.- help for skewed bimodal generator .-Skewed bimodal generator ------------------------

^skbim^ p v1 v2 v3 v4 option [obs] [seednum] [sk1] [ku1] [sk2] [ku2]

Description -----------

The program generates random numbers from a bimodal distribution. The two unimo > dal distributions that make up the bimodal can be normal or skewed-normal (see sknor for more det > ails). Different arguments can be inputted to the function, as specified by 'option'

p: the probability of the 1st unimodal distribution (obviously the prob for the > 2nd will be 1-p) option: takes values 1, 2 or 3 and defines v1-v4. for option=1: v1 the mean v2 the variance of the 1st unimodal, v3 the mean > v4 the variance of the 2nd unimodal for option=2: v1 the mean v2 the variance of the bimodal, v3 & v4 the varia > nces of the unimodals for option=3: v1 the mean v2 the variance of the bimodal, v3 the mean v4 th > e variance of the 1st unimodal [obs]: number of observations to be generated (optional; set to current dataset size if omitted) [seednum]: seed number to be used for drawing 'random' numbers - makes a differ > ence if your simulations occur on the same second (i.e. running a batch of simulations) [sk1]: skewness of the 1st unimodal (optional; set to 0 if omitted) [ku1]: kurtosis of the 1st unimodal (optional; set to 3 if omitted) [sk2]: skewness of the 2nd unimodal (optional; set to 0 if omitted) [ku2]: kurtosis of the 2nd unimodal (optional; set to 3 if omitted)

Remarks -------

When using options 2 and 3, parameters of the distribution may not be computed > for certain sets of inputted values. for option 1 it must hold that: var - p*var(d1) - (1-p)*var(d2)>=0 for option 2 it must hold that: var > p*var(d1) + p(1-p)*(m1-m2)^2 where var the variance of the bimodal; m1,m2,v1,v2 means and variances of t > he unimodals

For the default skewness and kurtosis values the generated distributions are no > rmal. The program is based on a method described in "Ramberg et al, A probability Di > stribution and its uses in fitting data, Technometrics, 1979". The method uses a formula with 4 parameters and parameter values which generate data of various degrees of skewness and kurtosis have been provided in the pape > r. Only a few of those have been included in this program the command, but it can easily be updated with more. The included pairs of values are (sk,ku): (0,1.75), (0,3), (0,3.2), (0,3.3), (0 > ,3.4), (0,3.6), (0,4), (0,4.4), (0,5), (0,6), (0,7), (0,8), (0,9), (0.5,3), (0.5,4), ( > 0.5,5), (0.5,6), (0.5,7), (0.5,8), (0.5,9), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), ( > 1,10), (1.5,6), (1.5,7), (1.5,8), (1.5,9), (1.5,10), (1.5,11), (1.5,12), (2,9), (2,10) > , (2,11), (2,12), (2,13), (2,14), (2,15), For sk=2 and ku=9 the most extreme distribution is generated. Negative skewness values can be inputted for left-skew distributions (the oppos > ite values of the ones listed above)

Examples --------

. ^skbim 0.5 5 1 7 1.5 1^ . ^skbim 0.3 5 2 1.2 1.5 2 10000^ . ^skbim 0.3 5 2 1.2 1.5 2 10000 1234^ . ^skbim 0.5 9 3.5 8 2 3 10000 1234 1.5 6 -0.5 9^

Keywords -------- bimodal, normal, distribution, skewness (skew, skewed), kurtosis

Author -------

Evangelos Kontopantelis National Primary Care Reserch and Development Centre University of Manchester M13 9PL e.kontopantelis@manchester.ac.uk

Also see -------- Ramberg et al paper: http://www.jstor.org/view/00401706/ap040083/04a00080/0 STB: STB-41 sg44.1, STB-28 sg44