ufitstab.skew {alphastable} R Documentation

## ufitstab.skew

### Description

using the EM algorithm, it estimates the parameters of skew stable distribution.

### Usage

`ufitstab.skew(y, alpha0, beta0, sigma0, mu0, param)`

### Arguments

 `y` vector of observations `alpha0` initial value of tail index parameter to start the EM algorithm `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

For any skew stable distribution we give a new representation by the following. Suppose `Y~ S_{0}(alpha, beta, sigma, mu), P~ S_{1}(alpha/2,1,(cos(pi*alpha/4))^(2/alpha),0)`, and `V~ S_{1}(alpha,1,1,0)`. Then, `Y=eta*(2P)^(1/2)*N+theta*V+ mu-lambda`, where `eta=sigma*(1-|beta|)^(1/alpha)`, `theta=sigma*sign(beta)*|beta|^(1/alpha)`, `lambda=sigma*beta*tan(pi*alpha/2)`, and `N~N(0,1)` follows a skew stable distribution. All random variables `N`, `P`, and `V` are mutually independent.

### Value

 `alpha` estimated value of the tail index parameter `beta` estimated value of the skewness parameter `sigma` estimated value of the scale parameter `mu` estimated value of the location parameter

### Note

Daily price returns of Abbey National shares between 31/7/91 and 8/10/91 (including `n=50` business days). By assuming that `p_{t}` denotes the price at `t`-th day, the price return at `t`-th day is defined as `(p_{t-1}-p_{t})/p_{t-1}`; for `t=2,...,n`, see Buckle (1995). We note that the EM algorithm is robust with respect to the initial values.

### References

Buckle, D. J. (1995). Bayesian inference for stable distributions, Journal of the American Statistical Association, 90(430), 605-613.

### Examples

```# For example, We use the daily price returns of Abbey National shares. Using the initial
# values as alpha_{0}=0.8, beta_{0}=0, sigma_{0}=0.25, and mu_{0}=0.25.
price<-c(296,296,300,302,300,304,303,299,293,294,294,293,295,287,288,297,
305,307,304,303,304,304,309,309,309,307,306,304,300,296,301,298,
295,295,293,292,307,297,294,293,306,303,301,303,308,305,302,301,
297,299)
x<-c()
n<-length(price)
for(i in 2:n){x[i]<-(price[i-1]-price[i])/price[i-1]}
library("stabledist")
ufitstab.skew(x[2:n],0.8,0,0.25,0.25,1)
```

[Package alphastable version 0.2.1 Index]