cdfsmd {lmomco}

Cumulative Distribution Function of the Singh–Maddala Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the Singh–Maddala (Burr Type XII) distribution given parameters (a, b, and q) of the distribution computed by parsmd. The cumulative distribution function is

F(x) = 1 - \biggl(1 + \bigl[ (x - \xi) / a \bigr]^b \biggl)^{-q}\mbox{,}

where F(x) is the nonexceedance probability for quantile x with 0 \le x \le \infty, \xi is a location parameter, a is a scale parameter (a > 0), b is a shape parameter (b > 0), and q is another shape parameter (q > 0).

Usage

cdfsmd(x, para)

Arguments

x

A real value vector.

para

The parameters from parsmd or vec2par.

Value

Nonexceedance probability (F) for x.

Author(s)

W.H. Asquith

References

Kumar, D., 2017, The Singh–Maddala distribution—Properties and estimation: International Journal of System Assurance Engineering and Management, v. 8, no. S2, 15 p., doi:10.1007/s13198-017-0600-1.

Shahzad, M.N., and Zahid, A., 2013, Parameter estimation of Singh Maddala distribution by moments: International Journal of Advanced Statistics and Probability, v. 1, no. 3, pp. 121–131, doi:10.14419/ijasp.v1i3.1206.

See Also

pdfsmd, quasmd, lmomsmd, parsmd

Examples

# The SMD approximating the normal and use x=0
tau4_of_normal <- 30 * pi^-1 * atan(sqrt(2)) - 9 # from theory
cdfsmd(0, parsmd( vec2lmom( c( -pi, pi, 0, tau4_of_normal ) ) ) ) # 0.7138779
pnorm( 0, mean=-pi, sd=pi*sqrt(pi))                               # 0.7136874

## Not run: 
t3 <- 0.6
t4 <- (t3 * (1 + 5 * t3))/(5 + t3) # L-kurtosis of GPA from lmrdia()
paraA <- parsmd( vec2lmom( c( -1000, 200, t3, t4-0.02 ) ) )
paraB <- parsmd( vec2lmom( c( -1000, 200, t3, t4      ) ) )
paraC <- parsmd( vec2lmom( c( -1000, 200, t3, t4+0.02 ) ) )
FF <- nonexceeds(); x <- quasmd(FF, paraA)
plot( x, prob2grv(cdfsmd(x, paraA)), col="red", type="l",
      xlab="Quantile", ylab="Gumbel Reduced Variate, prob2grv()")
lines(x, prob2grv(cdfsmd(x, paraB)), col="green")
lines(x, prob2grv(cdfsmd(x, paraC)), col="blue" ) # 
## End(Not run)