gconv function

Convolution of Two Probability Generating Functions

Computes the convolution of two probability generating functions using the convolve function in the stats package. The convolve function uses the fast fourier transform.

gconv(g1,g2)

Arguments

• g1: A probability generating function. A vector g1 for a random variable X taking values 0, 1, 2, ..., length(g1)-1, where g1[i] = Pr(X=i-1)For example, g1 = c(2/3, 1/3) is the generating function of a binary random variable X with Pr(X=0)=2/3, Pr(X=1)=1/3. The random variable that is 0 with probability 1 has g1=1.
• g2: Another probability generating function for a random variable Y. For a fair die, g2 = c(0, 1/6, 1/6, 1/6, 1/6, 1/6, 1/6).

Returns

The probability generating function of X+Y when X and Y are independent.

Examples

gconv(c(2/3,1/3),c(2/3,1/3))

gconv(1,c(2/3,1/3))

round(gconv(c(0, 1/6, 1/6, 1/6, 1/6, 1/6, 1/6),
     c(0, 1/6, 1/6, 1/6, 1/6, 1/6, 1/6)),3)
#
# Compute the exact distribution of Quade's treated-control
# statistic forI=3 blocks of size J=3.
#
# Block with range rank = 1
rk1<-c(0,1/3,1/3,1/3)
names(rk1)<-0:3
rk1
#
# Block with range rank = 2
rk2<-c(0,0,1/3,0,1/3,0,1/3)
names(rk2)<-0:6
rk2
#
# Block with range rank = 3
rk3<-c(0,0,0,1/3,0,0,1/3,0,0,1/3)
names(rk3)<-0:9
rk3
#
# Convolution of rk1 and rk2
round(gconv(rk1,rk2),3)
1/(3^2)
#
# Convolution of rk1, rk2 and rk3
round(gconv(gconv(rk1,rk2),rk3),3)
1/(3^3)

Note

The gconv function is a slight modification of a similar function in the sensitivity2x2xk package.

References

Pagano, M. and Tritchler, D. (1983) doi:10.2307/2288653 On obtaining permutation distributions in polynomial time. Journal of the American Statistical Association, 78, 435-440.

Rosenbaum, P. R. (2020) doi:10.1007/978-3-030-46405-9 Design of Observational Studies. New York: Springer. Chapter 3 Appendix: Exact Computations for Sensitivity Analysis, page 103.