# cauchypoly: Cauchy-polynomial quantile mixture In Lmoments: L-Moments and Quantile Mixtures

## Description

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

## Usage

 ```1 2 3 4 5 6 7 8``` ```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) ```

## Arguments

 `x` vector of quantiles `cp` vector of probabilities `n` number of observations `param` vector of parameters

## 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.)

## Value

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

## Author(s)

Juha Karvanen juha.karvanen@iki.fi

## References

Karvanen, J. 2006. Estimation of quantile mixtures via L-moments and trimmed L-moments, Computational Statistics & Data Analysis 51, (2), 947–959. http://www.bsp.brain.riken.jp/publications/2006/karvanen_quantile_mixtures.pdf.

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```#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'); ```

### Example output ```
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Lmoments documentation built on May 2, 2019, 2:04 a.m.