Type:	Package
Title:	Maximum Likelihood Estimation for Univariate Densities
Version:	1.5.0
Description:	User-friendly maximum likelihood estimation (Fisher (1921) <doi:10.1098/rsta.1922.0009>) of univariate densities.
License:	MIT + file LICENSE
Encoding:	UTF-8
LazyData:	true
Depends:	R (≥ 2.10)
Imports:	assertthat, extraDistr, tibble, logitnorm, actuar, nakagami, fGarch, rlang, intervals, Rfast, pracma, sads
Suggests:	testthat, knitr, rmarkdown, markdown, copula, dplyr, covr
RoxygenNote:	7.3.2
VignetteBuilder:	knitr
URL:	https://github.com/JonasMoss/univariateML, https://jonasmoss.github.io/univariateML/
BugReports:	https://github.com/JonasMoss/univariateML/issues
NeedsCompilation:	no
Packaged:	2025-03-04 12:48:34 UTC; jmoss
Author:	Jonas Moss [aut, cre], Thomas Nagler [ctb], Chitu Okoli [ctb]
Maintainer:	Jonas Moss <jonas.gjertsen@gmail.com>
Repository:	CRAN
Date/Publication:	2025-03-04 13:30:02 UTC

univariateML

Description

An R-package for fast, easy, and reliable maximum likelihood estimation for a selection of parametric univariate densities.

Details

Data analysis often requires the estimation of univariate densities. Maximum likelihood estimation is sensible for almost every univariate density used in practice. Moreover, the maximum likelihood estimator is typically consistent and efficient.

The purpose of this package is to

• Support maximum likelihood estimation of a large selection of densities.

• Supports plenty of generics such as plot and AIC to aid your data analysis.

Read the vignettes to learn more about univariateML: browseVignettes(package = "univariateML")

Author(s)

Maintainer: Jonas Moss jonas.gjertsen@gmail.com (ORCID)

Other contributors:

• Thomas Nagler mail@tnagler.com [contributor]

• Chitu Okoli chitu.okoli@skema.edu [contributor]

See Also

Useful links:

• https://github.com/JonasMoss/univariateML

• https://jonasmoss.github.io/univariateML/

• Report bugs at https://github.com/JonasMoss/univariateML/issues

Maximum likelihood estimated distribution

Description

Density, distribution function, quantile function and random generation for a univariate distribution estimated by maximum likelihood.

Usage

dml(x, obj, log = FALSE)

pml(q = q, obj, lower.tail = TRUE, log.p = FALSE)

qml(p = p, obj, lower.tail = TRUE, log.p = FALSE)

rml(n = n, obj)

Arguments

Details

dml is the density, pml is the distribution function, qml is the quantile function, and rml is the random variable generator.

These functions work like their counterparts in stats, e.g. Normal. The univariateML object contains both maximum likelihood estimates and the identity of the model these estimates were calculated under. These functions are wrappers around underlying density, distribution, quantile and random generation functions where unknown parameters are filled with the maximum likelihood estimates. See the example.

Value

dml gives the density, pml gives the distribution function, qml gives the quantile function, and rml generates random deviates.

Examples

## Simple example
obj <- mlnorm(airquality$Wind)
dml(0.5, obj) == dnorm(0.5, mean = obj[1], sd = obj[2])

obj <- mlbetapr(airquality$Wind)

# Plot the logarithm of the beta prime distribution.
plot(function(x) dml(x, obj, log = TRUE),
  from = 0, to = 20,
  main = "Logarithm of Density", ylab = NA, lwd = 2
)

Probability Plots Using Maximum Likelihood Estimates

Description

Make quantile-quantile plots and probability-probability plots using maximum likelihood estimation.

Usage

ppmlplot(y, obj, plot.it = TRUE, datax = FALSE, ...)

ppmlline(...)

ppmlpoints(y, obj, plot.it = TRUE, datax = TRUE, ...)

qqmlplot(y, obj, plot.it = TRUE, datax = FALSE, ...)

qqmlline(y, obj, datax = FALSE, probs = c(0.25, 0.75), qtype = 7, ...)

qqmlpoints(y, obj, plot.it = TRUE, datax = TRUE, ...)

Arguments

Details

qqmlplot produces a quantile-quantile plot (Q-Q plot) of the values in y with respect to the distribution defined by obj, which is either a univariateML object or a function returning a univariateML object when called with y. qqmlline adds a line to a <U+201C>theoretical<U+201D>, quantile-quantile plot which passes through the probs quantiles, by default the first and third quartiles. qqmlpointsbehaves like stats::points and adds a Q-Q plot to an existing plot.

ppmlplot, ppmlline, and ppmlpoints produce probability-probability plots (or P-P plots). They behave similarly to the quantile-quantile plot functions.

This function is modeled after qqnorm.

Quantile-quantile plots and probability-probability plots are only supported for continuous distributions.

Graphical parameters may be given as arguments to all the functions below.

Value

For qqmlplot, qqmlpoints, ppmlplot, and ppmlpoints, a list with components x (plotted on the x axis) and y (plotted on the y axis). qqmlline and ppmlline returns nothing.

References

M. B. Wilk, R. Gnadadesikan, Probability plotting methods for the analysis for the analysis of data, Biometrika, Volume 55, Issue 1, March 1968, Pages 1<U+2013>17, https://doi.org/10.1093/biomet/55.1.1

Examples

## Make a single probability plot with a line.

obj <- mlgamma(Nile)
qqmlplot(Nile, obj)
qqmlline(Nile, obj)

## Make multiple probability plots. datax = TRUE must be used to make this
## look good.

ppmlplot(airquality$Wind, mlgamma, main = "Many P-P plots")
ppmlpoints(airquality$Wind, mlexp, col = "red")
ppmlpoints(airquality$Wind, mlweibull, col = "purple")
ppmlpoints(airquality$Wind, mllnorm, col = "blue")

Abalone data

Description

Physical measurements of 4177 abalones, a species of sea snail.

Usage

abalone

Format

A tibble with 4,177 observations and 9 variables:

sex

Sex of the abalone, F is female, M male, and I infant.

length

Longest shell measurement.

diameter

Diameter perpendicular to length.

height

Height with with meat in shell.

whole_weight

Grams whole abalone.

shucked_weight

Grams weight of meat.

viscera_weight

Grams gut weight (after bleeding).

shell_weight

Grams after being dried.

rings

+1.5 gives the age in years.

Details

See the web page https://archive.ics.uci.edu/ml/datasets/Abalone for more information about the data set.

Source

Dua, D. and Graff, C. (2019). UCI Machine Learning Repository https://archive.ics.uci.edu/ml/. Irvine, CA: University of California, School of Information and Computer Science.

References

Ko, V., Hjort, N. L., & Hobaek Haff, I. (2019). Focused information criteria for copulas. Scandinavian Journal of Statistics.

Examples

abalone

Parametric Bootstrap on Distributions Fitted with Maximum Likelihood

Description

The parametric bootstrap is a resampling technique using random variates from a known parametric distribution. In this function the distribution of the random variates is completely determined by the unvariateML object object.

Usage

bootstrapml(
  object,
  reps = 1000,
  map = identity,
  reducer = stats::quantile,
  ...
)

Arguments

Details

For each bootstrap iteration a maximum likelihood estimate is calculated using the ⁠ml***⁠ function specified by object. The resulting numeric vector is then passed to map. The values returned by map is collected in an array and the reducer is called on each row of the array.

By default the map function is the identity and the default reducer is the quantile function taking the argument probs, which defaults to c(0.025, 0.975). This corresponds to a 95\ basic percentile confidence interval and is also reported by confint()

Note: The default confidence intervals are percentile intervals, not empirical intervals. These confidence intervals will in some cases have poor coverage as they are not studentized, see e.g. Carpenter, J., & Bithell, J. (2000).

Value

The transposed map-reduced bootstrap samples.

References

Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press.

Carpenter, J., & Bithell, J. (2000). Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Statistics in medicine, 19(9), 1141-1164.

See Also

confint() for an application of bootstrapml.

Examples

set.seed(1)
object <- mlgamma(mtcars$qsec)

## Calculate c(0.025, 0.975) confidence interval for the gamma parameters.
bootstrapml(object)

#            2.5%      97.5%
# shape 68.624945 160.841557
# rate   3.896915   9.089194

## The mean of a gamma distribution is shape/rate. Now we calculate a
## parametric bootstrap confidence interval for the mean with confidence
## limits c(0.05, 0.95)

bootstrapml(object, map = function(x) x[1] / x[2], probs = c(0.05, 0.95))

#       5%      95%
# 17.33962 18.31253

## Print a histogram of the bootstrapped estimates from an exponential.
object <- mlexp(mtcars$qsec)
hist(bootstrapml(object, reducer = identity))

Confidence Intervals for Maximum Likelihood Estimates

Description

Computes a confidence interval for one or more parameters in a unvariateML object.

Usage

## S3 method for class 'univariateML'
confint(object, parm = NULL, level = 0.95, Nreps = 1000, ...)

Arguments

Details

confint.univariateML is a wrapper for bootstrapml() that computes confidence intervals for the main parameters of object. The main parameters of object are the members of names(object). For instance, the main parameters of an object obtained from mlnorm are mean and sd. The confidence intervals are parametric bootstrap percentile intervals with limits (1-level)/2 and 1 - (1-level).

Value

A matrix or vector with columns giving lower and upper confidence limits for each parameter in parm.

See Also

stats::confint() for the generic function and bootstrapml() for the function used to calculate the confidence intervals.

Examples

object <- mlinvgauss(airquality$Wind)
confint(object) # 95% confidence interval for mean and shape
confint(object, "mean") # 95% confidence interval for the mean parameter
# confint(object, "variance") # Fails since 'variance isn't a main parameter.

Frequencies of butterflies collected in Malaya

Description

Species abundance data from 1943; a classical application of the logarithmic series distribution.

Usage

corbet

Format

A vector of size 501 containing integer observations between 1 and 24.

References

Fisher, R. A., Corbet, A. S., & Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. The Journal of Animal Ecology, 12(1), 42. https://doi.org/10.2307/1411

Examples

corbet

Decorate estimator

Description

The decorator constructs an ⁠ml***⁠. It first checks the input arguments, then call the defined estimator (with elements estimates and logLik), and the optional parameter support, then adds the required attributes.

Usage

decorator(name)

Arguments

Value

The proper ⁠ml***⁠ function.

Mortality data from ancient Egypt

Description

Age at death of 141 Roman era Egyptian mummies.

Usage

egypt

Format

A tibble with 141 observations and 2 variables:

age

Age at death.

sex

Sex of deceased; 82 males and 49 females.

Details

This data was collected by Spiegelberg (1901) and analyzed by Karl Pearson (1902) in the first volume of Biometrika. It was analyzed by Claeskens & Hjort (2008) and the data is based on their transcription.

References

Spiegelberg, W. (1901). Aegyptische und Griechische Eigennamen aus Mumientiketten der R<U+00F6>mischen Kaiserzeit.

Pearson, K. (1902). On the change in expectation of life in man during a period of circa 2000 years. Biometrika, 1(2), 261-264.

Claeskens, G., & Hjort, N. L. (2008). Model selection and model averaging. Cambridge University Press.

See Also

The source of the data is https://feb.kuleuven.be/public/u0043181/modelselection/datasets/egyptlives_data.txt

Examples

egypt

Returns appropriate starting value

Description

Returns appropriate starting value

Usage

get_start(default, name, ...)

Arguments

Value

Default value if name is not present

Inverse digamma function

Description

Calculates the inverse digamma function using Newton–Raphson. Works for y > -500. Uses Newton–Raphson with relative tolerance of eps^0.25.

Usage

inverse_digamma(y)

Arguments

Details

The number of iterations are few, 1 for most input values, especially those that are large. The starting value is the lower bound found by Batir (2017).

References

Batir, N. (2017). Inequalities for the inverses of the polygamma functions. arXiv. http://arxiv.org/abs/1705.06547

Merges two lists.

Description

Merges two lists.

Usage

listmerge(x, y, type = c("merge", "template"))

Arguments

Value

A merged list where conflicts are solved in favor of y. Does not preserve ordering.

Beta distribution maximum likelihood estimation

Description

Uses stat::nlm to estimate the parameters of the Beta distribution.

Usage

mlbeta(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Beta distribution see Beta.

For type, the option none is fastest.

Value

mlbeta returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape1 and shape2 and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Chapter 25. Wiley, New York.

See Also

Beta for the Beta density, nlm for the optimizer this function uses.

Examples

AIC(mlbeta(USArrests$Rape / 100))

Beta prime distribution maximum likelihood estimation

Description

This function does not estimate the scale parameter for the BetaPrime distribution. Transforms the data and uses stat::nlm to estimate the parameters of the Beta distribution.

Usage

mlbetapr(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Beta prime distribution see BetaPrime.

For type, the option none is fastest.

Value

mlbetapr returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape1 and shape2 and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Chapter 25. Wiley, New York.

See Also

BetaPrime for the Beta prime density, nlm for the optimizer this function uses, mlbeta for the Beta distribution maximum likelihood estimator.

Examples

AIC(mlbetapr(USArrests$Rape))

Binomial distribution maximum likelihood estimation

Description

For the density function of the Binomial distribution see Binomial.

Usage

mlbinom(x, na.rm = FALSE, ...)

Arguments

Details

The estimator computes both the size and prob parameter by default. Be aware that the likelihood will often be unbounded. According to Olkin et al. (1981), the likelihood is unbounded when \hat{\mu}/\hat{\sigma}^2 \leq 1, where \hat{\sigma}^2 is the biased sample variance. When the likelihood is unbounded,the maximum likelihood estimator can be regarded as a Poisson with lambda parameter equal to the mean of the observation.

When \hat{\mu}/\hat{\sigma}^2 \leq 1 and size is not supplied by the user, an error is cast. If size is provided and size < max(x), an error is cast.

The maximum likelihood estimator of size is unstable, and improvements exist. See, e.g., Carroll and Lomard (1985) and DasGupta and Rubin (2005).

Value

mlbinom returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for size and prob and the following attributes:

References

Olkin, I., Petkau, A. J., & Zidek, J. V. (1981). A comparison of n Estimators for the binomial distribution. Journal of the American Statistical Association, 76(375), 637-642. https://doi.org/10.1080/01621459.1981.10477697

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

Carroll, R. J., & Lombard, F. (1985). A Note on N Estimators for the Binomial Distribution. Journal of the American Statistical Association, 80(390), 423-426. https://doi.org/10.1080/01621459.1985.10478134

DasGupta, A., & Rubin, H. (2005). Estimation of binomial parameters when both n,p are unknown. Journal of Statistical Planning and Inference, 130(1-2), 391-404. https://doi.org/10.1016/j.jspi.2004.02.019

See Also

Binomial for the density.

Examples

# The likelihood will often be unbounded.
## Not run: 
mlbinom(ChickWeight$weight)

## End(Not run)
# Provide a size
mlbinom(ChickWeight$weight, size = 400)

# Or use mlpoiss, the limiting likelihood of the binomial.
mlpois(ChickWeight$weight)

Burr distribution maximum likelihood estimation

Description

The maximum likelihood estimator fails to exist when the data contains no values strictly smaller than 1. Then the likelihood converges to the likelihood of the Pareto distribution in this case.

Usage

mlburr(x, na.rm = FALSE, ...)

Arguments

Details

This function estimates the only the shape parameters of the Burr distribution. The shape is set to 1.

For the density function of the Burr distribution see Burr.

Value

mlburr returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape1 and shape2 and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 20. Wiley, New York.

See Also

Burr for the Burr density.

Examples

mlburr(abalone$length)

Cauchy distribution maximum likelihood estimation

Description

Calculates the estimates using nlm and an exponential transform of the location parameter. If n < 5, an exact solution is reported. In the edge case where no maximum likelihood estimator exists and error is thrown.

Usage

mlcauchy(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Cauchy distribution see Cauchy.

Value

mlcauchy returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for location and scale and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 16. Wiley, New York.

See Also

Cauchy for the Cauchy density, nlm for the optimizer this function uses.

Examples

mlcauchy(airquality$Temp)

Discrete uniform distribution maximum likelihood estimation

Description

For the density function of the Discrete uniform distribution see DiscreteUniform.

Usage

mldunif(x, na.rm = FALSE, ...)

Arguments

Value

mldunif returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for min and max and the following attributes:

References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

DiscreteUniform for the density.

Examples

mldunif(corbet)

Exponential distribution maximum likelihood estimation

Description

The maximum likelihood estimate of rate is the inverse sample mean.

Usage

mlexp(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the exponential distribution see Exponential.

Value

mlexp returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for rate and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 19. Wiley, New York.

See Also

Exponential for the exponential density.

Examples

mlexp(precip)

Gamma distribution maximum likelihood estimation

Description

Uses Newton-Raphson to estimate the parameters of the Birnbaum–Saunders distribution. The parameter mu is set to 0, hence only alpha and beta are estmated.

Usage

mlfatigue(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Birnbaum–Saunders distribution see BirnbaumSaunders.

Value

mlfatigue returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for alpha and beta, with mu=0, and the following attributes:

References

From, S. G., & Li, L. (2006). Estimation of the parameters of the Birnbaum<U+2013>Saunders distribution. Communications in Statistics: Theory and Methods, 35(12), 2157<U+2013>2169. https://doi.org/10.1080/03610920600853563

Examples

mlfatigue(precip)

Gamma distribution maximum likelihood estimation

Description

Uses Newton-Raphson to estimate the parameters of the Gamma distribution.

Usage

mlgamma(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Gamma distribution see GammaDist.

Value

mlgamma returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape and rate and the following attributes:

References

Choi, S. C, and R. Wette. "Maximum likelihood estimation of the parameters of the gamma distribution and their bias." Technometrics 11.4 (1969): 683-690.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 17. Wiley, New York.

See Also

GammaDist for the Gamma density.

Examples

mlgamma(precip)

Generalized Error distribution maximum likelihood estimation

Description

Joint maximum likelihood estimation as implemented by fGarch::gedFit.

Usage

mlged(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Student t-distribution see ged.

Value

mlged returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for the parameters mean, sd, nu and the following attributes:

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347<U+2013>370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint.

See Also

ged for the Student t-density.

Examples

mlged(precip)

Geometric distribution maximum likelihood estimation

Description

For the density function of the Geometric distribution see Geometric.

Usage

mlgeom(x, na.rm = FALSE, ...)

Arguments

Value

mlgeom returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for prob and the following attributes:

References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

Geometric for the density.

Examples

mlgeom(corbet)

Gompertz distribution maximum likelihood estimation

Description

Uses Newton-Raphson to estimate the parameters of the Gompertz distribution.

Usage

mlgompertz(x, na.rm = FALSE, ...)

Arguments

Details

For some data sets the maximum likelihood estimator of b fails to exist since the root of the profile maximum likelihood equation is non-positive. The value 1e-06 is returned in this case, along with a warning.

For the density function of the Gompertz distribution see Gompertz.

Value

mlgompertz returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for a and b and the following attributes:

References

Lenart, A. (2012). The Gompertz distribution and Maximum Likelihood Estimation of its parameters - a revision. MPIDR WORKING PAPER WP 2012-008. https://www.demogr.mpg.de/papers/working/wp-2012-008.pdf

See Also

Gompertz for the Gompertz density.

Examples

mlgompertz(precip)

Gumbel distribution maximum likelihood estimation

Description

Uses Newton-Raphson to estimate the parameters of the Gumbel distribution.

Usage

mlgumbel(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Gumbel distribution see Gumbel.

Value

mlgumbel returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for mu and s and the following attributes:

shape and sigma.

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Chapter 22. Wiley, New York.

See Also

Gumbel for the Gumbel density.

Examples

mlgumbel(precip)

Inverse Burr distribution maximum likelihood estimation

Description

The maximum likelihood estimator fails to exist when the data contains no values strictly greater than 1. Then the likelihood converges to the likelihood of the Pareto distribution in this case.

Usage

mlinvburr(x, na.rm = FALSE, ...)

Arguments

Details

mlinvburr(x) calls mlburr(1/x) internally.

For the density function of the Inverse Burr distribution see Inverse Burr.

Value

mlburr returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape1 and shape2 and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 20. Wiley, New York.

See Also

Inverse Burr for the Inverse Burr density.

Examples

mlburr(abalone$length)

Inverse Gamma distribution maximum likelihood estimation

Description

Transforms the data and uses Newton-Raphson to estimate the parameters of the Gamma distribution.

Usage

mlinvgamma(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the inverse Gamma distribution see InvGamma.

Value

A named numeric vector with maximum likelihood estimates for alpha and beta.

References

Choi, S. C, and R. Wette. "Maximum likelihood estimation of the parameters of the gamma distribution and their bias." Technometrics 11.4 (1969): 683-690.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 17. Wiley, New York.

Witkovsky, V. (2001). "Computing the Distribution of a Linear Combination of Inverted Gamma Variables". Kybernetika. 37 (1): 79<U+2013>90

See Also

InvGamma for the Inverse Gamma density.

Examples

mlinvgamma(precip)

Inverse Gaussian (Wald) maximum likelihood estimation

Description

The maximum likelihood estimate of mean is the empirical mean and the maximum likelihood estimate of 1/shape is the difference between the mean of reciprocals and the reciprocal of the mean.

Usage

mlinvgauss(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Inverse Gamma distribution see InverseGaussian.

Value

mlinvgauss returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for mean and shape and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 15. Wiley, New York.

See Also

InverseGaussian for the Inverse Gaussian density.

Examples

mlinvgauss(precip)

Inverse Weibull distribution maximum likelihood estimation

Description

The maximum likelihood estimate of shape and rate are calculated by calling mlweibull on the transformed data.

Usage

mlinvweibull(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the log normal distribution see InverseWeibull.

Value

mlinvweibull returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape and rate and the following attributes:

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Dutang, C., Goulet, V., & Pigeon, M. (2008). actuar: An R package for actuarial science. Journal of Statistical Software, 25(7), 1-37.

See Also

InverseWeibull for the Inverse Weibull density.

Examples

mlinvweibull(precip)

Kumaraswamy distribution maximum likelihood estimation

Description

Uses Newton-Raphson to estimate the parameters of the Kumaraswamy distribution.

Usage

mlkumar(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Kumaraswamy distribution see Kumaraswamy.

Value

mlkumar returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for a and b and the following attributes:

References

Jones, M. C. "Kumaraswamy's distribution: A beta-type distribution with some tractability advantages." Statistical Methodology 6.1 (2009): 70-81.

Kumaraswamy, Ponnambalam. "A generalized probability density function for double-bounded random processes." Journal of Hydrology 46.1-2 (1980): 79-88.

See Also

Kumaraswamy for the Kumaraswamy density.

Examples

AIC(mlkumar(USArrests$Rape / 100))

Laplace distribution maximum likelihood estimation

Description

The maximum likelihood estimate of mu is the sample median while the maximum likelihood estimate of sigma is mean absolute deviation from the median.

Usage

mllaplace(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Laplace distribution see Laplace.

Value

mllaplace returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for mu and sigma and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Chapter 24. Wiley, New York.

See Also

Laplace for the Laplace density.

Examples

mllaplace(precip)

Log-gamma distribution maximum likelihood estimation

Description

The maximum likelihood estimate of shapelog and ratelog are calculated by calling mlgamma() on the transformed data.

Usage

mllgamma(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the log normal distribution see Loggamma.

Value

mllgamma returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shapelog and ratelog and the following attributes:

References

Hogg, R. V. and Klugman, S. A. (1984), Loss Distributions, Wiley.

Dutang, C., Goulet, V., & Pigeon, M. (2008). actuar: An R package for actuarial science. Journal of Statistical Software, 25(7), 1-37.

See Also

Loggamma for the log normal density.

Examples

mllgamma(precip)

Logarithmic series distribution maximum likelihood estimation

Description

For the density function of the Logarithmic series distribution see Logarithmic series. For an example data set, see corbet.

Usage

mllgser(x, na.rm = FALSE, ...)

Arguments

Value

mllgser returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for theta.

References

Fisher, R. A., Corbet, A. S., & Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. The Journal of Animal Ecology, 12(1), 42. https://doi.org/10.2307/1411

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

Logarithmic series for the density.

Examples

theta_hat <- mllgser(corbet)

# The corbert data contains observations from 1 to 24.
observed <- table(corbet)

# The chi square test evaluated at the maximum likelihood is highly significant.
expected <- extraDistr::dlgser(1:24, theta_hat)
chisq.test(observed, p = expected / sum(expected))

# But chi square test evaluated at 0.997 (used in Corbet) is not.
expected <- extraDistr::dlgser(1:24, 0.997)
chisq.test(observed, p = expected / sum(expected))

# The chi square for `dzipf` is similar.
expected <- sads::dzipf(1:24, mlzipf(corbet)[1], mlzipf(corbet)[2]) * length(corbet)
chisq.test(observed, p = expected / sum(expected))

Log-logistic distribution maximum likelihood estimation

Description

The maximum likelihood estimate of shape and rate are calculated by transforming the data back to the logistic model and applying mllogis.

Usage

mlllogis(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the log-logistic distribution see Loglogistic

Value

mllogis returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape and rate and the following attributes:

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Dutang, C., Goulet, V., & Pigeon, M. (2008). actuar: An R package for actuarial science. Journal of Statistical Software, 25(7), 1-37.

See Also

Loglogistic for the log-logistic density.

Examples

mllnorm(precip)

Log-normal distribution maximum likelihood estimation

Description

The maximum likelihood estimate of meanlog is the empirical mean of the log-transformed data and the maximum likelihood estimate of sdlog is the square root of the biased sample variance based on the log-transformed data.

Usage

mllnorm(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the log normal distribution see Lognormal.

Value

mllnorm returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for meanlog and sdlog and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 14. Wiley, New York.

See Also

Lognormal for the log normal density.

Examples

mllnorm(precip)

Logistic distribution maximum likelihood estimation

Description

Calculates the estimates using nlm with an exponential transform of the location parameter.

Usage

mllogis(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the logistic distribution see Logistic.

Value

mllogis returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for location and scale and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Chapter 23. Wiley, New York.

See Also

Logistic for the Logistic density, nlm for the optimizer this function uses.

Examples

mllogis(precip)

Logit-Normal distribution maximum likelihood estimation

Description

The maximum likelihood estimate of mu is the empirical mean of the logit transformed data and the maximum likelihood estimate of sigma is the square root of the logit transformed biased sample variance.

Usage

mllogitnorm(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the logit-normal distribution see dlogitnorm.

Value

mllogitnorm returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for mu and sigma and the following attributes:

References

Atchison, J., & Shen, S. M. (1980). Logistic-normal distributions: Some properties and uses. Biometrika, 67(2), 261-272.

See Also

Normal for the normal density.

Examples

AIC(mllogitnorm(USArrests$Rape / 100))

Lomax distribution maximum likelihood estimation

Description

Uses Newton-Raphson to estimate the parameters of the Lomax distribution.

Usage

mllomax(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Lomax distribution see Lomax.

The likelihood estimator of the Lomax distribution is unbounded when mean(x^2) < 2*mean(x)^2. When this happens, the likelihood converges to an exponential distribution with parameter equal to the mean of the data. This is the natural limiting case for the Lomax distribution, and it is reasonable to use mlexp in this case.

Value

mllomax returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for lambda and kappa and the following attributes:

References

Kleiber, Christian; Kotz, Samuel (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley Series in Probability and Statistics, 470, John Wiley & Sons, p. 60

See Also

Lomax for the Lomax density.

Examples

set.seed(3)
mllomax(extraDistr::rlomax(100, 2, 4))

# The maximum likelihood estimator may fail if the data is exponential.
## Not run: 
set.seed(5)
mllomax(rexp(10))

## End(Not run)

Nakagami distribution maximum likelihood estimation

Description

The maximum likelihood estimates of shape and scale are calculated by calling mlgamma on the transformed data.

Usage

mlnaka(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Nakagami distribution see Nakagami.

Value

mlgamma returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape and rate and the following attributes:

References

Choi, S. C, and R. Wette. "Maximum likelihood estimation of the parameters of the gamma distribution and their bias." Technometrics 11.4 (1969): 683-690.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 17. Wiley, New York.

See Also

Nakagami for the Nakagami distribution. GammaDist for the closely related Gamma density. See mlgamma for the machinery underlying this function.

Examples

mlgamma(precip)

Negative binomial distribution maximum likelihood estimation

Description

For the density function of the Negative binomial distribution see Negative binomial.

Usage

mlnbinom(x, na.rm = FALSE, ...)

Arguments

Value

mlnbinom returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for size and prob and the following attributes:

References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

Negative binomial for the density.

Examples

mlnbinom(corbet)

Normal distribution maximum likelihood estimation

Description

The maximum likelihood estimate of mean is the empirical mean and the maximum likelihood estimate of sd is the square root of the biased sample variance.

Usage

mlnorm(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the normal distribution see Normal.

Value

mlnorm returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for mean and sd and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 13. Wiley, New York.

See Also

Normal for the normal density.

Examples

mlnorm(precip)

Paralogistic distribution maximum likelihood estimation

Description

This function estimates the only the shape parameters of the Paralogistic distribution. The rate is set to 1.

Usage

mlparalogis(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Paralogistic distribution see Paralogistic.

Value

mlparalogis returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape and the following attributes:

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley. Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

Paralogistic for the paralogistic density.

Examples

mlparalogis(abalone$length)

Pareto distribution maximum likelihood estimation

Description

The maximum likelihood estimate of b is the minimum of x and the maximum likelihood estimate of a is 1/(mean(log(x)) - log(b)).

Usage

mlpareto(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Pareto distribution see Pareto.

Value

mlpareto returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for a and b and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 20. Wiley, New York.

See Also

Pareto for the Pareto density.

Examples

mlpareto(precip)

Poisson distribution maximum likelihood estimation

Description

The maximum likelihood estimate of lambda is the empirical mean.

Usage

mlpois(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Poisson distribution see Poisson.

Value

mlpois returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for lambda and the following attributes:

References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

Poisson for the Poisson density.

Examples

mlpois(ChickWeight$weight)

Power distribution maximum likelihood estimation

Description

The maximum likelihood estimate of alpha is the maximum of x + epsilon (see the details) and the maximum likelihood estimate of beta is 1/(log(alpha)-mean(log(x))).

Usage

mlpower(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the power distribution see PowerDist. The maximum likelihood estimator of alpha does not exist, strictly speaking. This is because x is supported c(0, alpha) with an open endpoint on alpha in the extraDistr implementation of dpower. If the endpoint was closed, max(x) would have been the maximum likelihood estimator. To overcome this problem, we add a possibly user specified epsilon to max(x).

Value

mlpower returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for alpha and beta and the following attributes:

References

Arslan, G. "A new characterization of the power distribution." Journal of Computational and Applied Mathematics 260 (2014): 99-102.

See Also

PowerDist for the power density. extraDistr::Pareto for the closely related Pareto distribution.

Examples

mlpower(precip)

Rayleigh distribution maximum likelihood estimation

Description

Calculates the sigma parameter as the square root of half the empirical second moment.

Usage

mlrayleigh(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Rayleigh distribution see Rayleigh.

Value

mlrayleigh returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for sigma and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 18. Wiley, New York.

See Also

Rayleigh for the Rayleigh density.

Examples

mlrayleigh(precip)

Skew Generalized Error distribution maximum likelihood estimation

Description

Joint maximum likelihood estimation as implemented by fGarch::sgedFit.

Usage

mlsged(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Student t-distribution see sged.

Value

mlsged returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for the parameters mean, sd, nu, xi, and the following attributes:

References

Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347<U+2013>370.

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint.

See Also

sged for the Student t-density.

Examples

mlsged(precip)

Skew Normal distribution maximum likelihood estimation

Description

Joint maximum likelihood estimation as implemented by fGarch::snormFit.

Usage

mlsnorm(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Student t distribution see dsnorm.

Value

mlsnorm returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for the parameters mean, sd, xi and the following attributes:

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint.

See Also

dsnorm for the Student-t density.

Examples

mlsnorm(precip)

Skew Student t-distribution maximum likelihood estimation

Description

Joint maximum likelihood estimation as implemented by fGarch::sstdFit.

Usage

mlsstd(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the skew Student t-distribution see sstd.

Value

mlsstd returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for the parameters mean, sd, nu, xi and the following attributes:

References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint.

See Also

sstd for the Skew Student t-density.

Examples

mlsstd(precip)

Student-t distribution maximum likelihood estimation

Description

Joint maximum likelihood estimation as implemented by fGarch::stdFit.

Usage

mlstd(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the Student t-distribution see std.

Value

mlstd returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for the parameters mean, sd, nu and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 13. Wiley, New York.

See Also

std for the Student-t density.

Examples

mlstd(precip)

Uniform distribution maximum likelihood estimation

Description

The estimates are min(x) and max(x).

Usage

mlunif(x, na.rm = FALSE, ...)

Arguments

Details

For the density function of the logistic distribution see Uniform.

Value

mlunif returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for min and max and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Chapter 26. Wiley, New York.

See Also

Uniform for the uniform density.

Examples

mlunif(precip)

Weibull distribution maximum likelihood estimation

Description

For the density function of the Weibull distribution see Weibull.

Usage

mlweibull(x, na.rm = FALSE, ...)

Arguments

Details

Uses mlgumbel to estimate the parameters of the Weibull distribution.

Value

mlweibull returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for shape and scale and the following attributes:

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 21. Wiley, New York.

See Also

Weibull for the Weibull density.

Examples

BIC(mlweibull(precip))

Zero-inflated Poisson distribution maximum likelihood estimation

Description

For the density function of the zeero-inflated Poisson distribution see Zero-inflated Poisson distribution.

Usage

mlzip(x, na.rm = FALSE, ...)

Arguments

Value

mlzip returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for lambda and pi and the following attributes:

References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

Zero-inflated Poisson distribution for the density.

Examples

mlzip(corbet)

Zipf distribution maximum likelihood estimation

Description

For the density function of the Zipf distribution see Zipf.

Usage

mlzipf(x, na.rm = FALSE, ...)

Arguments

Details

This function follows the same format as every other function in the package, but most applications of Zipf's law use rank-abundance data. See, e.g., sads::fitzipf for estimation of this sort of data.

Value

mlzipf returns an object of class univariateML. This is a named numeric vector with maximum likelihood estimates for N and s and the following attributes:

References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Wiley-Blackwell.

See Also

Zipf for the density.

Examples

AIC(mlzipf(corbet)) # 2729.536
AIC(mllgser(corbet)) # 2835.883

Fit multiple models and select the best fit

Description

Selects the best model by log-likelihood, aic, or bic.

Usage

model_select(
  x,
  models = univariateML_models,
  criterion = c("AIC", "BIC", "logLik"),
  na.rm = FALSE,
  type = c("both", "discrete", "continuous"),
  return = c("best", "all"),
  ...
)

Arguments

Value

The return value depends on the return argument. For return = "best" (default), model_select returns an object of class univariateML

For return = "all", model_select returns a tibble data frame with the following columns:

. So, the model with the lowest aic will have d_aic of 0; the d_aic of all the other models shows how much higher their aics are from the minimum. The same goes with d_loglik and d_bic.

See Also

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 1, Chapter 17. Wiley, New York.

Examples

# Select among all possible continuous models.
model_select(precip, type = "continuous")

# View possible models to fit.
print(univariateML_models)

# Try out only gamma, Weibull, and exponential.
model_select(precip, c("gamma", "weibull", "exp"))

# Fit the discrete `corbet` data to all available discrete models
model_select(corbet, type = "discrete", return = "all")

One-dimensional Newton–Raphson

Description

One-dimensional Newton–Raphson

Usage

newton_raphson_1d(f_over_df, param0, ...)

Arguments

Value

Newton–Raphson estimate.

Plot, Lines and Points Methods for Maximum Likelihood Estimates

Description

The plot, lines, and points methods for univariateML objects.

Usage

## S3 method for class 'univariateML'
plot(x, range = NULL, kind = c("d", "p", "q"), ...)

## S3 method for class 'univariateML'
lines(x, range = NULL, kind = c("d", "p", "q"), ...)

## S3 method for class 'univariateML'
points(x, range = NULL, kind = c("d", "p", "q"), ...)

Arguments

Value

An invisible copy of x.

Examples

plot(mlweibull(datasets::precip), main = "Annual Precipitation in US Cities")
lines(mlgamma(datasets::precip), lty = 2)
rug(datasets::precip)

Wrangles arguments for use in the plot, lines and points functions.

Description

Wrangles arguments for use in the plot, lines and points functions.

Usage

plot_wrangler(x, range, points = FALSE, kind, ...)

Arguments

Wrangles arguments for use in ppml and qqml functions.

Description

Wrangles arguments for use in ppml and qqml functions.

Usage

ppqq_wrangler(y, obj, datax, pp, ...)

Arguments

Simulate n observations from a ⁠ml***⁠ string using default parameters.

Description

Simulate n observations from a ⁠ml***⁠ string using default parameters.

Usage

simulate_default(dens, n)

Arguments

Data and function to 'univariateML'

Description

Data and function to 'univariateML'

Usage

to_univariateML(y, obj)

Arguments

Construct univariateML object.

Description

Construct univariateML object.

Usage

univariateML_construct(estimates, name, params)

Arguments

Value

Object of class univariateML

Metadata for univariateML models.

Description

Metadata for univariateML models.

Usage

univariateML_metadata

Format

An object of class list of length 42.

Implemented models

Description

Vector of all supported models in univariateML.

Usage

univariateML_models

Format

An object of class character of length 42.

Details

The currently supported models are mlbeta, mlbetapr, mlbinom, mlburr, mlcauchy, mldunif, mlexp, mlfatigue, mlgamma, mlged, mlgeom, mlgompertz, mlgumbel, mlinvburr, mlinvgamma, mlinvgauss, mlinvweibull, mlkumar, mllaplace, mllgamma, mllgser, mlllogis, mllnorm, mllogis, mllogitnorm, mllomax, mlnaka, mlnbinom, mlnorm, mlparalogis, mlpareto, mlpois, mlpower, mlrayleigh, mlsged, mlsnorm, mlsstd, mlstd, mlunif, mlweibull, mlzip, mlzipf

Transform a univariateML object to a quantile, CDF, density or random variate generation.

Description

Transform a univariateML object to a quantile, CDF, density or random variate generation.

Usage

univariateML_to_function(obj, type = c("d", "p", "q", "r", "ml"))

Arguments

Value

A string

Transform a univariateML object to a string specifying quantile, CDF, density or random variate generation.

Description

Transform a univariateML object to a string specifying quantile, CDF, density or random variate generation.

Usage

univariateML_to_string(obj, type = c("d", "p", "q", "r", "ml"))

Arguments

Value

A string