# ihs: The Inverse Hyperbolic Sine Distribution In ihs: Inverse Hyperbolic Sine Distribution

## Description

Density, distribution function, quantile function and random generation for the inverse hyperbolic sine distribution.

## Usage

 1 2 3 4 5 6 dihs(x, mu = 0, sigma = SIGCONST, lambda = 0, k = 1, log = FALSE) pihs(q, mu = 0, sigma = SIGCONST, lambda = 0, k = 1, lower.tail = TRUE, log.p = FALSE) qihs(p, mu = 0, sigma = SIGCONST, lambda = 0, k = 1, lower.tail = TRUE, log.p = FALSE) rihs(n, mu = 0, sigma = SIGCONST, lambda = 0, k = 1) 

## Arguments

 x, q vector of quantiles. p vector of probabilities. n number of observations. If length(n) > 1, the length is taken to be the number required. mu vector of means. The default value is 0. sigma vector of standard deviations. The default value is sqrt((exp(2)-1)/2). lambda vector of skewness parameters. If lambda < 0, the distribution is skewed to the left. If lambda > 0, the distribution is skewed to the right. If lambda = 0, then the distribution is symmetric. k vector of parameters. This parameter controls the skewness of the distribution. log, log.p logical; if TRUE, probabilities p are given as log(p). lower.tail logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x].

## Details

If mu, sigma, lambda, or k are not specified they assume the default values of mu = 0, sigma is approximately sqrt((exp(2)-1)/2), lambda = 0, and k = 1. These default values give the distribution sinh(z), where z is a standard normal random variable.

An inverse hyperbolic sine random variable Y is defined by the transformation

Y = a + b*sinh( λ + Z/k)

where Z is a standard normal random variable, and a, b, λ, and k control the mean, variance, skewness, and kurtosis respectively. Thus the inverse hyperbolic sine distribution has density

f(x) = \frac{k e^{(-k^2/2) (log ( \frac{x-a}{b} + (\frac{(x-a)^2}{b^2} + 1)^{1/2}) - λ )^2}}{√{2 π ((a-x)^2+b^2)}}

and if we reparametrize the distribution so that the parameters include the mean (μ) and the standard deviation (σ) instead of a and b, then we let

b = \frac{2 σ}{√{( e^{2 λ + k^{-2}} + e^{-2 λ + k^{-2}} + 2 ) ( e^{k^{-2}} - 1 )}}

a = μ - \frac{b}{2} (( e^{λ} - e^{-λ} ) e^{\frac{1}{2 k^2}} )

Thus if μ = 0, σ = √{\frac{e^2-1}{2}}, λ = 0, and k = 1, then Y = sinh(Z).

## Value

dihs gives the density, pihs gives the distribution function, qihs gives the quantile function, and rihs generates random deviates.

The length of the result is determined by n for rihs, 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.

sigma <= 0 and k <= 0 are errors and return NaN.

## Author(s)

Carter Davis, carterdavis@byu.edu

## Source

dihs is calculated from the definition (in ‘Details’). [pqr]ihs are based on the relationship to the normal.

## References

Hansen, C., McDonald, J. B., and Theodossiou, P. (2007) "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models" Economics - The Open-Access, Open-Assessment E-Journal, volume 1, 1-20.

Hansen, C., McDonald, J. B., and Newey, W. K. (2010) "Instrumental Variables Regression with Flexible Distributions" Journal of Business and Economic Statistics, volume 28, 13-25.

Distributions for other standard distributions such as dnorm for the normal distribution and dlnorm for the log-normal distribution, which is also a transformation of a normal random variable.

## Examples

  1 2 3 4 5 6 7 8 9 10 require(graphics) ### This shows how default values of the IHS compare ### to a standard normal. x = seq(-5,5,by=0.05) plot(x, dnorm(x), type='l') lines(x, dihs(x), col='blue') pihs(0) pihs(0, lambda = -0.5) 

### Example output

Loading required package: maxLik