Package dng. November 22, 2017

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1 Version Date Title Distributions and Gradients Type Package Author Feng Li, Jiayue Zeng Maintainer Jiayue Zeng Depends R (>= 3.0.0) Package dng November 22, 2017 Provides density, distribution function, quantile function and random generation for the splitt distribution, and computes the mean, variance, skewness and kurtosis for the splitt distribution (Li, F, Villani, M. and Kohn, R. (2010) <doi: /j.jspi >). License GPL (>= 2) URL Encoding UTF-8 LazyData true Imports stats, Rcpp (>= ) LinkingTo Rcpp NeedsCompilation yes Repository CRAN Date/Publication :16:29 UTC R topics documented: dsplitt psplitt qsplitt rsplitt split-t Index 10 1

2 2 dsplitt dsplitt Split-t distribution Density function for the split student-t distribution. Usage dsplitt(x, mu, df, phi, lmd, logarithm) Arguments x mu df vector of quantiles. vector of location parameter. (The mode of the density) degrees of freedom (> 0, maybe non-integer). df = Inf is allowed. phi vector of scale parameters (>0). lmd Details logarithm vector of skewness parameters (>0). If is 1, reduced to symmetric student t distribution. logical; if TRUE, probabilities p are given as log(p). The random variable y follows a split-t distribution with ν>0 degrees of freedom, y~t(µ, φ, λ, ν), if its density function is of the form CK(µ, φ, ν, )I(y µ) + CK(µ, λφ, ν)i(y > µ), where, K(µ, φ, ν, ) = [ν/(ν + (y µ) 2 /φ 2 )] (ν+1)/2 is the kernel of a student t density with variance φ 2 ν/(ν 2) and c = 2[(1 + λ)φ( ν)beta(ν/2, 1/2)] 1 is the normalization constant. Value dsplitt gives the density. Invalid arguments will result in return value NaN, with a warning. The length of the result is determined by n for rsplitt, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

3 psplitt 3 Author(s) Feng Li, Jiayue Zeng References Li, F., Villani, M., & Kohn, R. (2009). Flexible modeling of conditional distributions using smooth mixtures of asymmetric student t densities. Journal of Statistical Planning & Inference, 140(12), See Also splitt_mean(), splitt_var(),splitt_skewness() and splitt_kurtosis() for numerical characteristics of the Split-t distribution. Examples n <- 5 x <- c(0.25,0.5,0.75) q <- c(0.25,0.5,0.75) p <- c(0.25,0.5,0.75) mu <- c(0,1,2) df <- rep(10,3) phi <- c(0.5,1,2) lmd <- c(1,2,3) dsplitt0 <- dsplitt(x, mu, df, phi, lmd, logarithm = TRUE) psplitt Distribution function of Split-t distribution Usage Density, distribution function, quantile function and random generation for the split student-t distribution. psplitt(q, mu, df, phi, lmd) Arguments q mu df vector of quantiles. vector of location parameter. (The mode of the density) degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.

4 4 psplitt phi vector of scale parameters (>0). lmd vector of skewness parameters (>0). If is 1, reduced to symmetric student t distribution. Details The random variable y follows a split-t distribution with ν>0 degrees of freedom, y~t(µ, φ, λ, ν), if its density function is of the form CK(µ, φ, ν, )I(y µ) + CK(µ, λφ, ν)i(y > µ), where, K(µ, φ, ν, ) = [ν/(ν + (y µ) 2 /φ 2 )] (ν+1)/2 is the kernel of a student t density with variance φ 2 ν/(ν 2) and is the normalization constant. c = 2[(1 + λ)φ( ν)beta(ν/2, 1/2)] 1 Value psplitt gives the distribution function. (dsplitt, psplitt, qsplitt and rsplitt are all vectors.) Invalid arguments will result in return value NaN, with a warning. The length of the result is determined by n for rsplitt, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used. Author(s) Feng Li, Jiayue Zeng References Li, F., Villani, M., & Kohn, R. (2009). Flexible modeling of conditional distributions using smooth mixtures of asymmetric student t densities. Journal of Statistical Planning & Inference, 140(12), See Also splitt_mean(), splitt_var(),splitt_skewness() and splitt_kurtosis() for numerical characteristics of the Split-t distribution.

5 qsplitt 5 Examples n <- 5 x <- c(0.25,0.5,0.75) q <- c(0.25,0.5,0.75) p <- c(0.25,0.5,0.75) mu <- c(0,1,2) df <- rep(10,3) phi <- c(0.5,1,2) lmd <- c(1,2,3) psplitt0 <- psplitt(q, mu, df, phi, lmd) qsplitt quantile function of Split-t distribution Usage Density, distribution function, quantile function and random generation for the split student-t distribution. qsplitt(p, mu, df, phi, lmd) Arguments p mu df vector of probabilities. vector of location parameter. (The mode of the density) degrees of freedom (> 0, maybe non-integer). df = Inf is allowed. phi vector of scale parameters (>0). lmd Details vector of skewness parameters (>0). If is 1, reduced to symmetric student t distribution. The random variable y follows a split-t distribution with ν>0 degrees of freedom, y~t(µ, φ, λ, ν), if its density function is of the form where, CK(µ, φ, ν, )I(y µ) + CK(µ, λφ, ν)i(y > µ), K(µ, φ, ν, ) = [ν/(ν + (y µ) 2 /φ 2 )] (ν+1)/2 is the kernel of a student t density with variance φ 2 ν/(ν 2) and is the normalization constant. c = 2[(1 + λ)φ( ν)beta(ν/2, 1/2)] 1

6 6 rsplitt Value qsplitt gives the quantile function. Invalid arguments will result in return value NaN, with a warning. The length of the result is determined by n for rsplitt, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used. Author(s) Feng Li, Jiayue Zeng References Li, F., Villani, M., & Kohn, R. (2009). Flexible modeling of conditional distributions using smooth mixtures of asymmetric student t densities. Journal of Statistical Planning & Inference, 140(12), See Also splitt_mean(), splitt_var(),splitt_skewness() and splitt_kurtosis() for numerical characteristics of the Split-t distribution. Examples n <- 5 x <- c(0.25,0.5,0.75) q <- c(0.25,0.5,0.75) p <- c(0.25,0.5,0.75) mu <- c(0,1,2) df <- rep(10,3) phi <- c(0.5,1,2) lmd <- c(1,2,3) qsplitt0 <- qsplitt(p, mu, df, phi, lmd) rsplitt random generation for Split-t distribution Usage Density, distribution function, quantile function and random generation for the split student-t distribution. rsplitt(n, mu, df, phi, lmd)

7 rsplitt 7 Arguments n mu df number of observations. If length(n) > 1, the length is taken to be the number required. vector of location parameter. (The mode of the density) degrees of freedom (> 0, maybe non-integer). df = Inf is allowed. phi vector of scale parameters (>0). lmd Details Value vector of skewness parameters (>0). If is 1, reduced to symmetric student t distribution. The random variable y follows a split-t distribution with ν>0 degrees of freedom, y~t(µ, φ, λ, ν), if its density function is of the form where, CK(µ, φ, ν, )I(y µ) + CK(µ, λφ, ν)i(y > µ), K(µ, φ, ν, ) = [ν/(ν + (y µ) 2 /φ 2 )] (ν+1)/2 is the kernel of a student t density with variance φ 2 ν/(ν 2) and is the normalization constant. c = 2[(1 + λ)φ( ν)beta(ν/2, 1/2)] 1 rsplitt generates random deviates. (dsplitt, psplitt, qsplitt and rsplitt are all vectors.) Invalid arguments will result in return value NaN, with a warning. The length of the result is determined by n for rsplitt, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used. Author(s) Feng Li, Jiayue Zeng References Li, F., Villani, M., & Kohn, R. (2009). Flexible modeling of conditional distributions using smooth mixtures of asymmetric student t densities. Journal of Statistical Planning & Inference, 140(12), See Also splitt_mean(), splitt_var(),splitt_skewness() and splitt_kurtosis() for numerical characteristics of the Split-t distribution.

8 8 split-t Examples n <- 5 x <- c(0.25,0.5,0.75) q <- c(0.25,0.5,0.75) p <- c(0.25,0.5,0.75) mu <- c(0,1,2) df <- rep(10,3) phi <- c(0.5,1,2) lmd <- c(1,2,3) rsplitt0 <- rsplitt(n, mu, df, phi, lmd) split-t Moments of the Split-t distribution Usage Compute the mean, variance, skewness and kurtosis for the split student-t distribution with df degrees of freedom. splitt_mean(mu, df, phi, lmd) splitt_var(df, phi, lmd) splitt_skewness(df, phi, lmd) splitt_kurtosis(df, phi, lmd) Arguments mu df vector of location parameter. (The mode of the density) degrees of freedom (> 0, maybe non-integer). df = Inf is allowed. phi vector of scale parameters (> 0). lmd Details vector of skewness parameters (> 0). If is 1, reduced to symmetric student t distribution. The random variable y follows a split-t distribution with ν>0 degrees of freedom, y~t(µ, φ, λ, ν), if its density function is of the form where, CK(µ, φ, ν, )I(y µ) + CK(µ, λφ, ν)i(y > µ), K(µ, φ, ν, ) = [ν/(ν + (y µ) 2 /φ 2 )] (ν+1)/2 is the kernel of a student t density with variance φ 2 ν/(ν 2) and c = 2[(1 + λ)φ( ν)beta(ν/2, 1/2)] 1

9 split-t 9 is the normalization constant. If y~t(µ, φ, λ, ν) then,,,, Value E(y) = µ + h V (y) = (1 + λ 3 )/(1 + λ)ν/(ν 2)φ 2 h 2 E[y E(y)] 3 = 2h 3 + 2hφ 2 (λ 2 + 1)ν/(ν 3) 3hφ 2 (λ 3 + 1)/(λ + 1)ν/(ν 2) E[y E(y)] 4 = (3ν 2 φ 4 (1+λ 5 ))/((1+λ)(ν 2)(ν 4)) 4h 4 +(6h ( 2)(1+λ 3 )νφ 2 )/((1+λ)(ν 2)) (8h 2 (λ 2 νφ 2 ))/(ν 3). splitt_mean gives the mean. splitt_var gives the variance. splitt_skewness gives the skewness. splitt_kurtosis gives the kurtosis. (splitt_mean, splitt_var,splitt_skeness and splitt_kurtosis are all vectors.) Invalid arguments will result in return value NaN, with a warning. Author(s) Feng Li, Jiayue Zeng References Li, F., Villani, M., & Kohn, R. (2009). Flexible modeling of conditional distributions using smooth mixtures of asymmetric student t densities. Journal of Statistical Planning & Inference, 140(12), See Also dsplitt(), psplitt(), qsplitt() and rsplitt() for the split-t distribution. Examples mu <- c(0,1,2) df <- rep(10,3) phi <- c(0.5,1,2) lmd <- c(1,2,3) mean0 <- splitt_mean(mu, df, phi, lmd) var0 <- splitt_var(df, phi, lmd) skewness0 <- splitt_skewness(df, phi, lmd) kurtosis0 <- splitt_kurtosis(df, phi, lmd)

10 Index Topic asymmetric student-t dsplitt, 2 psplitt, 3 qsplitt, 5 rsplitt, 6 split-t, 8 Topic distribution dsplitt, 2 psplitt, 3 qsplitt, 5 rsplitt, 6 split-t, 8 dng_dsplitt (dsplitt), 2 dng_psplitt (psplitt), 3 dng_qsplitt (qsplitt), 5 dng_rsplitt (rsplitt), 6 dng_splitt_kurtosis (split-t), 8 dng_splitt_mean (split-t), 8 dng_splitt_skewness (split-t), 8 dng_splitt_var (split-t), 8 dsplitt, 2, 9 psplitt, 3, 9 qsplitt, 5, 9 rsplitt, 6, 9 split-t, 8 splitt_kurtosis, 3, 4, 6, 7 splitt_kurtosis (split-t), 8 splitt_mean, 3, 4, 6, 7 splitt_mean (split-t), 8 splitt_skewness, 3, 4, 6, 7 splitt_skewness (split-t), 8 splitt_var, 3, 4, 6, 7 splitt_var (split-t), 8 10