Package {zmctp}

Type: Package
Title: Zero-Modified Complex 'Tri-Parametric' Pearson Distribution for Overdispersed Count Data
Version: 0.1.1
Description: Implements zero-modified versions of the Complex 'Tri-Parametric' Pearson distribution for overdispersed count data. The package addresses limitations of existing implementations when the parameter b approaches zero. It provides distribution functions, maximum likelihood estimation, and diagnostic tools for modeling count data with excess zeros. The methodology is based on 'Rodriguez-Avi' and coauthors (2003) <doi:10.1007/s00362-002-0134-7>.
License: GPL-3
Encoding: UTF-8
Depends: R (≥ 3.5.0)
Imports: stats, graphics
Suggests: testthat (≥ 3.0.0), knitr, rmarkdown, spelling
VignetteBuilder: knitr
RoxygenNote: 7.3.3
URL: https://github.com/roladoja/zmctp
BugReports: https://github.com/roladoja/zmctp/issues
Maintainer: Rasheedat Oladoja <roladoja@ttu.edu>
Language: en-US
Config/testthat/edition: 3
NeedsCompilation: no
Packaged: 2026-07-08 22:22:33 UTC; kemmi
Author: Rasheedat Oladoja ORCID iD [aut, cre]
Repository: CRAN
Date/Publication: 2026-07-09 08:10:02 UTC

zmctp: Zero-Modified Complex Triparametric Pearson Distribution

Description

The zmctp package extends the Complex Triparametric Pearson (CTP) distribution with zero-modified versions for handling overdispersed count data, particularly when the parameter b approaches zero.

Details

Main functions:

Author(s)

Maintainer: Rasheedat Oladoja roladoja@ttu.edu (ORCID)

See Also

Useful links:

Maximum Likelihood Estimation for the CTP Distribution

Description

Fits the Complex Triparametric Pearson (CTP) distribution to count data using maximum likelihood estimation.

Usage

ctp.fit(
  x,
  a_start = NULL,
  b_start = NULL,
  gama_start = NULL,
  method = "L-BFGS-B",
  penalty = 1e+10
)

Arguments

x

Numeric vector of nonnegative counts.

a_start

Optional starting value for parameter a.

b_start

Optional starting value for parameter b.

gama_start

Optional starting value for parameter gamma.

method

Optimization method (default: "L-BFGS-B").

penalty

numeric penalty added for numerical stability when b → 0

Value

An object of class "ctpfit" containing:

estimates

Named vector of MLEs

se

Standard errors

vcov

Variance-covariance matrix

logLik

Log-likelihood

AIC

Akaike Information Criterion

BIC

Bayesian Information Criterion

pearson_chisq

Pearson's chi-squared statistic

wald_chisq

Wald's chi-squared statistic

fitted_freq

Data frame of observed vs expected frequencies

data

Original data

converged

Convergence status

Examples

set.seed(123)

x <- rctp(30, a = 1, b = 0.5, gama = 5)
fit <- ctp.fit(x)
print(fit)
plot(fit)

Probability Mass Function of the CTP Distribution

Description

Evaluates the probability mass function of the Complex TriParametric Pearson (CTP) distribution for nonnegative integer values.

The pmf is

p(x) = p(0)\frac{(a+ib)_x(a-ib)_x}{(\gamma)_x x!}, \quad x=0,1,\dots

and satisfies the recurrence

\frac{p(x+1)}{p(x)} = \frac{(a+x)^2+b^2}{(\gamma+x)(x+1)}.

Usage

dctp(x, a, b, gama, log = FALSE)

Arguments

x

Vector of nonnegative integers.

a

Parameter a.

b

Parameter b (must satisfy b >= 0).

gama

Parameter gamma.

log

Logical; if TRUE, returns log-probabilities.

Value

A numeric vector of probabilities.

Examples

dctp(0:5, a = 1, b = 0.5, gama = 6)

Probability Mass Function of the ZM-CTP Distribution

Description

The Zero-Modified CTP (ZM-CTP) distribution modifies the zero probability of the baseline CTP distribution.

Usage

dzictp(x, a, b, gama, omega, log = FALSE)

Arguments

x

Vector of nonnegative integers.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

omega

Zero-modification parameter, with 0 < omega < 1.

log

Logical; if TRUE, returns log-probabilities.

Value

Numeric vector of probabilities.

Examples

dzictp(0:5, a = 1, b = 0.5, gama = 6, omega = 0.3)

Theoretical moments and mode of the CTP distribution

Description

Theoretical moments and mode of the CTP distribution

Usage

mean_ctp(a, b, gama)

var_ctp(a, b, gama)

skew_ctp(a, b, gama)

mode_ctp(a, b, gama)

Arguments

a, b, gama

CTP parameters.

Details

mode_ctp() returns the unimodal-case mode (rounded). The distribution has two consecutive modes in the knife-edge case where ((a-1)^2+b^2)/(gama-2*a+1) is exactly an integer; this edge case is not separately flagged.

These functions do not validate that the parameter constraints required for a given moment to exist are satisfied (e.g. gama > 2*a+2 for the variance, gama > 2*a+3 for the skewness). Outside these regions the returned value may be Inf, NaN, or negative; see dctp for the corresponding existence warnings applied at the density level.

Examples

mean_ctp(a = 1, b = 0.5, gama = 6)
var_ctp(a = 1, b = 0.5, gama = 6)
skew_ctp(a = 1, b = 0.5, gama = 6)
mode_ctp(a = 1, b = 0.5, gama = 6)

Theoretical moments and mode of the ZM-CTP distribution

Description

Theoretical moments and mode of the ZM-CTP distribution

Usage

mean_zictp(a, b, gama, omega)

var_zictp(a, b, gama, omega)

skew_zictp(a, b, gama, omega)

mode_zictp(a, b, gama, omega)

Arguments

a, b, gama

CTP parameters.

omega

Zero-modification parameter.

Details

These functions do not validate that the parameter constraints required for a given moment to exist are satisfied (e.g. gama > 2*a+2 for the variance, gama > 2*a+3 for the skewness), the same caveat documented in mean_ctp.

Examples

# Quick illustration (fast, runs during R CMD check)
mean_zictp(a = 1, b = 0.5, gama = 6, omega = 0.3)
var_zictp(a = 1, b = 0.5, gama = 6, omega = 0.3)
skew_ctp(a = 1, b = 0.5, gama = 6)
skew_zictp(a = 1, b = 0.5, gama = 6, omega = 0.3)
mode_zictp(a = 1, b = 0.5, gama = 6, omega = 0.3)


# Empirical check against simulated data (slower, skipped on CRAN checks)
set.seed(2026)
x <- rzictp(50000, a = 1, b = 0.5, gama = 6, omega = 0.3)
skew_zictp(1, 0.5, 6, 0.3)                       # theoretical
mean((x - mean(x))^3) / sd(x)^3                  # empirical

Cumulative Distribution Function of the CTP Distribution

Description

Cumulative Distribution Function of the CTP Distribution

Usage

pctp(q, a, b, gama, lower.tail = TRUE, log.p = FALSE)

Arguments

q

Vector of quantiles.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

lower.tail

Logical; if TRUE, probabilities are P(X <= q), otherwise P(X > q).

log.p

Logical; if TRUE, probabilities are given on the log scale.

Value

Numeric vector of cumulative probabilities.

Examples

pctp(0:5, a = 1, b = 0.5, gama = 6)

Plot method for ctpfit objects

Description

Plot method for ctpfit objects

Usage

## S3 method for class 'ctpfit'
plot(x, type = c("frequency", "cdf", "qq"), ...)

Arguments

x

A ctpfit object

type

Type of plot: "frequency", "cdf", or "qq"

...

Additional graphical parameters

Value

No return value. Called for its side effect of producing a plot.

Plot method for zictpfit objects

Description

Creates diagnostic plots for Zero-Modified CTP distribution fits, including frequency comparisons, CDF plots, and Q-Q plots.

Usage

## S3 method for class 'zictpfit'
plot(x, type = c("frequency", "cdf", "qq"), ...)

Arguments

x

A zictpfit object from zictp.fit()

type

Type of plot: "frequency", "cdf", or "qq"

...

Additional graphical parameters

Value

No return value. Called for its side effect of producing a plot.

Print method for ctpfit objects

Description

Print method for ctpfit objects

Usage

## S3 method for class 'ctpfit'
print(x, ...)

Arguments

x

A ctpfit object

...

Additional arguments (ignored)

Value

Invisibly returns the original fitted model object. The function is called for its side effect of printing a model summary.

Print method for zictpfit objects

Description

Print method for zictpfit objects

Usage

## S3 method for class 'zictpfit'
print(x, ...)

Arguments

x

A zictpfit object

...

Additional arguments (ignored)

Value

Invisibly returns the original fitted model object. The function is called for its side effect of printing a model summary.

Cumulative Distribution Function of the ZM-CTP Distribution

Description

Cumulative Distribution Function of the ZM-CTP Distribution

Usage

pzictp(q, a, b, gama, omega, lower.tail = TRUE, log.p = FALSE)

Arguments

q

Vector of quantiles.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

omega

Zero-modification parameter.

lower.tail

Logical.

log.p

Logical.

Value

Numeric vector of cumulative probabilities.

Examples

pzictp(0:5, a = 1, b = 0.5, gama = 6, omega = 0.3)

Quantile Function of the CTP Distribution

Description

Quantile Function of the CTP Distribution

Usage

qctp(p, a, b, gama, lower.tail = TRUE, log.p = FALSE, max_q = 1000)

Arguments

p

Vector of probabilities.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

lower.tail

Logical.

log.p

Logical.

max_q

Maximum x to search.

Value

Numeric vector of quantiles.

Examples

qctp(c(0.25, 0.5, 0.75), a = 1, b = 0.5, gama = 6)

Quantile Function of the ZM-CTP Distribution

Description

Quantile Function of the ZM-CTP Distribution

Usage

qzictp(p, a, b, gama, omega, lower.tail = TRUE, log.p = FALSE, max_q = 1000)

Arguments

p

Vector of probabilities.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

omega

Zero-modification parameter.

lower.tail

Logical.

log.p

Logical.

max_q

Maximum x to search.

Value

Numeric vector of quantiles.

Examples

qzictp(c(0.25, 0.5, 0.75), a = 1, b = 0.5, gama = 6, omega = 0.3)

Random Generation from the CTP Distribution

Description

Random Generation from the CTP Distribution

Usage

rctp(n, a, b, gama)

Arguments

n

Number of observations.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

Value

Integer vector of random draws.

Examples

set.seed(123)
rctp(10, a = 1, b = 0.5, gama = 6)

Random Generation from the ZM-CTP Distribution

Description

Random Generation from the ZM-CTP Distribution

Usage

rzictp(n, a, b, gama, omega)

Arguments

n

Number of observations.

a

Parameter a.

b

Parameter b.

gama

Parameter gamma.

omega

Zero-modification parameter.

Value

Integer vector of random draws.

Examples

set.seed(123)
rzictp(10, a = 1, b = 0.5, gama = 6, omega = 0.3)

Summary method for ctpfit objects

Description

Summary method for ctpfit objects

Usage

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

Arguments

object

A ctpfit object

...

Additional arguments (ignored)

Value

Invisibly returns the original fitted model object. The function is called for its side effects, producing a formatted summary of parameter estimates, moments, and goodness-of-fit diagnostics.

Summary method for zictpfit objects

Description

Summary method for zictpfit objects

Usage

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

Arguments

object

A zictpfit object

...

Additional arguments (ignored)

Value

Invisibly returns the original fitted model object. The function is called for its side effects, producing a formatted summary of parameter estimates, moments, and goodness-of-fit diagnostics.

Maximum Likelihood Estimation for the Zero-Modified CTP Distribution

Description

Fits the Zero-Modified Complex TriParametric Pearson (ZM-CTP) distribution to count data using maximum likelihood estimation.

A logit reparameterization is used for omega to ensure 0 < omega < 1, and a log reparameterization is used for gamma so that gama = 2*a + 2 + exp(eta), which guarantees variance existence throughout optimization.

Usage

zictp.fit(
  x,
  a_start = NULL,
  b_start = NULL,
  gama_start = NULL,
  omega_start = NULL,
  method = "BFGS"
)

Arguments

x

Numeric vector of nonnegative counts.

a_start

Optional starting value for parameter a.

b_start

Optional starting value for parameter b.

gama_start

Optional starting value for parameter gamma.

omega_start

Optional starting value for omega (0 < omega < 1).

method

Optimization method (default: "BFGS").

Value

An object of class '"zictpfit"'.

Examples


set.seed(123)
x <- rzictp(30, a = 1, b = 0.5, gama = 5, omega = 0.3)
fit <- zictp.fit(x)
fit$estimates