alphastable (version 0.2.1)

ufitstab.cauchy: ufitstab.cauchy

Description

estimates the parameters of the Cauchy distribution. Given the initial values of the skewness, scale, and location parameters, it uses the EM algorithm to estimate the parameters of the Cauchy distribution.

Usage

ufitstab.cauchy(y, beta0, sigma0, mu0, param)

Arguments

y

vector of observations

beta0

initial value of skewness parameter to start the EM algorithm

sigma0

initial value of scale parameter to start the EM algorithm

mu0

initial value of location parameter to start the EM algorithm

param

kind of parameterization; must be 0 or 1 for S_0 and S_1 parameterizations, respectively

Value

beta

estimated value of the skewness parameter

sigma

estimated value of the scale parameter

mu

estimated value of the location parameter

Details

Generally the EM algorithm seeks for the ML estimations when the log-likelihood function is not tractable mathematically. This is done by considering an extra missing (or latent) variable when the conditional expectation of the complete data log-likelihood given observed data and a guess of unknown parameter(s) is maximized. So, first we look for a stochastic representation. The representation given by the following proposition is valid for Cauchy distribution. Suppose Y~S_0(1,beta,sigma,mu) and T~S_{1}(1,1,1,0) (here S_0 and S_1 refer to parameterizations S_0 and S_1, respectively). Then Y=sigma*(1-|beta|)*N/Z+sigma*beta*T+mu where N~Z~N(0,1). The random variables N, Z, and T are mutually independent.

References

Davidian, M. and Giltinan, D.M. (1995). Nonlinear Mixed Effects Models for Repeated Measurement Data, Chapman and Hall.

Examples

# NOT RUN {
# In the following example, using the initial values beta_0=0.5, sigma_0=5, and mu_0=10,
# we apply the EM algorithm to estimate the parameters of Cauchy distribution fitted to
# the earthquake data given by the vector y.
y<-c(7.5,  8.8,   8.9,   9.4,   9.7,   9.7,   10.5,  10.5,  12.0,  12.2,  12.8,  14.6,
     14.9,  17.6,  23.9,  25.0,  2.9,   3.2 ,  7.6,   17.0,  8.0,   10.0,  10.0,  8.0,
     19.0,  21.0,  13.0,  22.0,  29.0,  31.0,  5.8,   12.0,  12.1,  20.5,  20.5,  25.3,
     35.9,  36.1,  36.3,  38.5,  41.4,  43.6,  44.4,  46.1,  47.1,  47.7,  49.2,  53.1,
     4.0,   10.1,  11.1,  17.7,  22.5,  26.5,  29.0,  30.9,  37.8,  48.3,  62.0,  50.0,
     16.0,  62.0,  1.2,   1.6,   9.1,   3.7,   5.3,   7.4,   17.9,  19.2,  23.4,  30.0,
     38.9,  10.8,  15.7,  16.7,  20.8,  28.5,  33.1,  40.3,  8.0,   32.0,  30.0,  31.0,
     16.1,  63.6,  6.6,   9.3,   13.0,  17.3,  105.0, 112.0, 123.0, 5.0,   23.5,  26.0,
     0.5,   0.6,   1.3,   1.4,   2.6,   3.8,   4.0,   5.1,   6.2,   6.8,   7.5,   7.6,
     8.4,   8.5,   8.5,   10.6,  12.6,  12.7,  12.9,  14.0,  15.0,  16.0,  17.7,  18.0,
     22.0,  22.0,  23.0,  23.2,  29.0,  32.0,  32.7,  36.0,  43.5,  49.0,  60.0,  64.0,
     105.0, 122.0, 141.0, 200.0, 45.0,  130.0, 147.0, 187.0, 197.0, 203.0, 211.0, 17.0,
     19.6,  20.2,  21.1,  88.0,  91.0,  12.0,  148.0, 42.0,  85.0,  21.9,  24.2,  66.0,
     87.0,  23.4,  24.6,  25.7,  28.6,  37.4,  46.7,  56.9,  60.7,  61.4,  62.0,  64.0,
     82.0,  107.0, 109.0, 156.0, 224.0, 293.0, 359.0, 370.0, 25.4,  32.9,  92.2,  45.0,
     145.0, 300.0)
library("stabledist")
# }
# NOT RUN {
ufitstab.cauchy(y,0.5,5,10,0)
# }