Conversion functions between parameters a, k, w,
d, e used in Kiener distributions K2, K3 and K4.
aw2k(a, w)aw2d(a, w)
aw2e(a, w)
ad2e(a, d)
ad2k(a, d)
ad2w(a, d)
ae2d(a, e)
ae2k(a, e)
ae2w(a, e)
ak2d(a, k)
ak2e(a, k)
ak2w(a, k)
de2a(d, e)
de2k(d, e)
de2w(d, e)
dk2a(d, k)
dk2e(d, k)
dk2w(d, k)
dw2a(d, w)
dw2e(d, w)
dw2k(d, w)
ek2a(e, k)
ek2d(e, k)
ek2w(e, k)
ew2a(e, w)
ew2d(e, w)
ew2k(e, w)
kd2a(k, d)
kd2e(k, d)
kd2w(k, d)
ke2a(k, e)
ke2d(k, e)
ke2w(k, e)
kw2a(k, w)
kw2d(k, w)
kw2e(k, w)
a numeric value.
a numeric value.
a numeric value.
a numeric value.
a numeric value.
a (alpha) is the left tail parameter,
w (omega) is the right tail parameter,
d (delta) is the distortion parameter,
e (epsilon) is the eccentricity parameter.
k (kappa) is the harmonic mean of a and w and
describes a global tail parameter.
They are defined by:
\( aw2k(a, w) = k = \frac{2}{\frac{1}{a} + \frac{1}{w}} \)
\( aw2d(a, w) = d = \frac{-\frac{1}{a} +\frac{1}{w}}{2} \)
\( aw2e(a, w) = e = \frac{a-w}{a+w} \)
\( kd2a(k, d) = a = \frac{1}{\frac{1}{k} - d} \)
\( kd2w(k, d) = w = \frac{1}{\frac{1}{k} + d} \)
\( ke2a(k, e) = a = \frac{k}{1-e} \)
\( ke2w(k, e) = w = \frac{k}{1+e} \)
\( ke2d(k, e) = d = \frac{e}{k} \)
\( kd2e(k, d) = e = k * d \)
\( de2k(k, e) = k = \frac{e}{d} \)
The asymmetric Kiener distributions K2, K3, K4:
kiener2, kiener3, kiener4
aw2k(4, 6); aw2d(4, 6); aw2e(4, 6)
outer(1:6, 1:6, aw2k)
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