The GLD is an extension of the family of lambda distributions proposed by Tukey (see Tukey 1962). The latter family is defined by the quantile functionQ(u) with u∈ [0,1], that is, the inverse of the distribution function:
Q(u) =
{u𝜆−(1−u)𝜆
𝜆 𝜆≠0,
log u
1−u 𝜆=0. (6.6)
The parameter𝜆is referred to as a shape parameter andQ(u)is symmetric. It should be noted that this quantile function does not have a simple closed form for any param- eter values𝜆, except for𝜆=0, and hence the values of the density and distribution function have to be computed numerically. Incidentally, the density function can be expressed parametrically for all values of𝜆in terms of the quantile function, as in the equation above, and the reciprocal of the quantile density function, that is, the first derivative of (6.6). Tukey’s lambda distribution is termed a family of distributions, because many other statistical distributions can be approximated by it. For instance, if𝜆= −1 thenQ(u)behaves approximately as a Cauchy distribution, and if𝜆=1 a uniform[−1,1]distribution results. Indeed, an approximate distribution for a given data series can be discerned by plotting the probability plot correlation coefficient.
By splitting the parameter 𝜆 in (6.6) into distinct parameters, one obtains the GLD. Here, different parameterizations have been proposed in the literature. A four-parameter extension of the quantile function is due to Ramberg and Schmeiser (1974):
Q(u)RS=𝜆1+ u𝜆3− (1−u)𝜆4 𝜆2
. (6.7)
Tukey’s lambda distribution is recovered from this specification when𝜆1=0 and 𝜆2 =𝜆3=𝜆4=𝜆. The four parameters represent the location (𝜆1), the scale (𝜆2), and the shape characteristics (𝜆3and𝜆4) of a distribution. A symmetric distribution is given for𝜆3=𝜆4. The characteristics of this specification have been extensively investigated by Ramberg and Schmeiser (1974), Ramberg et al. (1979), King and MacGillivray (1999), and Karian and Dudewicz (2000), among others. As it turns out, not all parameter combinations yield a valid density/distribution function. The probability density function of the GLD at the pointx=Q(u)is given by
f(x) =f(Q(u)) = 𝜆2
𝜆3u𝜆3−1+𝜆4(1−u)𝜆4−1. (6.8) Valid parameter combinations for𝛌must yield the following, such that (6.8) qual- ifies as a density function:
f(x)≥0 (6.9)
k k and
∫ f(x)dx=1. (6.10)
Originally, only four regions of valid parameter constellations for𝜆3 and𝜆4were identified by Ramberg and Schmeiser (1974). In Karian et al. (1996) this scheme was amended by additional regions which are labeled “5” and “6” in Figure 6.2.
The distributions pertinent to these regions share the same boundaries as for adjacent regions. The parameter constellations for the four/six regions and the implied support boundaries of the GLD are replicated in Table 6.1.
−3 −2 −1 0 1 2 3
−3−2−10123
λ3
λ4
Region 1:
λ3 < −1, λ4 > 1
Region 4:
λ3 ≤ 0, λ4 ≤ 0
Region 2:
λ3 > 1, λ4 < –1 Region 3:
λ3 ≥ 0, λ4 ≥ 0 5
6
Figure 6.2 GLD: valid parameter combinations of𝜆3and𝜆4in non-shaded areas.
Table 6.1 Range of valid GLD parameter combinations.
Region 𝜆1 𝜆2 𝜆3 𝜆4 Minimum Maximum
1 and 5 all <0 <−1 >1 −∞ 𝜆1+ (1∕𝜆2)
2 and 6 all <0 >1 <−1 𝜆1− (1∕𝜆2) ∞
all >0 >0 >0 𝜆1− (1∕𝜆2) 𝜆1+ (1∕𝜆2)
3 all >0 =0 >0 𝜆1 𝜆1+ (1∕𝜆2)
all >0 >0 =0 𝜆1− (1∕𝜆2) 𝜆1
all <0 <0 <0 −∞ ∞
4 all <0 =0 <0 𝜆1 ∞
all <0 <0 =0 −∞ 𝜆1
k k Two observations can be made from Table 6.1. First, the support of the GLD dis-
tribution can change quite abruptly for slightly different parameter constellations.
Second, parameter constellations that fall in the third quadrant imply skewed and heavy-tailed distributions. These characteristics are part of the stylized facts about financial return series that were stated earlier. Recalling that the market risk mea- sures VaR and ES are quantile values, the GLD seems to be an ideal candidate for computing these measures. This will be shown further below.
In order to avoid the problem that the GLD is confined to certain parameter con- stellations for𝜆3and𝜆4, Freimer et al. (1988) proposed a different specification:
Q(u)FMKL=𝜆1+
u𝜆3−1
𝜆3 −(1−u)𝜆 𝜆4
4
𝜆2
. (6.11)
This specification yields valid density functions over the entire(𝜆3, 𝜆4)plane. The dis- tribution given by this specification will have finitekth-order moment if min(𝜆3, 𝜆4)>
−1∕k.
Recently, Chalabi et al. (2010, 2011) proposed a respecification of the GLD. This proposed approach is a combination of using robust estimators for location, scale, skewness, and kurtosis, and expressing the tail exponents𝜆3and𝜆4by more intuitive steepness and asymmetric parameters𝜉 and𝜒. The new parameterization takes the following form:
̂𝜇=u0.5, (6.12)
̂𝜎=u0.75−u0.25, (6.13)
𝜒= 𝜆3−𝜆4
√1+ (𝜆3−𝜆4)2
, (6.14)
𝜉=1
2 − 𝜆3+𝜆4
2×√
1+ (𝜆3+𝜆4)2
. (6.15)
Here, the bounds for𝜒and𝜉are−1< 𝜒 <1 and 0≤𝜉 <1, respectively. The quan- tile function of the GLD can then be written as
Q(u|̂𝜇, ̂𝜎, 𝜒, 𝜉) = ̂𝜇+ ̂𝜎 S(û |𝜒, 𝜉) −S(0.5|𝜒, 𝜉)̂
S(0.75|𝜒, 𝜉) −̂ S(0.25|𝜒, 𝜉)̂ . (6.16) The functionS(û |𝜒, 𝜉)is defined for the following cases:
S(û |𝜒, 𝜉)
⎧⎪
⎪⎨
⎪⎪
⎩
log(u) −log(1−u) 𝜒=0, 𝜉=0.5, log(u) −(1−u)2𝛼2𝛼−1 𝜒=2𝜉−1,
u2𝛼−1
2𝛼 −log(1−u) 𝜒=1−2𝜉,
u𝛼+𝛽−1
𝛼+𝛽 − (1−u)𝛼−𝛽−1
𝛼−𝛽 otherwise,
(6.17)
k k
–1.0 −0.5 0.0 0.5 1.0
0.00.20.40.60.81.0
χ
ξ
Infinite support
Finite support
Lower infinite support Upper infinite support
Figure 6.3 GLD shape plot.
where
𝛼=1 2
0.5−𝜉
√𝜉(1−𝜉), (6.18)
𝛽=1 2
√ 𝜒
1−𝜒2. (6.19)
In this parameterization the GLD has infinite support if the condition(|𝜒|+1)∕2≤ 𝜉is met. Akin to the shape triangle of the HYP, one can now construct a triangle which starts at𝜒=0 and has as corners(𝜉=1, 𝜒 = −1)and(𝜉=1, 𝜒 =1). All parameter combinations of𝜒 and𝜉 would thus give a distribution with infinite support. The GLD shape plot is shown in Figure 6.3.
As already hinted, the computation of VaR and ES can easily be achieved when a return series has been fitted to the GLD. Here, the formulas are expressed in terms of returns and not for the losses, which are expressed as positive numbers. The VaR for a given probability of error is given by
VaR𝛼=Q(u|𝛌)
=𝜆1+ 𝛼𝜆3− (1−𝛼)𝜆4
𝜆2 , (6.20)
and the formula for computing the ES for a given probability of error can be expressed as
ES𝛼=∫
VaR
−∞
xf(x|𝛌)dx=∫
𝛼
−∞
Q(u|𝛌)du
=𝜆1+ 1
𝜆2(𝜆3+1)𝛼𝜆3+1+ 1 𝜆2(𝜆4+1)
[(1−𝛼)𝜆4+1−1]
. (6.21)
k k Various estimation methods for finding optimal values for the parameter vector𝛌
have been proposed in the literature. Among these are
•the moment-matching approach
•the percentile-based approach
•the histogram-based approach
•the goodness-of-fit approach
•maximum likelihood and maximum product spacing.
The method of moment matching was suggested in the seminal papers of Ramberg and Schmeiser (1974) and Ramberg et al. (1979). The first four moments are matched to the distribution parameters𝜆1,ã ã ã, 𝜆4. For𝜆1=0, thekth moment of the GLD is defined as
𝔼(Xk) =𝜆−k2
∑k
i=0
(k i )
(−1)i𝛽(𝛼, 𝛾), (6.22)
where 𝛽(𝛼, 𝛾)denotes the beta function evaluated at𝛼=𝜆3(k−i) +1 and𝛾 =𝜆4i +1. A nonlinear system of four equations in four unknowns results, and has to be solved. Incidentally, this can be accomplished sequentially, by first determining esti- mates for𝜆3and𝜆4and then solving for the two remaining parameters (see Ramberg and Schmeiser 1974, Section 3). This approach is only valid in the parameter regions for which these moments exist (region 4), and the condition min(𝜆3, 𝜆4)<−1∕4 must be met. As an aside, because the estimates for the skewness and the kurtosis of a data set are very sensitive to outliers, the resulting parameter vector𝛌is affected likewise.
In order to achieve robust estimation results with respect to outlier sensitivity, Chalabi et al. (2010) suggested replacing the moment estimators by their robust counterparts.
The robust moments for location, scale, skewness, and kurtosis are defined as (see, for instance, Kim and White 2004):
𝜇r=𝜋1∕2, (6.23)
𝜎r=𝜋3∕4−𝜋1∕4, (6.24)
sr=𝜋3∕4+𝜋1∕4−2𝜋1∕2
𝜋3∕4−𝜋1∕4
, (6.25)
kr=𝜋7∕8−𝜋5∕8+𝜋3∕8−𝜋1∕8
𝜋6∕8−𝜋2∕8
. (6.26)
These statistics can be estimated by inserting the empirical quantilespq. It is shown in Chalabi et al. (2010) that the higher robust skewness and kurtosis moments only depend on𝜆3and𝜆4. Hence, a nonlinear system of two equations in two unknowns results, which has to be solved. The robust estimation of the GLD parameters has the further advantage of deriving a standardized distribution characterized by a zero median, a unit interquartile range, and the two shape parameters𝜆1and𝜆2:
k k Q(u|𝜆3, 𝜆4) =Q(u|𝜆∗1, 𝜆∗2, 𝜆3, 𝜆4),
𝜆∗2 =S𝜆
3,𝜆4(3∕4) −S𝜆
3,𝜆4(1∕4), 𝜆∗1 = −S𝜆
3,𝜆4(1∕2)∕𝜆∗2, (6.27) whereS𝜆
3,𝜆4(u)is equal to the numerator in (6.7).
Karian and Dudewicz (1999) proposed an estimation approach based on the em- pirical percentiles of the data. From the order statistics ̂𝜋p of the data the following four percentiles are defined, whereu∈ (0,0.25):
̂
p1= ̂𝜋0.5, (6.28)
̂
p2= ̂𝜋1−u− ̂𝜋u, (6.29)
̂
p3= ̂𝜋0.5− ̂𝜋u
̂𝜋1−u− ̂𝜋0.5, (6.30)
̂
p4= ̂𝜋0.75− ̂𝜋0.25
̂
p2 . (6.31)
Foru=0.1 these percentiles refer to the sample median (p̂1), the interdecile range (p̂2), the left–right tail weight ratio (̂p3), and a measure of relative tail weights of the left tail to the right tail (p̂4). These quantiles correspond to the following quantiles of the GLD:
p1=Q(0.5) =𝜆1+0.5𝜆3 −0.5𝜆4 𝜆2
, (6.32)
p2=Q(1−u) −Q(u) = (1−u)𝜆3−u𝜆4+ (1−u)𝜆4 −u𝜆3 𝜆2
, (6.33)
p3= Q(0.5) −Q(u)
Q(1−u) −Q(0.5) = (1−u)𝜆4 −u𝜆3+0.5𝜆3−0.5𝜆4
(1−u)𝜆3 −u𝜆4+0.5𝜆4−0.5𝜆3, (6.34) p4=Q(0.75) −Q(0.25)
p2 = 0.75𝜆3−0.25𝜆4 +0.75𝜆4 −0.25𝜆3
(1−u)𝜆3 −u𝜆4 + (1−u)𝜆4−u𝜆3 . (6.35) This nonlinear system of four equations in four unknowns has to be solved. Simi- lar to the moment-matching method, a sequential approach by first solving only the subsystem consisting ofp̂3 =p3andp̂4=p4for𝜆3and𝜆4can be applied.
Deriving estimates for𝛌from histograms was proposed by Su (2005, 2007). Within this approach the empirical probabilities are binned in a histogram and the resulting midpoint probabilities are fitted to the true GLD density. A drawback of this method is that the resultant estimates are dependent on the chosen number of bins.
The fourth kind of estimation method is based on goodness-of-fit statistics, such as the Kolmogorov–Smirnov, Cramér–von Mises, or Anderson–Darling statistics. These statistics measure the discrepancy between the hypothetical GLD and the empirical distribution, which is derived from the order statistics of the data in question. Parame- ter estimates can be employed when these statistics are minimized with respect to the parameters of the GLD. The determination of the parameter values can be achieved
k k with the starship method as proposed by Owen (1988) and adapted to the fitting of
the GLD by King and MacGillivray (1999). It consists of the following four steps:
1. Compute the pseudo-uniform variables of the data set.
2. Specify a valid range of values for𝜆1,…, 𝜆4and generate a four-dimensional grid of values that obey these bounds.
3. Calculate the goodness-of-fit statistics compared to the uniform(0,1)distri- bution.
4. Choose the grid point(𝜆1, 𝜆2, 𝜆3, 𝜆4)that minimizes the goodness-of-fit statis- tic as the estimate for𝛌.
Finally, the GLD parameters could also be estimated by the ML principle and/or the method of maximum product spacing. The latter method was proposed separately by Cheng and Amin (1983) and Ranneby (1984). This method is also based on the order statistics{x(1),x(2),…,x(N)}of the sample{x1,x2,…,xN}of sizeN. Next, the spacings between adjacent points are defined asD(x(i)|𝛌) =F(x(i)|𝛌) −F(x(i−1)|𝛌)for i=2,…,N. The objective is the maximization of the sum of the logarithmic spac- ings. Compared to the ML method, the maximum product spacing method has the advantage of not breaking down when the support of the distribution is not warranted for a given parameter combination.