fitctp {cpd}

Maximum-likelihood fitting of the CTP distribution

Description

Maximum-likelihood fitting of the Complex Triparametric Pearson (CTP) distribution with parameters a, b and \gamma. Generic methods are print, summary, coef, logLik, AIC, BIC and plot.

Usage

fitctp(x, astart = NULL, bstart = NULL, gammastart = NULL, 
          method = "L-BFGS-B", control = list(), ...)

Arguments

x

A numeric vector of length at least one containing only finite values.

astart

A starting value for the parameter a>0; by default NULL.

bstart

A starting value for the parameter b; by default NULL.

gammastart

A starting value for the parameter \gamma>max(0,2a); by default NULL.

method

The method to be used in fitting the model. See 'Details'.

control

A list of parameters for controlling the fitting process.

...

Additional parameters.

Details

If the starting values of the parameters a, b and \gamma are omitted (default option), they are computing by the method of moments (if possible; otherwise they must be entered).

The default method is "L-BFGS-B" (see details in optim function), but non-linear minimization (nlm) or those included in the optim function ("Nelder-Mead", "BFGS", "CG" and "SANN") may be used.

Standard error (SE) estimates for a, b and \gamma are provided by the default method; otherwise, SE for \gamma_0 where \gamma=exp(\gamma_0) is computed.

Value

An object of class 'fitCTP' is a list containing the following components:

• n, the number of observations,

• initialValues, a vector with the starting values used,

• coefficients, the parameter ML estimates of the CTP distribution,

• se, a vector of the standard error estimates,

• hessian, a symmetric matrix giving an estimate of the Hessian at the solution found in the optimization of the log-likelihood function,

• cov, an estimate of the covariance matrix of the model coefficients,

• corr, an estimate of the correlation matrix of the model estimates,

• loglik, the maximized log-likelihood,

• aic, Akaike Information Criterion, minus twice the maximized log-likelihood plus twice the number of parameters,

• bic, Bayesian Information Criterion, minus twice the maximized log-likelihood plus twice the number of parameters,

• code, a code that indicates successful convergence of the fitter function used (see nlm and optim helps),

• converged, logical value that indicates if the optimization algorithms succesfull,

• method, the name of the fitter function used.

Generic functions:

• print: The print of a 'fitCTP' object shows the ML parameter estimates and their standard errors in parenthesis.

• summary: The summary provides the ML parameter estimates, their standard errors and the statistic and p-value of the Wald test to check if the parameters are significant. This summary also shows the loglikelihood, AIC and BIC values, as well as the results for the chi-squared goodness-of-fit test and the Kolmogorov-Smirnov test for discrete variables. Finally, the correlation matrix between parameter estimates appears.

• coef: It extracts the fitted coefficients from a 'fitCTP' object.

• logLik: It extracts the estimated log-likelihood from a 'fitCTP' object.

• AIC: It extracts the value of the Akaike Information Criterion from a 'fitCTP' object.

• BIC: It extracts the value of the Bayesian Information Criterion from a 'fitCTP' object.

• plot: It shows the plot of a 'fitCTP' object. Observed and theoretical probabilities, empirical and theoretical cumulative distribution functions or empirical cumulative probabilities against theoretical cumulative probabilities are the three plot types.

References

Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ, Olmo-Jimenez MJ (2004). “A triparametric discrete distribution with complex parameters.” Stat. Pap., 45, 81-95. doi:10.1007/BF02778271.

Olmo-Jimenez MJ, Rodriguez-Avi J, Cueva-Lopez V (2018). “A review of the CTP distribution: a comparison with other over- and underdispersed count data models.” Journal of Statistical Computation and Simulation, 88(14), 2684-2706. doi:10.1080/00949655.2018.1482897.

See Also

Plot of observed and theoretical frequencies for a CTP fit: plot.fitCTP

Maximum-likelihood fitting for the CBP distribution: fitcbp.

Maximum-likelihood fitting for the EBW distribution: fitebw.

Examples

set.seed(123)
x <- rctp(500, -0.5, 1, 2)
fitctp(x)
summary(fitctp(x, astart = 1, bstart = 1.1, gammastart = 3))