users.jyu.fi/~jutakarv/papers/Karvanen_quantile_mixtures.pdf.

See Also

data2cauchypoly4 for the parameter estimation and dnormpoly for the normal-polynomial quantile mixture.

Examples

#Generates 500 random variables from the Cauchy-polynomial quantile mixture, 
#calculates the trimmed L-moments,
#estimates parameters via trimmed L-moments and 
#plots the true pdf and the estimated pdf together with the histogram of the data.
true_params <- t1lmom2cauchypoly4(c(0,1,0.075,0.343));
x <- rcauchypoly(500,true_params);
t1lmom <- t1lmoments(x);
estim_params <- t1lmom2cauchypoly4(t1lmom);
plotpoints <- seq(-10,10,by=0.01);
histpoints <- c(seq(min(x)-1,-20,length.out=50),seq(-10,10,by=0.5),seq(20,max(x)+1,length.out=50));
hist(x, breaks=histpoints, freq=FALSE, xlim=c(-10,10));
lines(plotpoints, dcauchypoly(plotpoints,estim_params), col='red');
lines(plotpoints, dcauchypoly(plotpoints,true_params), col='blue');