KAPCDF

KAPCDF (LET)

Library Function

Compute the kappa cumulative distribution function with shape parameters h and k.

The general form of the kappa distribution has the following probability density function:
\( F(x;k,h,\xi,\alpha) = \left( 1 - h \left( 1 - \frac{k(x - \xi)}{\alpha} \right) ^{1/k} \right) ^{1/h} \hspace{20pt} \alpha > 0 \)

with k and h denoting the shape parameters and \( \xi \) and \( \alpha \) denoting the location and scale parameters, respectively, and where F is the kappa cumulative distribution function.

The upper bound of x is

\( \begin{array}{ll} x < \xi + \alpha (1 - h^{-k}) & \mbox{ if } k > 0 \\ x < \infty & \mbox{ if } k \le 0 \end{array} \)

The lower bound of x is

\( \begin{array}{ll} x > \xi + \frac{\alpha (1 - h^{-k})}{k} & \mbox{ if } h > 0 \\ x > \frac{\xi \alpha}{k} & \mbox{ if } h \le 0, k < 0 \\ x > -\infty & \mbox{ if } h \le 0, k \ge 0 \end{array} \)

The standard form of the distribution is defined as \( \xi \) = 0 and \( \alpha \) = 1.

LET <y> = KAPCDF(<x>,<k>,<h>,<xi>,<alpha>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable;
            <k> is a number, parameter, or variable that specifies the first shape parameter;
            <h> is a number, parameter, or variable that specifies the second shape parameter;
            <xi> is a number, parameter, or variable that specifies the location parameter;
            <alpha> is a number, parameter, or variable that specifies the scale parameter;
            <y> is a variable or a parameter (depending on what <x> is) where the computed kappa cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The <xi> and <alpha> parameters are optional.

LET A = KAPCDF(3,0.5,2,0,1.5)
LET X2 = KAPCDF(X1,K,H)

Dataplot uses Hoskings CDFKAP routine to compute the kappa cumulative distribution function. Hoskings report and associated Fortran code can be downloaded from the Statlib archive at
http://lib.stat.cmu.edu/general/lmoments

None

None

KAPPDF	= Compute the kappa probability density function.
KAPPPF	= Compute the kappa percent point function.
MIEPDF	= Compute Miekle's beta-kappa probability density function.
GEVPDF	= Compute the generalized extreme value probability density function.
GEPPDF	= Compute the generalized Pareto probability density function.
GL5PDF	= Compute the Hosking's generalized logistic probability density function.

Hosking and Wallis (1997), "Regional Frequency Analysis", Cambridge University Press, Appendix A10.

J. R. M. Hosking (2000), "Research Report: Fortran Routines for use with the Method of L-Moments", IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598.

Hoskings (1990), "L-moments: Analysis and Estimation of Distribution using Linear Combinations of Order Statistics", Journal of the Royal Statistical Society, Series B, 52, pp. 105-124.

Distributional Modeling

2008/5

 
LET KP = DATA -0.5  0.1  0.5  1.0
LET H1 = -0.5
LET H2 = 0.1
LET H3 = 1
LET H4 = 2
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 95 95
MULTIPLOT SCALE FACTOR 2
TITLE CASE ASIS
TITLE OFFSET 2
X3LABEL
LINE COLOR BLACK RED BLUE GREEN
.
LOOP FOR KK = 1 1 4
   LET K = KP(KK)
   LET LL1 = KAPPPF(0.05,K,H1)
   LET UL1 = KAPPPF(0.95,K,H1)
   LET LL2 = KAPPPF(0.05,K,H2)
   LET UL2 = KAPPPF(0.95,K,H2)
   LET LL3 = KAPPPF(0.05,K,H3)
   LET UL3 = KAPPPF(0.95,K,H3)
   LET LL4 = KAPPPF(0.05,K,H4)
   LET UL4 = KAPPPF(0.95,K,H4)
   TITLE K = ^K, H = -0.5, 0.1, 1, 2
   PLOT KAPCDF(X,K,H1) FOR X = LL1  0.01  UL1  AND
   PLOT KAPCDF(X,K,H2) FOR X = LL2  0.01  UL2  AND
   PLOT KAPCDF(X,K,H3) FOR X = LL3  0.01  UL3  AND
   PLOT KAPCDF(X,K,H4) FOR X = LL4  0.01  UL4
END OF LOOP
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Kappa CDF Functions