fitebw {cpd}

Maximum-likelihood fitting of the EBW distribution

Description

Maximum-likelihood fitting of the Extended Biparametric Waring (EBW) distribution with parameters \alpha, \rho and \gamma. Generic methods are print, summary, coef, logLik, AIC, BIC and plot. The method to be used in fitting the model is "L-BFGS-B" which allows constraints for each variable (see details in optim funtion).

Usage

fitebw(x, alphastart = NULL, rhostart = NULL, gammastart = NULL, 
          method = "L-BFGS-B", control = list(),...)

Arguments

x

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

alphastart

A starting value for the parameter \alpha; by default NULL.

rhostart

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

gammastart

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

method

The method to be used in fitting the model. The default method is "L-BFGS-B" (optim).

control

A list of parameters for controlling the fitting process.

...

Additional parameters.

Details

If the starting value for \alpha is positive, the parameterization (\alpha,\rho) is used; otherwise, the parameterization (\alpha,\gamma) is used.

If the starting values of the parameters \alpha, \gamma or \rho 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 \alpha, \gamma or \rho are provided by the default method; otherwise, SE for \alpha_0 and \gamma_0 where \alpha=-exp(\alpha_0) and \gamma=exp(\gamma_0) (or for \alpha_0 and \rho_0 where \alpha=exp(\alpha_0) and \rho=exp(\rho_0)) are computed.

Value

An object of class 'fitEBW' 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 'fitEBW' 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 'fitEBW' object.

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

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

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

• plot: It shows the plot of a 'fitEBW' 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

Cueva-Lopez V, Olmo-Jimenez MJ, Rodriguez-Avi J (2021). “An Over and Underdispersed Biparametric Extension of the Waring Distribution.” Mathematics, 9(170), 1-15. doi:10.3390/math9020170.

See Also

Plot of observed and theoretical frequencies for a EBW fit: plot.fitEBW

Maximum-likelihood fitting for the CTP distribution: fitctp.

Maximum-likelihood fitting for the CBP distribution: fitcbp.

Examples

set.seed(123)
x <- rebw(500, 2, rho = 5)
fitebw(x)
summary(fitebw(x, alphastart = 1, rhostart = 5))