Mixed normals {essHist}

The mixture of normal distributions

Description

Density, distribution function and random generation for the mixture of normals with each component specified by mean and sd, and mixture weights by prob. paramExample gives detailed parameters for some examples specified by type.

Usage

dmixnorm(x, mean = c(0), sd = rep(1,length(mean)), 
          prob = rep(1,length(mean)), type = NULL, ...)
pmixnorm(x, mean = c(0), sd = rep(1,length(mean)), 
          prob = rep(1,length(mean)), type = NULL, ...)
rmixnorm(n, mean = c(0), sd = rep(1,length(mean)), 
          prob = rep(1,length(mean)), type = NULL)
paramExample(type)

Arguments

x

vector of locations.

n

integer; number of observations.

mean

vector of means for each mixture component.

sd

vector of standard deviations for each mixture component. Default is of unit variance for each component.

prob

vector of prior probability for each mixture component (i.e. mixture weights). All nonnegative values are allowed, and automatically recaled to ensure their sum equal to 1. Default is of equal probability for each component.

type

a (case insensitive) character string of example name; It includes examples from Marron & Wand (1992): "MW1", ..., "MW15", or equivalently "gauss", "skewed_unimodal", "strong_skewed", "kurtotic_unimodal", "outlier", "bimodal", "separated_bimodal", "skewed_bimodal", "trimodal", "claw", "double_claw", "asymmetric_claw", "asymmetric_double_claw", "smooth_comb", "discrete_comb"; It also includes "harp" example from Li et al. (2016).

...

further arguments passed to dnorm and pnorm.

Details

Users either provide, optionally, mean, sd and prob; or type. In case of providing type, the values of mean, sd and prob are ignored.

The default case is standard normal, the same as dnorm, pnorm and rnorm.

Value

dmixnorm gives the density, pmixnorm gives the distribution function, and rmixnorm generates random deviates.

The length of the result is determined by n for rmixnorm, and is the length of x for dmixnorm and pmixnorm.

paramExample gives a data frame with components mean, sd and prob.

References

Li, H., Munk, A., Sieling, H., and Walther, G. (2016). The essential histogram. arXiv:1612.07216.

Marron, J. S., & Wand, M. P. (1992). Exact mean integrated squred error. Ann. Statist., 20(2), 712–736.

See Also

Normal for standard normal distributions; Distributions for other standard distributions.

Examples

## Example harp 
type = "harp" 
#     generate random numbers
n = 500
Y = rmixnorm(n, type = type)
#     compute the density
x = seq(min(Y), max(Y), length.out = n)
f = dmixnorm(x, type = type)
#     compute the distribution
F = pmixnorm(x, type = type)
#     plots
op = par(mfrow = c(1,2))
plot(x, f, type = "l", main = "Harp Density")
rug(Y, col = 'red')
plot(x, F, type = "l", main = "Harp Distribution")
rug(Y, col = 'red')
par(op)