ufitstab.cauchy.mix {alphastable}

ufitstab.cauchy.mix

Description

estimates the parameters of a k-component mixture of Cauchy distributions. Assuming that k is known, given the vector of initial values of entire parameter space, it uses the EM algorithm to estimate the parameters of the k-component mixture of Cauchy distributions.

Usage

ufitstab.cauchy.mix(y, k, omega0, beta0, sigma0, mu0)

Arguments

y

vector of observations

k

number of components

omega0

initial value for weight vector to start the EM algorithm

beta0

initial value for skewness vector to start the EM algorithm

sigma0

initial value for scale vector to start the EM algorithm

mu0

initial value for location vector to start the EM algorithm

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 P are mutually independent.

Value

omega-bar

a k-component vector of estimated values for the weight vector

beta-bar

a k-component vector of estimated values for the skewness vector

sigma-bar

a k-component vector of estimated values for the scale vector

mu-bar

a k-component vector of estimated values for the location vector

Note

We use the survival times in days of 72 guinea pigs infected with different doses of tubercle bacilli, see Bjerkedal (1960). We note that the EM algorithm is robust with respect to the initial values.

Author(s)

Mahdi Teimouri, Adel Mohammadpour, and Saralees Nadarajah

References

Bjerkedal, T. (1960) Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, American Journal of Epidemiology, 72, 130-148.

Examples

# In the following, we give an example that fits a two-component mixture of Cauchy distributions
# to the survival times (in days) of 72 guinea pigs through the EM algorithm. For this, the initial
# values are: omega_0=(0.65,0.35), sigma_0=(20,50), beta_0=(0.20,0.05), and mu_0=(95,210).
library("stabledist")
y<-c(10,33,44,56,59,72,74,77,92,93,96,100,100,102,105,107,107,108,108,108,
     109,112,121,122,122,124,130,134,136,139,144,146,153,159,160,163,163,
     168,171,172,176,113,115,116,120,183,195,196,197,202,213,215,216,222,
     230,231,240,245,251,253,254,255,278,293,327,342,347,361,402,432,458,
     555)
ufitstab.cauchy.mix(y,2,c(0.65,0.35),c(0.20,0.05),c(20,50),c(95,210))