good (version 1.0.1)

dgood: Probability mass function for the Good distribution

Description

Probability mass function for the Good distribution with parameters z and s.

Usage

dgood ( x , z , s )

Arguments

x

vector of non-negative integer quantiles.

z

vector of first parameter for the Good distribution.

s

vector of second parameter for the Good distribution.

Value

dgood gives the probability mass function for the Good distribution with parameters z and s. x should be a vector of non-negative integer quantiles. If x is non-integer and/or negative, dgood returns \(0\) with a warning. z and s can be vectors with values within the interval \((0,1)\) and the reals respectively. If vector z has negative values and/or outside the interval \((0,1)\), dgood returns NaN with a warning.

If function polylog from package copula returns Inf (e.g., for large negative values of parameter s), dgood uses the approximation described above for probabilities, and additionally returns an informative warning.

Details

The Good distribution has the probability mass function (pmf): $$P(X=x)=(1/F(z,s)) \cdot (z^{(x+1)}/(x+1)^s),$$ where \(x = 0, 1, 2 \ldots\). Parameter z should be within the interval \((0,1)\), and parameter s in the reals. \(F(z,s)\) is the polylogarithm function: $$F(z,s)=\sum_{i=1}^{\infty} z^n/n^s,$$ and acts in the pmf as the normalizing constant.

If \(F(z,s)\) does not converge (e.g., for large negative values of the parameter s), the following approximation is used instead: $$F(z,s)\approx \Gamma(1-s) \cdot (-\log(z))^{(s-1)},$$ and dgood returns approximated probabilities: $$P(X=x) \approx \exp((x+1) \cdot \log(z) - s \cdot \log(x+1)-\log(\Gamma(1-s))-(s-1) \cdot \log(-\log(z))).$$

References

Good, J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika, 40: 237<U+2013>264.

Z<U+00F6>rnig, P. and Altmann, G. (1995). Unified representation of zipf distributions. Computational Statistics & Data Analysis, 19: 461<U+2013>473.

Kulasekera, K.B. and Tonkyn, D. (1992). A new distribution with applications to survival dispersal anddispersion. Communication in Statistics - Simulation and Computation, 21: 499<U+2013>518.

Doray, L.G. and Luong, A. (1997). Efficient estimators for the good family. Communications in Statistics - Simulation and Computation, 26: 1075<U+2013>1088.

Johnson, N.L., Kemp, A.W. and Kotz, S. Univariate Discrete Distributions. Wiley, Hoboken, 2005.

Kemp. A.W. (2010). Families of power series distributions, with particular reference to the lerch family. Journal of Statistical Planning and Inference, 140:2255<U+2013>2259.

Wood, D.C. (1992). The Computation of Polylogarithms. Technical report. UKC, University of Kent, Canterbury, UK (KAR id:21052).

See Also

See also polylog from copula, pgood, and qgood and rgood from good.

Examples

# NOT RUN {
# if x is not a non-negative integer, dgood returns 0 with a warning
dgood ( x = -3 , z = c ( 0.6 , 0.5 ) , s = -3 )
dgood ( x = 4.5 , z = c ( 0.6 , 0.5 ) , s = -3 )

# if z is not within 0 and 1, dgood returns NaN with a warning
dgood ( x = 4 , z = c ( 0.6 , 0.5 , -0.9 ) , s = -3 )

# if the approximation is used, dgood returns a warning
dgood ( x = 330 : 331 , z = c ( 0.6 , 0.5 ) , s = -170 )

dgood ( x = 4 , z = 0.6 , s = -3 )
dgood ( x = 4 , z = c ( 0.6 , 0.5 ) , s = -3 )
dgood ( x = 4 : 5 , z = c ( 0.6 , 0.5 ) , s = c ( -3 , -10 ) )
dgood ( x = 4 : 6 , z = c ( 0.6 , 0.5 ) , s = c ( -3 , -10 ) )
dgood ( x = 3 : 5 ,  z = c ( 0.6 , 0.5 , 0.9 , 0.4 ) , s = c ( -3 , -10 ) )

# }