pdfsmd {lmomco}

Probability Density Function of the Singh–Maddala Distribution

Description

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

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

where f(x) is the probability density 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

pdfsmd(x, para)

Arguments

x

A real value vector.

para

The parameters from parsmd or vec2par.

Value

Probability density (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

cdfsmd, 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
pdfsmd(0, parsmd( vec2lmom( c( -pi, pi, 0, tau4_of_normal ) ) ) ) # 0.061953
dnorm( 0, mean=-pi, sd=pi*sqrt(pi))                               # 0.06110337

## Not run: 
LMlo <- vec2lmom(c(10000, 1500, 0.3, 0.1))
LMhi <- vec2lmom(c(10000, 1500, 0.3, 0.6))
SMDlo <- parsmd(LMlo, snap.tau4=TRUE) # Tau4 snapped to 0.15077
SMDhi <- parsmd(LMhi, snap.tau4=TRUE) # Tau4 snapped to 0.25360
FF <- pnorm(seq(-6, 3, by=.01))
x <- sort(c(quasmd(FF, SMDlo), quasmd(FF, SMDhi)))
plot( x, pdfsmd(x, SMDlo), col="red", xlim=range(x), type="l")
lines(x, pdfsmd(x, SMDhi), col="blue") #
## End(Not run)