Student t Distribution

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Description

Estimating the parameters of a Student t distribution.

Usage

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studentt (ldf = "loglog", idf = NULL, tol1 = 0.1, imethod = 1)
studentt2(df = Inf, llocation = "identitylink", lscale = "loge",
          ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")
studentt3(llocation = "identitylink", lscale = "loge", ldf = "loglog",
          ilocation = NULL, iscale = NULL, idf = NULL,
          imethod = 1, zero = c("scale", "df"))

Arguments

llocation, lscale, ldf

Parameter link functions for each parameter, e.g., for degrees of freedom nu. See Links for more choices. The defaults ensures the parameters are in range. A loglog link keeps the degrees of freedom greater than unity; see below.

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

See CommonVGAMffArguments.

Details

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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Note

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.

Author(s)

T. W. Yee

References

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.

See Also

uninormal, cauchy1, logistic, huber2, sc.studentt2, TDist, simulate.vlm.

Examples

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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)

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