ufitstab.cauchy {alphastable}

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

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.

Value

beta

estimated value of the skewness parameter

sigma

estimated value of the scale parameter

mu

estimated value of the location parameter

Note

The set of data considered here is large recorded intensities (in Richter scale) of the earthquake at seismometer locations in western North America between 1940 and 1980, see Davidian and Giltinan (1995). Among the features, we focus on the 182 distances from the seismological measuring station to the epicenter of the earthquake (in km) as the variable of interest. This set of data can be found in package nlme. We note that ufitstab.cauchy() is robust with respect to the initial values.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

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

Examples

# 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")
ufitstab.cauchy(y,0.5,5,10,0)