Title: | Skewed Generalized T Distribution Tree |
---|---|
Description: | Density, distribution function, quantile function and random generation for the skewed generalized t distribution. This package also provides a function that can fit data to the skewed generalized t distribution using maximum likelihood estimation. |
Authors: | Carter Davis |
Maintainer: | Carter Davis <[email protected]> |
License: | GPL (>= 3) |
Version: | 2.0 |
Built: | 2024-12-22 06:36:01 UTC |
Source: | CRAN |
Density, distribution function, quantile function and random generation for the skewed generalized t distribution.
dsgt(x, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE, log = FALSE) psgt(quant, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE, lower.tail = TRUE, log.p = FALSE) qsgt(prob, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE, lower.tail = TRUE, log.p = FALSE) rsgt(n, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE)
dsgt(x, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE, log = FALSE) psgt(quant, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE, lower.tail = TRUE, log.p = FALSE) qsgt(prob, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE, lower.tail = TRUE, log.p = FALSE) rsgt(n, mu = 0, sigma = 1, lambda = 0, p = 2, q = Inf, mean.cent = TRUE, var.adj = TRUE)
x , quant
|
vector of quantiles. |
prob |
vector of probabilities. |
n |
number of observations. If |
mu |
vector of parameters. Note that if |
sigma |
vector of variance parameters. The default is 1. The variance of the distribution increases as |
lambda |
vector of skewness parameters. Note that |
p , q
|
vector of parameters. Smaller values of |
mean.cent |
logical; if TRUE, |
var.adj |
logical or a positive scalar. If |
log , log.p
|
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
If mu
, sigma
, lambda
, p
, or q
are not specified they assume the default values of mu = 0
, sigma = 1
, lambda = 0
, p = 2
, and q = Inf
. These default values yield a standard normal distribution.
See vignette('sgt')
for the probability density function, moments, and various special cases of the skewed generalized t distribution.
dsgt
gives the density,
psgt
gives the distribution function,
qsgt
gives the quantile function, and
rsgt
generates random deviates.
The length of the result is determined by n
for
rsgt
, and is the maximum of the lengths of the
numerical arguments for the other functions.
The numerical arguments other than n
are recycled to the
length of the result. Only the first elements of the logical
arguments are used.
sigma <= 0
, lambda <= -1
, lambda >= 1
, p <= 0
, and q <= 0
are errors and return NaN
. Also, if mean.cent
is TRUE
but codep*q <= 1, the result is an error and NaNs
are produced. Similarly, if var.adj
is TRUE
but codep*q <= 2, the result is an error and NaNs
are produced.
Carter Davis, [email protected]
For psgt
, based on
a transformation of the cumulative probability density function that uses the incomplete beta function or incomplete gamma function.
For qsgt
, based on
solving for the inverse of the psgt
function that uses the inverse of the incomplete beta function or incomplete gamma function.
For rsgt
, the algorithm simply uses the qsgt
function with probabilities that are uniformly distributed.
Hansen, C., McDonald, J. B., and Newey, W. K. (2010) "Instrumental Variables Regression with Flexible Distributions" Journal of Business and Economic Statistics, volume 28, 13-25.
Kerman, S. C., and McDonald, J. B. (2012) "Skewness-Kurtosis Bounds for the Skewed Generalized T and Related Distributions" Statistics and Probability Letters, volume 83, 2129-2134.
Theodossiou, Panayiotis (1998) "Financial Data and the Skewed Generalized T Distribution" Management Science, volume 44, 1650-1661.
Distributions for other standard distributions which are special cases of the skewed generalized t distribution, including dt
for the t distribution, dnorm
for the normal distribution, and dunif
for the uniform distribution. Other special cases of the skewed generalized t distribution include the generalized t distribution in the gamlss.dist
package, the skewed t distribution in the skewt
package, the exponential power distribution (also known as the generalized error distribution) in the normalp
package, and the Laplace distribution in the rmutil
package. Also see beta
for the beta function.
require(graphics) ### This shows how to get a normal distribution x = seq(-4,6,by=0.05) plot(x, dnorm(x, mean=1, sd=1.5), type='l') lines(x, dsgt(x, mu=1, sigma=1.5), col='blue') ### This shows how to get a cauchy distribution plot(x, dcauchy(x, location=1, scale=1.3), type='l') lines(x, dsgt(x, mu=1, sigma=1.3, q=1/2, mean.cent=FALSE, var.adj = sqrt(2)), col='blue') ### This shows how to get a Laplace distribution plot(x, dsgt(x, mu=1.2, sigma=1.8, p=1, var.adj=FALSE), type='l', col='blue') ### This shows how to get a uniform distribution plot(x, dunif(x, min=1.2, max=2.6), type='l') lines(x, dsgt(x, mu=1.9, sigma=0.7, p=Inf, var.adj=FALSE), col='blue')
require(graphics) ### This shows how to get a normal distribution x = seq(-4,6,by=0.05) plot(x, dnorm(x, mean=1, sd=1.5), type='l') lines(x, dsgt(x, mu=1, sigma=1.5), col='blue') ### This shows how to get a cauchy distribution plot(x, dcauchy(x, location=1, scale=1.3), type='l') lines(x, dsgt(x, mu=1, sigma=1.3, q=1/2, mean.cent=FALSE, var.adj = sqrt(2)), col='blue') ### This shows how to get a Laplace distribution plot(x, dsgt(x, mu=1.2, sigma=1.8, p=1, var.adj=FALSE), type='l', col='blue') ### This shows how to get a uniform distribution plot(x, dunif(x, min=1.2, max=2.6), type='l') lines(x, dsgt(x, mu=1.9, sigma=0.7, p=Inf, var.adj=FALSE), col='blue')
This function allows data to be fit to the skewed generalized t distribution using maximum likelihood estimation. This function uses the maxLik
package to perform its estimations.
sgt.mle(X.f, mu.f = mu ~ mu, sigma.f = sigma ~ sigma, lambda.f = lambda ~ lambda, p.f = p ~ p, q.f = q ~ q, data = parent.frame(), start, subset, method = c("Nelder-Mead", "BFGS"), itnmax = NULL, hessian.method="Richardson", gradient.method="Richardson", mean.cent = TRUE, var.adj = TRUE, ...)
sgt.mle(X.f, mu.f = mu ~ mu, sigma.f = sigma ~ sigma, lambda.f = lambda ~ lambda, p.f = p ~ p, q.f = q ~ q, data = parent.frame(), start, subset, method = c("Nelder-Mead", "BFGS"), itnmax = NULL, hessian.method="Richardson", gradient.method="Richardson", mean.cent = TRUE, var.adj = TRUE, ...)
X.f |
A formula specifying the data, or the function of the data with parameters, that should be used in the maximisation procedure. |
mu.f , sigma.f , lambda.f , p.f , q.f
|
formulas including variables and
parameters that specify the functional form of the parameters in the skewed generalized t log-likelihood function. |
data |
an optional data frame in which to evaluate the variables in |
start |
a named list or named numeric vector of starting estimates for every parameter. |
subset |
an optional vector specifying a subset of observations to be used in the fitting process. |
method |
A list of the optimization methods to be used, which is passed directly to the |
itnmax |
If provided as a vector of the same length as |
hessian.method |
method used to calculate the hessian of the final estimates, either "Richardson" or "complex". This method is passed to the |
gradient.method |
method used to calculate the gradient of the final estimates, either "Richardson", "simple", or "complex". This method is passed to the |
mean.cent , var.adj
|
arguments passed to the skewed generalized t distribution function (see |
... |
further arguments that are passed to the |
The parameter names are taken from start
. If there is a name of a parameter or some data found on the right-hand side of one of the formulas but not found in data
and not found in start
, then an error is given.
This function simply uses the optimx
function in the optimx
package to maximize the skewed generalized t distribution log-likelihood function. It takes the method that returned the highest log-likelihood, and saves these results as the final estimates.
sgt.mle
returns a list of class "sgtest"
.
A list of class "sgtest"
has the following components:
maximum |
log-likelihood value of estimates (the last calculated value if not converged) of the method that achieved the greatest log-likelihood value. |
estimate |
estimated parameter value with the method that achieved the greatest log-likelihood value. |
convcode |
|
niter |
The amount of iterations that the method which achieved the the greatest log-likelihood value used to reach its estimate. |
best.method.used |
name of the method that achieved the greatest log-likelihood value. |
optimx |
A |
gradient |
vector, gradient value of the estimates with the method that achieved the greatest log-likelihood value. |
hessian |
matrix, hessian of the estimates with the method that achieved the greatest log-likelihood value. |
varcov |
variance/covariance matrix of the maximimum likelihood estimates |
std.error |
standard errors of the estimates |
Carter Davis, [email protected]
Davis, Carter, James McDonald, and Daniel Walton (2015). "A Generalized Regression Specification using the Skewed Generalized T Distribution" working paper.
The optimx
package and its documentation. The sgt.mle
simply uses its functions to maximize the skewed generalized t log-likelihood. Also, the sgt.mle
function uses the numDeriv
package to compute the final hessian and gradients of the estimates.
# SINGLE VARIABLE ESTIMATION: ### generate random variable set.seed(7900) n = 1000 x = rsgt(n, mu = 2, sigma = 2, lambda = -0.25, p = 1.7, q = 7) ### Get starting values and estimate the parameter values start = list(mu = 0, sigma = 1, lambda = 0, p = 2, q = 10) result = sgt.mle(X.f = ~ x, start = start, method = "nlminb") print(result) print(summary(result)) # REGRESSION MODEL ESTIMATION: ### Generate Random Data set.seed(1253) n = 1000 x1 = rnorm(n) x2 = runif(n) y = 1 + 2*x1 + 3*x2 + rnorm(n) data = as.data.frame(cbind(y, x1, x2)) ### Estimate Linear Regression Model reg = lm(y ~ x1 + x2, data = data) coef = as.numeric(reg$coefficients) rmse = summary(reg)$sigma start = c(b0 = coef[1], b1 = coef[2], b2 = coef[3], g0 = log(rmse)+log(2)/2, g1 = 0, g2 = 0, d0 = 0, d1 = 0, d2 = 0, p = 2, q = 10) ### Set up Model X.f = X ~ y - (b0 + b1*x1 + b2*x2) mu.f = mu ~ 0 sigma.f = sigma ~ exp(g0 + g1*x1 + g2*x2) lambda.f = lambda ~ (exp(d0 + d1*x1 + d2*x2)-1)/(exp(d0 + d1*x1 + d2*x2)+1) ### Estimate Regression with a skewed generalized t error term ### This estimates the regression model from the Davis, ### McDonald, and Walton (2015) paper cited in the references section ### q is in reality infinite since the error term is normal result = sgt.mle(X.f = X.f, mu.f = mu.f, sigma.f = sigma.f, lambda.f = lambda.f, data = data, start = start, var.adj = FALSE, method = "nlm") print(result) print(summary(result))
# SINGLE VARIABLE ESTIMATION: ### generate random variable set.seed(7900) n = 1000 x = rsgt(n, mu = 2, sigma = 2, lambda = -0.25, p = 1.7, q = 7) ### Get starting values and estimate the parameter values start = list(mu = 0, sigma = 1, lambda = 0, p = 2, q = 10) result = sgt.mle(X.f = ~ x, start = start, method = "nlminb") print(result) print(summary(result)) # REGRESSION MODEL ESTIMATION: ### Generate Random Data set.seed(1253) n = 1000 x1 = rnorm(n) x2 = runif(n) y = 1 + 2*x1 + 3*x2 + rnorm(n) data = as.data.frame(cbind(y, x1, x2)) ### Estimate Linear Regression Model reg = lm(y ~ x1 + x2, data = data) coef = as.numeric(reg$coefficients) rmse = summary(reg)$sigma start = c(b0 = coef[1], b1 = coef[2], b2 = coef[3], g0 = log(rmse)+log(2)/2, g1 = 0, g2 = 0, d0 = 0, d1 = 0, d2 = 0, p = 2, q = 10) ### Set up Model X.f = X ~ y - (b0 + b1*x1 + b2*x2) mu.f = mu ~ 0 sigma.f = sigma ~ exp(g0 + g1*x1 + g2*x2) lambda.f = lambda ~ (exp(d0 + d1*x1 + d2*x2)-1)/(exp(d0 + d1*x1 + d2*x2)+1) ### Estimate Regression with a skewed generalized t error term ### This estimates the regression model from the Davis, ### McDonald, and Walton (2015) paper cited in the references section ### q is in reality infinite since the error term is normal result = sgt.mle(X.f = X.f, mu.f = mu.f, sigma.f = sigma.f, lambda.f = lambda.f, data = data, start = start, var.adj = FALSE, method = "nlm") print(result) print(summary(result))
Summary the maximum-likelihood estimation including standard errors and t-values.
## S3 method for class 'MLE' summary(object, ...) ## S3 method for class 'mult.MLE' summary(object, ...)
## S3 method for class 'MLE' summary(object, ...) ## S3 method for class 'mult.MLE' summary(object, ...)
object |
object of class |
... |
currently not used. |
summary.MLE
returns an
object of class 'summary.MLE'
with the following components:
parameters |
names of parameters used in the estimation procedure. |
type |
type of maximisation. |
iterations |
number of iterations. |
code |
code of success. |
message |
a short message describing the code. |
loglik |
the loglik value in the maximum. |
estimate |
numeric matrix, the first column contains the parameter estimates, the second the standard errors, third t-values and fourth corresponding probabilities. |
fixed |
logical vector, which parameters are treated as constants. |
NActivePar |
number of free parameters. |
constraints |
information about the constrained optimization.
Passed directly further from |
summary.mult.MLE
returns a list of class 'summary.mult.MLE'
with components of class 'summary.MLE'
.
Carter Davis, [email protected]
the maxLik
CRAN package
### Showing how to fit a simple vector of data to the skewed ### generalized t distribution. require(graphics) require(stats) set.seed(123456) x = rt(100, df=10) X.f = X ~ x start = list(mu = 0, sigma = 2, lambda = 0, p = 2, q = 12) result = sgt.mle(X.f = X.f, start = start, finalHessian = "BHHH") sumResult = summary(result) print(result) coef(result) print(sumResult) ### Note that the t distribution is a special case of the ### skewed generalized t distribution
### Showing how to fit a simple vector of data to the skewed ### generalized t distribution. require(graphics) require(stats) set.seed(123456) x = rt(100, df=10) X.f = X ~ x start = list(mu = 0, sigma = 2, lambda = 0, p = 2, q = 12) result = sgt.mle(X.f = X.f, start = start, finalHessian = "BHHH") sumResult = summary(result) print(result) coef(result) print(sumResult) ### Note that the t distribution is a special case of the ### skewed generalized t distribution
Summary the maximum-likelihood estimation.
## S3 method for class 'sgtest' summary(object, ...)
## S3 method for class 'sgtest' summary(object, ...)
object |
object of class |
... |
currently not used. |
summary.sgtest
returns an
object of class 'summary.sgtest'
with the following components:
maximum |
log-likelihood value of estimates (the last calculated value if not converged) of the method that achieved the greatest log-likelihood value. |
estimate |
estimated parameter value with the method that achieved the greatest log-likelihood value. |
convcode |
|
niter |
The amount of iterations that the method which achieved the the greatest log-likelihood value used to reach its estimate. |
best.method.used |
name of the method that achieved the greatest log-likelihood value. |
optimx |
A |
gradient |
vector, gradient value of the estimates with the method that achieved the greatest log-likelihood value. |
hessian |
matrix, hessian of the estimates with the method that achieved the greatest log-likelihood value. |
varcov |
variance/covariance matrix of the maximimum likelihood estimates |
std.error |
standard errors of the estimates |
z.score |
the z score of the estimates |
p.value |
the p-values of the estimates |
summary.table |
a |
Carter Davis, [email protected]
the optimx
CRAN package
# SINGLE VARIABLE ESTIMATION: ### generate random variable set.seed(7900) n = 1000 x = rsgt(n, mu = 2, sigma = 2, lambda = -0.25, p = 1.7, q = 7) ### Get starting values and estimate the parameter values start = list(mu = 0, sigma = 1, lambda = 0, p = 2, q = 10) result = sgt.mle(X.f = ~ x, start = start, method = "nlminb") print(result) print(summary(result))
# SINGLE VARIABLE ESTIMATION: ### generate random variable set.seed(7900) n = 1000 x = rsgt(n, mu = 2, sigma = 2, lambda = -0.25, p = 1.7, q = 7) ### Get starting values and estimate the parameter values start = list(mu = 0, sigma = 1, lambda = 0, p = 2, q = 10) result = sgt.mle(X.f = ~ x, start = start, method = "nlminb") print(result) print(summary(result))