# ufitstab.cauchy.mix: ufitstab.cauchy.mix In alphastable: Inference for Stable Distribution

## 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

 `1` ```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)

 ``` 1 2 3 4 5 6 7 8 9 10``` ```# 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)) ```