Student t Distribution
Estimating the parameters of a Student t distribution.
studentt (ldf = "logloglink", idf = NULL, tol1 = 0.1, imethod = 1)
studentt2(df = Inf, llocation = "identitylink", lscale = "loglink",
ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")
studentt3(llocation = "identitylink", lscale = "loglink", ldf = "logloglink",
ilocation = NULL, iscale = NULL, idf = NULL,
imethod = 1, zero = c("scale", "df"))llocation, lscale, ldf |
Parameter link functions for each parameter,
e.g., for degrees of freedom nu.
See |
ilocation, iscale, idf |
Optional initial values. If given, the values must be in range. The default is to compute an initial value internally. |
tol1 |
A positive value, the tolerance for testing whether an initial value is 1. Best to leave this argument alone. |
df |
Numeric, user-specified degrees of freedom. It may be of length equal to the number of columns of a response matrix. |
imethod, zero |
The Student t density function is
f(y;nu) = (gamma((nu+1)/2) / (sqrt(nu*pi) gamma(nu/2))) * (1 + y^2 / nu)^{-(nu+1)/2}
for all real y.
Then E(Y)=0 if nu>1 (returned as the fitted values),
and Var(Y)= nu/(nu-2)
for nu > 2.
When nu=1 then the Student t-distribution
corresponds to the standard Cauchy distribution,
cauchy1.
When nu=2 with a scale parameter of sqrt(2) then
the Student t-distribution
corresponds to the standard (Koenker) distribution,
sc.studentt2.
The degrees of freedom can be treated as a parameter to be estimated,
and as a real and not an integer.
The Student t distribution is used for a variety of reasons
in statistics, including robust regression.
Let Y = (T - mu) / sigma where
mu and sigma are the location
and scale parameters respectively.
Then studentt3 estimates the location, scale and
degrees of freedom parameters.
And studentt2 estimates the location, scale parameters
for a user-specified degrees of freedom, df.
And studentt estimates the degrees of freedom parameter only.
The fitted values are the location parameters.
By default the linear/additive predictors are
(mu, log(sigma), log log(nu))^T
or subsets thereof.
In general convergence can be slow, especially when there are covariates.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
studentt3() and studentt2() can handle multiple responses.
Practical experience has shown reasonably good initial values are
required. If convergence failure occurs try using arguments
such as idf.
Local solutions are also possible, especially when
the degrees of freedom is close to unity or
the scale parameter is close to zero.
A standard normal distribution corresponds to a t distribution
with infinite degrees of freedom. Consequently, if the data is close
to normal, there may be convergence problems; best to use
uninormal instead.
T. W. Yee
Student (1908). The probable error of a mean. Biometrika, 6, 1–25.
Zhu, D. and Galbraith, J. W. (2010). A generalized asymmetric Student-t distribution with application to financial econometrics. Journal of Econometrics, 157, 297–305.
tdata <- data.frame(x2 = runif(nn <- 1000))
tdata <- transform(tdata, y1 = rt(nn, df = exp(exp(0.5 - x2))),
y2 = rt(nn, df = exp(exp(0.5 - x2))))
fit1 <- vglm(y1 ~ x2, studentt, data = tdata, trace = TRUE)
coef(fit1, matrix = TRUE)
fit2 <- vglm(y1 ~ x2, studentt2(df = exp(exp(0.5))), data = tdata)
coef(fit2, matrix = TRUE) # df inputted into studentt2() not quite right
fit3 <- vglm(cbind(y1, y2) ~ x2, studentt3, data = tdata, trace = TRUE)
coef(fit3, matrix = TRUE)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.