Version:	1.0-2
Date:	2022-04-19
Title:	Fitting and Testing Generalized Logistic Distributions
Description:	Tools for the generalized logistic distribution (Type I, also known as skew-logistic distribution), encompassing basic distribution functions (p, q, d, r, score), maximum likelihood estimation, and structural change methods.
LazyLoad:	yes
Depends:	R (≥ 2.10.0), zoo
Suggests:	strucchange, fxregime, lattice
Imports:	graphics, stats, sandwich
License:	GPL-2 | GPL-3
NeedsCompilation:	no
Packaged:	2022-04-19 07:43:01 UTC; zeileis
Author:	Achim Zeileis [aut, cre], Thomas Windberger [aut]
Maintainer:	Achim Zeileis <Achim.Zeileis@R-project.org>
Repository:	CRAN
Date/Publication:	2022-04-19 12:12:37 UTC

Harmonised Index of Consumer Prices (1990–2010, OECD)

Description

Time series data with HICP (Harmonised Index of Consumer Prices) for 21 countries (plus EU) for 1990–2010 as provided by the OECD; and corresponding seasonally adjusted inflation ratios.

Usage

  data("HICP")

  data("hicps")  

Format

Monthly multiple "zooreg" time series with "yearmon" index from Jan 1990 (HICP) or Feb 1990 (hicps) to Dec 2010 for 21 countries (plus EU).

Details

HICP contains the raw unadjusted Harmonised Index of Consumer Prices as provided by the OECD from which unadjusted inflation rates can be easily computed (see examples).

As the different countries have rather different seasonal patterns which vary over time (especially in the 2000s), they will typically require seasonal adjustment before modeling. Hence, a seasonally adjusted version of the inflation rate series is provided as hicps, where X-12-ARIMA (version 0.3) has been employed for adjusted. An alternative seasonal adjustment can be easily computed use stl (see examples).

Source

Organisation for Economic Co-operation and Development (OECD)

https://stats.oecd.org/

References

Wikipedia (2010). "Harmonised Index of Consumer Prices – Wikipedia, The Free Encyclopedia." https://en.wikipedia.org/wiki/Harmonised_Index_of_Consumer_Prices, accessed 2010-06-10.

Windberger T, Zeileis A (2014). Structural Breaks in Inflation Dynamics within the European Monetary Union. Eastern European Economics, 52(3), 66–88.

Examples

## price series
data("HICP", package = "glogis")

## corresponding raw unadjusted inflation rates (in percent)
hicp <- 100 * diff(log(HICP))

## seasonal adjustment of inflation rates (via STL)
hicps1 <- do.call("merge", lapply(1:ncol(hicp), function(i) {
  z <- na.omit(hicp[,i])
  coredata(z) <- coredata(as.ts(z) - stl(as.ts(z), s.window = 13)$time.series[, "seasonal"])
  z
}))
colnames(hicps1) <- colnames(hicp)

## load X-12-ARIMA adjusted inflation rates
data("hicps", package = "glogis")

## compare graphically for one country (Austria)
plot(hicp[, "Austria"], lwd = 2, col = "lightgray")
lines(hicps1[, "Austria"], col = "red")
lines(hicps[, "Austria"], col = "blue")
legend("topleft", c("unadjusted", "STL", "X-12-ARIMA"), lty = c(1, 1, 1),
  col = c("lightgray", "red", "blue"), bty = "n")

## compare graphically across all countries (via lattice)
if(require("lattice")) {
trellis.par.set(theme = canonical.theme(color = FALSE))
xyplot(merge(hicp, hicps1, hicps), 
  screen = names(hicp)[rep(1:ncol(hicp), 3)],
  col = c("lightgray", "red", "blue")[rep(1:3, each = ncol(hicp))],
  lwd = c(2, 1, 1)[rep(1:3, each = ncol(hicp))])
}

Segmented Fitting of the Generalized Logistic Distribution

Description

Fitting univariate generalized logisitc distributions (Type I: skew-logistic with location, scale, and shape parameters) to segments of time series data.

Usage

## S3 method for class 'glogisfit'
breakpoints(obj, h = 0.15, breaks = NULL, ic = "LWZ",
  hpc = "none", ...)

## S3 method for class 'breakpoints.glogisfit'
refit(object, ...)

## S3 method for class 'breakpoints.glogisfit'
coef(object, log = TRUE, ...)

## S3 method for class 'breakpoints.glogisfit'
fitted(object, type = c("mean", "variance", "skewness"), ...)

## S3 method for class 'breakpoints.glogisfit'
confint(object, parm = NULL, level = 0.95, breaks = NULL, 
  meat. = NULL, ...)

Arguments

Details

To test whether sequences (typically time series) of observations follow the same generalized logistic distribution, the stability of the parameters can be tested. If there is evidence for parameter instability, breakpoints can be estimated to find segments with stable parameters.

The methods from the strucchange and fxregime packages are leveraged. For testing, the generalized M-fluctuation tests from strucchange can directly be employed using gefp. For breakpoint estimation, the methods documented here provide a user interface to some internal functionality from the fxregime packages. They employ the (unexported) workhorse function gbreakpoints which is modeled after breakpoints from the strucchange package but employing user-defined estimation methods.

Optional support for high performance computing is available in the breakpoints method based on the foreach package for the dynamic programming algorithm. If hpc = "foreach" is to be used, a parallel backend should be registered before. See breakpoints for more information.

Value

breakpoints.glogisfit returns an object of class "breakpoints.glogisfit" that inherits from "gbreakpointsfull".

References

Windberger T, Zeileis A (2014). Structural Breaks in Inflation Dynamics within the European Monetary Union. Eastern European Economics, 52(3), 66–88.

Zeileis A, Shah A, Patnaik I (2010). Testing, Monitoring, and Dating Structural Changes in Exchange Rate Regimes. Computational Statistics and Data Analysis, 54(6), 1696–1706. doi: 10.1016/j.csda.2009.12.005.

See Also

glogisfit, fxregimes, breakpoints

Examples

## artifical data with one structural change
set.seed(1071)
x <- c(rglogis(50, -1, scale = 0.5, shape = 3), rglogis(50, 1, scale = 0.5, shape = 1))
x <- zoo(x, yearmon(seq(2000, by = 1/12, length = 100)))

## full sample estimation
gf <- glogisfit(x)

if(require("strucchange")) {

## structural change testing
gf_scus <- gefp(gf, fit = NULL)
plot(gf_scus, aggregate = FALSE)
plot(gf_scus, functional = meanL2BB)
sctest(gf_scus)
sctest(gf_scus, functional = meanL2BB)


## breakpoint estimation
gf_bp <- breakpoints(gf)
plot(gf_bp)
summary(gf_bp)
breakdates(gf_bp)
coef(gf_bp)
confint(gf_bp)

## fitted model
plot(x)
lines(gf_bp)
lines(fitted(gf_bp, type = "mean"), col = 4)
lines(confint(gf_bp))

}

The Generalized Logistic Distribution (Type I: Skew-Logitic)

Description

Density, distribution function, quantile function and random generation for the logistic distribution with parameters location and scale.

Usage

dglogis(x, location = 0, scale = 1, shape = 1, log = FALSE)
pglogis(q, location = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)
qglogis(p, location = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)
rglogis(n, location = 0, scale = 1, shape = 1)
sglogis(x, location = 0, scale = 1, shape = 1)

Arguments

Details

If location, scale, or shape are omitted, they assume the default values of 0, 1, and 1, respectively.

The generalized logistic distribution with location = \mu, scale = \sigma, and shape = \gamma has distribution function

F(x) = \frac{1}{(1 + e^{-(x-\mu)/\sigma})^\gamma}%

.

The mean is given by location + (digamma(shape) - digamma(1)) * scale, the variance by (psigamma(shape, deriv = 1) + psigamma(1, deriv = 1)) * scale^2) and the skewness by (psigamma(shape, deriv = 2) - psigamma(1, deriv = 2)) / (psigamma(shape, deriv = 1) + psigamma(1, deriv = 1))^(3/2)).

[dpq]glogis are calculated by leveraging the [dpq]logis and adding the shape parameter. rglogis uses inversion.

Value

dglogis gives the probability density function, pglogis gives the cumulative distribution function, qglogis gives the quantile function, and rglogis generates random deviates. sglogis gives the score function (gradient of the log-density with respect to the parameter vector).

References

Johnson NL, Kotz S, Balakrishnan N (1995) Continuous Univariate Distributions, volume 2. John Wiley & Sons, New York.

Shao Q (2002). Maximum Likelihood Estimation for Generalised Logistic Distributions. Communications in Statistics – Theory and Methods, 31(10), 1687–1700.

Windberger T, Zeileis A (2014). Structural Breaks in Inflation Dynamics within the European Monetary Union. Eastern European Economics, 52(3), 66–88.

Examples

## PDF and CDF
par(mfrow = c(1, 2))
x <- -100:100/10
plot(x, dglogis(x, shape = 2), type = "l", col = 4, main = "PDF", ylab = "f(x)")
lines(x, dglogis(x, shape = 1))
lines(x, dglogis(x, shape = 0.5), col = 2)
legend("topleft", c("generalized (0, 1, 2)", "standard (0, 1, 1)",
  "generalized (0, 1, 0.5)"), lty = 1, col = c(4, 1, 2), bty = "n")
plot(x, pglogis(x, shape = 2), type = "l", col = 4, main = "CDF", ylab = "F(x)")
lines(x, pglogis(x, shape = 1))
lines(x, pglogis(x, shape = 0.5), col = 2)

## artifical empirical example
set.seed(2)
x <- rglogis(1000, -1, scale = 0.5, shape = 3)
gf <- glogisfit(x)
plot(gf)
summary(gf)

Fitting the Generalized Logistic Distribution

Description

Fit a univariate generalized logisitc distribution (Type I: skew-logistic with location, scale, and shape parameters) to a sample of observations.

Usage

glogisfit(x, ...)
## Default S3 method:
glogisfit(x, weights = NULL, start = NULL, fixed = c(NA, NA, NA),
  method = "BFGS", hessian = TRUE, ...)
## S3 method for class 'formula'
glogisfit(formula, data, subset, na.action, weights, x = TRUE, ...)

## S3 method for class 'glogisfit'
plot(x, main = "", xlab = NULL, fill = "lightgray",
  col = "blue", lwd = 1, lty = 1, xlim = NULL, ylim = NULL,
  legend = "topright", moments = FALSE, ...)

## S3 method for class 'glogisfit'
summary(object, log = TRUE, breaks = NULL, ...)
## S3 method for class 'glogisfit'
coef(object, log = TRUE, ...)
## S3 method for class 'glogisfit'
vcov(object, log = TRUE, ...)

Arguments

Details

glogisfit estimates the generalized logistic distribution (Type I: skew-logistic) as given by dglogis. Optimization is performed numerically by optim using analytical gradients. For obtaining numerically more stable results the scale and shape parameters are specified in logs. Starting values are chosen as c(0, 0, 0), i.e., corresponding to a standard (symmetric) logistic distribution. If these fail, better starting values are obtained by running a Nelder-Mead optimization on the original problem (without logs) first.

A large list of standard extractor methods is supplied to conveniently compute with the fitted objects, including methods to the generic functions print, summary, plot (reusing hist and lines), coef, vcov, logLik, residuals, and estfun and bread (from the sandwich package).

The methods for coef, vcov, summary, and bread report computations pertaining to the scale/shape parameters in logs by default, but allow for switching back to the original levels (employing the delta method).

Visualization employs a histogramm of the original data along with lines for the estimated density.

Further structural change methods for "glogisfit" objects are described in breakpoints.glogisfit.

Value

glogisfit returns an object of class "glogisfit", i.e., a list with components as follows.

References

Shao Q (2002). Maximum Likelihood Estimation for Generalised Logistic Distributions. Communications in Statistics – Theory and Methods, 31(10), 1687–1700.

Windberger T, Zeileis A (2014). Structural Breaks in Inflation Dynamics within the European Monetary Union. Eastern European Economics, 52(3), 66–88.

See Also

dglogis, dlogis, breakpoints.glogisfit

Examples

## simple artificial example
set.seed(2)
x <- rglogis(1000, -1, scale = 0.5, shape = 3)
gf <- glogisfit(x)
plot(gf)
summary(gf)

## query parameters and associated moments
coef(gf)
coef(gf, log = FALSE)
gf$parameters
gf$moments