Package 'sgt' reference manual

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:	2024-11-22 06:40:37 UTC
Source:	CRAN

The Skewed Generalized T Distribution

Description

Density, distribution function, quantile function and random generation for the skewed generalized t distribution.

Usage

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)

Arguments

`x`, `quant`	vector of quantiles.
`prob`	vector of probabilities.
`n`	number of observations. If `length(n) > 1`, the length is taken to be the number required.
`mu`	vector of parameters. Note that if `mean.cent == TRUE`, `mu` is the mean of the distribution. Otherwise, `mu` is the mode of the distribution.
`sigma`	vector of variance parameters. The default is 1. The variance of the distribution increases as `sigma` increases. Must be strictly positive.
`lambda`	vector of skewness parameters. Note that `-1 < lambda < 1`. If `lambda < 0`, the distribution is skewed to the left. If `lambda > 0`, the distribution is skewed to the right. If `lambda = 0`, then the distribution is symmetric.
`p`, `q`	vector of parameters. Smaller values of `p` and `q` result in larger values for the kurtosis of the distribution. Allowed to be infinite. Note that `p > 0`, `q > 0`, otherwise `NaNs` will be produced.
`mean.cent`	logical; if TRUE, `mu` is the mean of the distribution, otherwise `mu` is the mode of the distribution. May only be used if `p*q > 1`, otherwise `NaNs` will be produced.
`var.adj`	logical or a positive scalar. If `TRUE`, then `sigma` is rescaled so that `sigma` is the variance. If `FALSE`, then `sigma` is not rescaled. If `var.adj` is a positive scalar, then `sigma` is rescaled by `var.adj`. May only be used if `p*q > 2`, otherwise `NaNs` will be produced.
`log`, `log.p`	logical; if TRUE, probabilities p are given as log(p).
`lower.tail`	logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$ .

Details

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.

Value

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.

Author(s)

Carter Davis, [email protected]

Source

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.

References

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.

Examples

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')

Maximum Likelihood Estimation with the Skewed Generalized T Distribution

Description

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.

Usage

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, ...)

Arguments

`X.f`	A formula specifying the data, or the function of the data with parameters, that should be used in the maximisation procedure. `X` should be on the left-hand side and the right-hand side should be the data or function of the data that should be used.
`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. `mu`, `sigma`, `lambda`, `p`, and `q` should be on the left-hand side of these formulas respectively.
`data`	an optional data frame in which to evaluate the variables in `formula` and `weights`. Can also be a list or an environment.
`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 `optimx` function in the `optimx` package. See `?optimx` for a list of methods that can be used. Note that the method that achieves the highest log-likelihood value is the method that is printed and reported. The default method is to use both "Nelder-Mead" and the "BFGS" methods.
`itnmax`	If provided as a vector of the same length as `method`, gives the maximum number of iterations or function values for the corresponding method. If a single number is provided, this will be used for all methods.
`hessian.method`	method used to calculate the hessian of the final estimates, either "Richardson" or "complex". This method is passed to the `hessian` function in the `numDeriv` package. See `?hessian` for details.
`gradient.method`	method used to calculate the gradient of the final estimates, either "Richardson", "simple", or "complex". This method is passed to the `grad` function in the `numDeriv` package. See `?grad` for details.
`mean.cent`, `var.adj`	arguments passed to the skewed generalized t distribution function (see `?dsgt`).
`...`	further arguments that are passed to the `control` argument in the `optimx` function in the `optimx` package. See `?optimx` for a list of arguments that can be used in the `control` argument.

Details

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.

Value

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`	`convcode` returned from the `optimx` function in the `optimx` package of the method that achieved the greatest log-likelihood value. See `?optimx` for the different `convcode` values.
`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 `data.frame` of class `"optimx"` that contains the results of the `optimx` maximization for every method (not just the method that achieved the highest log-likelihood value). See `?optimx` for details.
`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

Author(s)

Carter Davis, [email protected]

References

Davis, Carter, James McDonald, and Daniel Walton (2015). "A Generalized Regression Specification using the Skewed Generalized T Distribution" working paper.

Examples

# 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 with the Skewed Generalized T Distribution

Description

Summary the maximum-likelihood estimation including standard errors and t-values.

Usage

## 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, ...)

Arguments

`object`	object of class `'MLE'` or of class `'mult.MLE'`, usually a result from maximum-likelihood estimation.
`...`	currently not used.

Value

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 `maxim`-object. `NULL` if unconstrained maximization.

summary.mult.MLE returns a list of class 'summary.mult.MLE' with components of class 'summary.MLE'.

Author(s)

Carter Davis, [email protected]

Examples

### 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 with the Skewed Generalized T Distribution

Description

Summary the maximum-likelihood estimation.

Usage

## S3 method for class 'sgtest'
summary(object, ...)
## S3 method for class 'sgtest'
summary(object, ...)

Arguments

`object`	object of class `'sgtest'`, usually a result from maximum-likelihood estimation.
`...`	currently not used.

Value

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`	`convcode` returned from the `optimx` function in the `optimx` package of the method that achieved the greatest log-likelihood value. See `?optimx` for the different `convcode` values.
`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 `data.frame` of class `"optimx"` that contains the results of the `optimx` maximization for every method (not just the method that achieved the highest log-likelihood value). See `?optimx` for details.
`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 `data.frame` containing the estimates, standard errors, z scores, and p-values of the estimates.

Author(s)

Carter Davis, [email protected]

Examples

# 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))

Package 'sgt'

Help Index

The Skewed Generalized T Distribution

Description

Usage

Arguments

Details

Value

Author(s)

Source

References

See Also

Examples

Maximum Likelihood Estimation with the Skewed Generalized T Distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Summary the Maximum-Likelihood Estimation with the Skewed Generalized T Distribution

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Summary the Maximum-Likelihood Estimation with the Skewed Generalized T Distribution

Description

Usage

Arguments

Value

Author(s)

See Also

Examples