Bell {countDM}

MLE of the Bell distribtion

Description

Evaluates the maximum likelihood estimate of the Bell distribtion. The PMF of the Bell distribution is as follows:

f(X=x\mid\theta)=\frac{\theta^{x}e^{e^{\theta}+1}B_{x}}{x!};\qquad x=0,1,2,\,\dots,

where \theta>0 denotes the Bell parameter and B_{x} is the Bell number and it is given by

B_{n}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{k^{n}}{k!}.

The Bell number B_{n} in the above equation is the nth moment of the Poisson distribution with parameter equal to 1.

Usage

bell_mle (x)
mle.bell (x, theta)

Arguments

x

A vector of (non-negative integer) discrete values.

theta

A vector of (non-negative integer) values.

Details

The function allows to estimate the unknown parameter of the Bell distribution with loglik value using a Newton-Raphson algorithm.

Value

bell_mle gives the maximum liklihood estimate of parameter theta. loglik gives value of the maximised log-likelihood. The mle.bell gives the maximum liklihood estimate with standard error and AIC,

Author(s)

Muhammad Imran and M.H. Tahir.

R implementation and documentation: Muhammad Imran [e-mail] and M.H. Tahir <[e-mail]>.

References

Castellares, F., Ferrari, S. L., & Lemonte, A. J. (2018). On the Bell distribution and its associated regression model for count data. Applied Mathematical Modelling, 56, 172-185.

See Also

mle_borel, mle_poisson

Examples

x <- data_sbirth
bell_mle (x)
mle.bell (x, 1.2)