Generalized Hermite distribution

Description

Probability mass, distribution and quantile functions; random generation; and regression models for the generalized Hermite distribution.

Details

The package implements probability mass function dhermite, distribution function phermite, quantile function qhermite and random generation rhermite for the generalized Hermite distribution. The probability mass function is usually parametrized in terms of the mean \mu and the index of dispersion d = \frac{\sigma^2}{\mu}:

P(X=x) = P(X=0) \frac{\mu^x (m-d)^x}{(m-1)^x} \sum_{j=0}^{[x/m]} \frac{(d-1)^j (m-1)^{(m-1)j}}{m^j \mu^{(m-1)j} (m-d)^{mj} (x-mj)!j!} where P(X=0) = exp(\mu (-1+ \frac{d-1}{m})), m is the degree of the generalized Poisson distribution and [x/m] is the integer part of x/m.

The package is able to fit Hermite regression models as well, by means of the function glm.hermite, also in the presence of covariates.

Author(s)

David Moriña, Manuel Higueras, Pedro Puig and María Oliveira

Mantainer: David Moriña Soler <david.morina@uab.cat>

References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

See Also

Distributions for some other distributions, qhermite, phermite, rhermite, hermite-package, glm.hermite

Probability mass function for the generalized Hermite distribution

Description

Probability mass function for the generalized Hermite distribution with parameters a, b and m.

Usage

  dhermite(x, a, b, m=2)

Arguments

Value

Probability for a generalized Hermite random varible with parameters a, b and m of taking x counts.

Author(s)

David Moriña, Manuel Higueras, Pedro Puig and María Oliveira

References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

See Also

Distributions for some other distributions, qhermite, phermite, rhermite, hermite-package, glm.hermite

Examples

d <- dhermite(3, 0.8, 0.3)

Maximum likelihood estimation and Hermite regression

Description

glm.hermite is used to fit generalized linear models with count responses following a Hermite distribution, specified by giving a symbolic description of the linear predictor. A summary method providing the most meaningful information on the fitted model is available for objects of class glm.hermite.

Usage

  glm.hermite(formula, data, link="log", start=NULL, m = NULL)

Arguments

Value

glm.hermite returns an object of class glm.hermite, which is a list including the following components:

• coefs the vector of coefficients.

• data an optional data frame containing the variables in the model.

• loglik log-likelihood of the fitted model.

• vcov covariance matrix of all coefficients in the model (derived from the Hessian of the maxLik output).

• hessHessian matrix, returned by the maxLik output.

• fitted.values the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.

• wLikelihood ratio test statistic.

• pvalLikelihood ratio test p-value.

Author(s)

María Oliveira, Manuel Higueras, David Moriña and Pere Puig

References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

See Also

Distributions for some other distributions, qhermite, phermite, rhermite, hermite-package

Examples

data <- c(rep(0,122), rep(1,40), rep(2,14), rep(3,16), rep(4,6), rep(5,2))
mle1 <- glm.hermite(data~1, link="log", start=NULL, m=3)
mle1

Internal hermite functions

Description

Internal hermite functions

Usage

int.hermite(x, a, b, m)
edg(y, a, b, m=2)
cofi(p, a, b, m=2)
## S3 method for class 'glm.hermite'
summary(object, ...)
## S3 method for class 'glm.hermite'
print.summary(x, ...)

Details

These functions are not to be called by the user

See Also

hermite-package, dhermite, phermite, qhermite, rhermite, glm.hermite

High-LET Radiation Exposure

Description

This data corresponds to an experimental simulation of in vitro whole body irradiation for high-LET radiation exposure, where peripheral blood samples were exposed to 10 different doses of 1480MeV oxygen ions. For each dose, the number of dicentrics chromosomes per blood cell were scored.

Usage

hi_let

Format

A data frame with 7413 rows and 3 columns.

Source

DiGiorgio M. et al. (2004) Chromosome aberrations induced in human lymphocytes by heavycharged particles in track segment mode. Radiation Protection Dosimetry, 108, 47-53.

References

DiGiorgio M., Edwards A. A., Moquet J. E., Finnon P., Hone P. A., Lloyd D. C., Kreiner A. J., Schuff J. A., Taja M. R., Vallerga M. B., López F. O. and Burlón A., Debray M. E., Valda A. (2004) Chromosome aberrations induced in human lymphocytes by heavycharged particles in track segment mode. Radiation Protection Dosimetry, 108, 47-53.

Hot 100 hits

Description

This data corresponds to the 965 "number 1" hits on the Hot 100 chart over the period January 1955 to December 2003. For a recording that reaches the number one spot, Weeks measures the number of weeks that it stays at number one. The covariates are: Elvis = 1 if the recording was by Elvis Presley, = 0 otherwise; Beatles = 1 if the recording was by the Beatles, = 0 otherwise; Group = 1 if the recording was by a band, = 0 otherwise; Female = 1 if the artist was a solo female, = 0 otherwise; Male = 1 if the artist was a solo male, = 0 otherwise; Inst = 1 if the recording was purely instrumental, = 0 otherwise; and NonCon = 1 if the recording topped the charts in nonconsecutive weeks, = 0 otherwise.

Usage

hot100

Format

A data frame with 965 rows and 9 columns.

Source

http://web.uvic.ca/~dgiles/downloads/data/hot100.xls

References

Giles, D. E. (2006) Superstardom in the US popular music industry revisited. Economics Letters, 92(1):68–74. Giles, D. E. (2007) Modeling inflated count data. In Y. Berbers and W. Zwaenepoel, editors, Proceedings of the MODSIM 2007 International Congress on Modelling and Simulation, pages 919–925. L. Oxley and D. Kulasiri, Eds., Modelling and Simulation Society of Australia and New Zealand

Distribution function for the generalized Hermite distribution

Description

Distribution function for the generalized Hermite distribution with parameters a, b and m.

Usage

  phermite(q, a, b, m=2, lower.tail=TRUE)

Arguments

Value

Probability for a generalized Hermite random varible with parameters a, b and m to be lower (or greater) than q.

Author(s)

David Moriña, Manuel Higueras, Pedro Puig and María Oliveira

References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

See Also

Distributions for some other distributions, dhermite, qhermite, rhermite, hermite-package, glm.hermite

Examples

d <- phermite(4, 0.8, 0.3, m=3)

Quantile function for the generalized Hermite distribution

Description

Quantile function for the generalized Hermite distribution with parameters a, b and m.

Usage

  qhermite(p, a, b, m=2, lower.tail=TRUE)

Arguments

Value

The smallest integer x such that P(X \le x) \ge p (or such that P(X \le x) \ge 1-p if lower.tail is set to FALSE), where X is a generalized Hermite random variable with parameters a, b and m.

Author(s)

David Moriña, Manuel Higueras, Pedro Puig and María Oliveira

References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

See Also

Distributions for some other distributions, dhermite, phermite, rhermite, hermite-package, glm.hermite

Examples

d <- qhermite(0.9999987, 0.8, 0.3, m=3)

Random generation for the generalized Hermite distribution

Description

Random generation for the generalized Hermite distribution with parameters a, b and m.

Usage

rhermite(n, a, b, m=2)

Arguments

Value

A vector containing n random deviates from a generalized Hermite distribution.

Author(s)

David Moriña, Manuel Higueras, Pedro Puig and María Oliveira

References

Kemp C D, Kemp A W. Some Properties of the Hermite Distribution. Biometrika 1965;52 (3-4):381–394.

McKendrick A G Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 1926;44:98–130.

Kemp A W, Kemp C D. An alternative derivation of the Hermite distribution. Biometrika 1966;53 (3-4):627–628.

Patel Y C. Even Point Estimation and Moment Estimation in Hermite Distribution. Biometrics 1976;32 (4):865–873.

Gupta R P, Jain G C. A Generalized Hermite distribution and Its Properties. SIAM Journal on Applied Mathematics 1974;27:359–363.

Bekelis, D. Convolutions of the Poisson laws in number theory. In Analytic & Probabilistic Methods in Number Theory: Proceedings of the 2nd International Conference in Honour of J. Kubilius, Lithuania 1996;4:283–296.

Zhang J, Huang H. On Nonnegative Integer-Valued Lévy Processes and Applications in Probabilistic Number Theory and Inventory Policies. American Journal of Theoretical and Applied Statistics 2013;2:110–121.

Kotz S. Encyclopedia of statistical sciences. John Wiley 1982-1989.

Kotz S. Univariate discrete distributions. Norman L. Johnson 2005.

Puig P. (2003). Characterizing Additively Closed Discrete Models by a Property of Their Maximum Likelihood Estimators, with an Application to Generalized Hermite Distributions. Journal of the American Statistical Association 2003; 98:687–692.

See Also

Distributions for some other distributions, dhermite, phermite, qhermite, hermite-package, glm.hermite

Examples

rnd <- rhermite(1000, 0.8, 0.3)