Ediweibull function

First and second order moments

First and second order moments of the discrete inverse Weibull distribution

Ediweibull(q, beta, eps = 1e-04, nmax = 1000)

Arguments

• q: the value of the $q$ parameter
• beta: the value of the $\beta$ parameter
• eps: error threshold for the approximated computation of the moments
• nmax: a first maximum value of the support considered for the approximated computation of the moments

Details

For a discrete inverse Weibull distribution we have $E(X;q,\beta)=\sum_{x=0}^{+\infty} 1-F(x;q, \beta)$ and $E(X^2;q,\beta)=2\sum_{x=1}^{+\infty} x(1-F(x;q, \beta))+E(X;q, \beta)$. The expected values are numerically computed considering a truncated support: integer values smaller than or equal to $\min(nmax;F^{-1}(1-eps;q,\beta))$, where $F^{-1}$ is the inverse of the cumulative distribution function (implemented by the function qdiweibull). Increasing the value of nmax or decreasing the value of eps improves the approximation, but slows down the calculation speed

Returns

a list comprising the (approximate) first and second order moments of the discrete inverse Weibull distribution. Note that the first moment is finite iff $\beta$ is greater than 1; the second order moment is finite iff $\beta$ is greater than 2

References

Khan M.S., Pasha G.R., Pasha A.H. (2008) Theoretical Analysis of Inverse Weibull Distribution, WSEAS Trabsactions on Mathematics 2(7): 30-38

Examples

# Ex.1
q<-0.75
beta<-1.25
Ediweibull(q, beta)
# Ex.2 
q<-0.5
beta<-2.5
Ediweibull(q, beta)
# Ex.3
q<-0.4
beta<-4
Ediweibull(q, beta)