The Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Mathematics Generalized Lambda Distribution and Estimation Parameters Presented by A-NASSER L. ALJAZAR Supervised by Professor: Mohammed S. Elatrash Assistant Professor:Mahmoud k. Okasha Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Islamic University of Gaza June 2005
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Generalized Lambda Distribution and Estimation Parameters
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The Islamic University of Gaza
Deanery of Higher Studies
Faculty of Science
Department of Mathematics
Generalized Lambda Distributionand
Estimation Parameters
Presented by
A-NASSER L. ALJAZAR
Supervised by
Professor: Mohammed S. Elatrash
Assistant Professor:Mahmoud k. Okasha
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Master of Science
Islamic University of Gaza
June 2005
Dedicated
To
my son
II
Acknowledgement
All my special thank after thanking God, my mother, to Department
of Mathematics in Islamic University of Gaza, special thanks to
Professor: Mohammed Eltrash
Assistant Professor: Mahmoud Okasha
for many interesting suggestions and remarks on my thesis.
Words can never express how I am grateful to my family and my wife
for their endless love and support in good and bad times. I wish to
thank all my friends for their Cooperations.
Finally, I would like to thank everyone who supported me
III
Abstract
Generalized Lambda Distribution and
Estimation Parameters
.
Generalized lambda distribution (GLD) is a very useful mean to
testing and fitting of data to well known distributions. Since the GLD
is defined by its quantile function, it can provide a simple and effective
algorithm for generating random variates. In this thesis we study the
defection of GLD and plotting the function. The purpose of this thesis
is to estimate the four parameters of the GLD by using three methods,
the method of moments, the method of least squares and percentiles
method. Numerical examples were used to estimate the parameters of
the GLD . Finally we applied one method on real data.
IV
Table of Contents Table of contents v Introduction 1
1. The Generalized lambda Family of Distributions 4 1.1 History and Background…………………………………………….......5 1.2 Definition of the Generalized lambda distributions……………………..9 1.3 The parameter space of the GLD………………………………………..13 1.4 Shape characteristics of the FMKL Parameterization…………………..17 2. The Moments of the GLD 22 2.1 Karian and dudewicz (2000) approach:………………………………...22
2.2 The moments of the FMKL parameterizations of the GLD Approach: 28 3. Estimating the parameters of the generalized lambda distribution 32 3.1 Introduction ………………………………………………………………32 3.2 Fitting the GLD by the Use of Tables……………………………………34 3.3 Example………………………………………………………………….36 3.4 Estimating the parameters of the generalized lambda distribution: the least squares method……………………………………………………………….40 3.5 Example…………………………………………………………………..43 4. Estimating the parameters of the generalized lambda distribution 45 4.1 The Use of percentiles…………………………………………………….47 4.2 Estimation of GLD parameters through a method of percentiles…………51 4.3 Example…………………………………………………………………...53
5. GLD approximations to some well known distributions 56 5.1 GLD approximations to some well known distributions by using a method of moment………………………………………………………..56 5.1.1 The normal distribution………………………………………56 5.1.2 The uniform distribution……………………………………...59 5.1.3 The exponential distribution………………………………….60 5.2 GLD approximations of some well – known distribution by using a method on percentiles……………………………………………………………….62 5.2.1 The normal distribution……………………………………….62 5.2.2 The uniform distribution………………………………………65 5.2.3 The exponential distribution…………………………………..65 5.3 Application………………………………………………………………67 Appendices 69 References 82
Introduction
Fitting a probability distribution to data is an important task in any statistical data
analysis. The data to be modeled may consist of observed events, such as a financial
time series, or it may comprise simulation results. When fitting data, one typically
first selects a general class, or family, of distributions and then finds values for the
distributional parameters that best match the observed data. Rachev and Mittnik
(2000),demonstrated that the usual approach to distribution fitting is to fit as many
distributions as possible and use goodness-of-fit tests to determine the best fit. This
method, the empirical method, is subjective and is not always conclusive. However,
except in the case of data, there is no single accepted rule for selecting one distribution
over another.
The Generalized Lambda Distribution (GLD), originally proposed by Ramberg
and Schmeiser (1974) at this time which is called ”RS” distribution , is a four-
parameter generalization of Tukey’s Lambda family (Hastings et al. 1947) that has
proved useful in a number of different applications. Since it can assume a wide variety
of shapes, the GLD offers risk managers great flexibility in modelling a broad range
of financial data. Due to its versatility, however, obtaining appropriate parameters
for the GLD can be a challenging problem. An excellent synopsis of the GLD, its ap-
plications and parameter estimation methods appear in Karian and Dudewicz (2000).
The initial, and still the most popular, approach for estimating the GLD parameters
is based on matching the first four moments of the empirical data. This is undoubt-
edly due in part to the availability of published tables that provide parameter values
for given levels of skewness and kurtosis (see, e.g., Ramberg et al. (1979); Karian and
1
2
Dudewicz (2000)).
However, different parameter values can give rise to the same moments and so,
while the tabulated parameters may match or closely approximate the first four mo-
ments, they may in fact fail to adequately represent the actual distribution of the data.
Thus, as is well noted in the literature, a goodness-of-fit test should be performed
to establish the validity of the results. If this test fails, or if the levels of skewness
and kurtosis are outside of the tabulated values, it is necessary to use numerical pro-
cedures to find suitable parameters. Such procedures, which typically involve the
downhill simplex method (Nelder and Mead 1965) or some variant thereof, require as
input an initial estimate of the parameters. If multiple local optima exist, the solu-
tion returned is contingent on this estimate. Thus, several attempts may be required
before obtaining parameter values that are acceptable from a goodness-of-fit perspec-
tive. Unlike previous approaches, King and MacGillivray (1999) assess the quality
of the GLD directly by performing goodness-of-fit tests for specified combinations of
parameter values
Instead of matching moments, Ozturk and Dale (1985) minimize the total squared
differences between the data and the expected values of order statistics implied by
the GLD. The NelderMead downhill simplex algorithm is used to find the optimal
parameters. The method, which is called ”least squares method,” successfully fits a
set of data for which tabulated moment-matching values are unavailable. As with
moment matching, the resulting distribution must be assessed using a goodness-of-fit
test, and several trials may be required before finding an acceptable solution. Instead
of matching moments or least squares methods. Karian and Dudewicz (2000) devel-
ops a method for fitting a GLD distribution to data that is based on percentiles rather
than moments. This approach makes a larger portion of the GLD family accessible
for data fitting and eases some of the computational difficulties encountered in the
method of moments. Examples of the use of the proposed system are included.
The generalized lambda distribution (GLD) is very useful in fitting data and approx-
imating many well known distributions. Since the GLD is defined by its a quintiles
3
function, it can provide a simple and effective algorithm for generating random vari-
ates. The purpose of this dissertation is to estimate the four parameters of the GLD
by using three methods; the method of moments , the method of least squares and the
method of percentiles . We use the moment-matching ,least squares or the method
of percentiles to obtain a candidate set of parameters.
This thesis will proceed as follows: we start with introduction to the history
and mathematical background of the GLD. This is followed by a discussion of how to
estimate the unknown parameter with the method of moments, least squares method,
and percentiles method. In particular we shall be looking at the application of the
GLD and approximating some well known probability distributions as well as the
quality of fit and solves some Examples.
Chapter 1
The Generalized lambda Family ofDistributions
Much of modern human endeavor, in wide-ranging fields that include science, technol-
ogy, medicine, engineering, management, and virtually all of the areas that comprise
human knowledge involves the construction of statistical models to describe the fun-
damental variables in those areas. The most basic and widely used model, called
the probability distribution, relates the values of the fundamental variables to their
probability of occurrence.
The problem of this thesis is how to model (or, how to fit) a continuous probabil-
ity distribution to data. The area of fitting distributions to data has seen explosive
growth in recent years. Consequently, few individuals are well versed in the new
results that have become available. In many cases these recent developments have
solved old problems with the fitting process .
The Generalized Lambda Distribution (GLD) is a four-parameter generalization origi-
nally proposed by Ramberg and Schmeiser (1974) of the one-parameter TukeyLambda
distribution introduced by Hastings et al in 1947. Since then the flexibility of the
GLD in assuming a wide variety of shapes has seen it being used extensively to fit and
4
5
model a wide range of differing phenomena to continuous probability distributions,
from applications in meteorology and modeling financial data, to Monte Carlo simu-
lation studies. The GLD is defined by an inverse distribution function or percentile
(quantile) function. This is the function Q(u) where u takes values between 0 and
1, which gives us the value of x such that F(x) = u, where F(x) is the cumulative
distribution function (c.d.f). From this it is easy to derive the probability density
function (p.d.f) for the GLD using differentiation by parts, however the cumulative
distribution function needs to be calculated numerically. The most popular method
for estimating the GLD parameters is to match the first four moments of the empirical
data to that of the GLD. The popularity of this method is partly due to the avail-
ability of extensive tables that provide parameter values for given values of skewness
and kurtosis see Ramberg (1979) and Karian and Dudewicz (2000). In our case we
will use the tables to find the parameter values and will be calculating them directly.
From values of given values of skewness and kurtosis see Ramberg (1979) and Karian
and Dudewicz (2000).
1.1 History and Background
The search for a method of fitting continuous probability distributions to data is quite
old. Pearson (1895) gave a four-parameter system of probability density function and
fitted the parameters by what he called the ”method of moments (Pearson (1894))
6
Tukey’s Lambda family of distributions is defined by the quantile function Q(u) ori-
gins in the one-parameter lambda distribution proposed by John Tukey (1960)
Q(u) =
uλ−(1−u)λ
λ, λ 6= 0
log(u)1−u
, λ = 0 , u 6= 1where 1 ≥ u ≥ 0.
Tukey’s lambda distribution was generalized, for the purpose of generating random
varieties for Monte Carlo simulation studies, to the four-parameter generalized lambda
distribution, or GLD, by Ramberg and Schmeiser (1972 - 1974) subsequently, and
Mykytka (1979).
Since the early 1970s the GLD has been applied to fitting phenomena in many
fields of endeavor with continuous probability density functions.
In an early application of the GLD (at the time called the RS (for Ramberg-
Schmeiser) distribution), Ricer (1980) dealt with construction industry data. His
concern was to correct for the deviations from normality, which occur in construction
data, especially in expectancy pricing in a competitive bidding environment, finding
such quantities as the optimum markup. In another important application area,
meteorology, it is recognized that many variables have used empirical distributions
as an alternative, fitting of solar radiation data with the GLD was successful due to
the flexibility and generality of the GLD, which could successfully be used to fit the
wide variety of curve shapes observed.” In many applications, this means that we
can use the GLD to describe data with a single functional form by specifying its four
parameter values for each case, instead of giving the basic data (which is what the
empirical distribution essentially does) for each case, Karian and Dudewicz (2000).
Before defining the Generalized Lambda Distribution (GLD) family we review
some basic notions from statistics.
7
Definition 1.1.1. . [2] Probability space
A probability space is triplet (Ω ,= , P [ . ]) where Ω is sample space ,= is a collection
of events( each subset of Ω) ,P[ . ]is a probability function with domain =
Definition 1.1.2. . [2] Random Variable ”r.v”
For a given probability space (Ω ,= , P [ . ]) , a random variable, denoted by X or X(.),
is a function with domain Ω. The function X(.) must be such that the set Ar, defined
by Ar = ω : X(ω) ≤ r belong to = for every real number r.
Definition 1.1.3. . [2] Discrete and continuous random variable:
If X can take on only a few discrete values (such as 0 or 1 for failure or success, or
0,1,2,3,...as the number of occurrences of some event of interest), then X is called a
discrete random variable.
If the outcome of interest X can take on values in a continuous range (such as all
values greater than zero and less than one), then X is called a continuous random
variable.
Definition 1.1.4. . [ 2] Cumulative distribution function :
One way of specifying the chances of occurrence of the various values that are pos-
sible. This is, called cumulative distribution function (c.d.f.) of random variable X
, denoted by FX( . ) ,is defined to be that function with domain the real line which
satisfies
FX(x) = p (X < x) −∞ < x < ∞
Definition 1.1.5. .[2]Probability density function:
A second way of specifying the chances of occurrence of the various values of X is to
8
give what is called the probability density function (p.d.f.) of x . This is a function
fX(x) that is fX(x) > 0 for all x, integrates to 1 over the range −∞ < x < ∞ , and
such that for all x,
FX(x) =
∫ x
−∞fX(t)dt
Definition 1.1.6. . [2] Quantiles function:
A third way of specifying the chances of occurrence of the various values of X is to
give what is called the inverse distribution function, or quantiles function (p.f.), of X.
This is the function QX(y) which, for each y between 0 and 1, tells us the value of x
such that
FX(x) = y : QX(y) = The value of x such that FX(x) = y , 0 < y < 1
We see that there are three ways to specify the chances of occurrence of a r.v. We
give the c.d.f. , p.d.f. , and p.f. for a r.v : with the general normal distribution with
mean µ, and variance σ2, N(µ, σ2). The p.d.f. is
f(x) =1√2πσ
exp−(x−µ)2
2σ2
and the percentile function (p.f.) can be obtained as follows:
FX(x) = p (X ≤ x) = y iff
p (X − µ
σ≤ x− µ
σ) = y = p (Z ≤ x− µ
σ) iff
QZ(y) = x−µσ
iff
x = µ + σ QZ(y)
9
therefor QX(y) = x The value of x such that
FX(x) = y = µ + σQZ(y)
We should note that, in addition to QX(y) , there are several notations in common
use for the p.f. one usually finds the notation F−1X (x).
1.2 Definition of the Generalized Lambda Distri-
butions
Definition 1.2.1. .[7]
The generalized lambda distribution family GLD with parameters λ1, λ2, λ3, λ4,
GLD(λ1, λ2, λ3, λ4), is most easily specified in terms of its percentile function
Q(y) = λ1 +yλ3 − (1− y)λ4
λ2
(1.2.1)
Where 0 < y < 1. The parameters λ1 and λ2 are, respectively, location and scale
parameters, λ3 and λ4 determine the skewness and kurtosis of the GLD(λ1, λ2, λ3, λ4).
Note that
Definition 1.2.2. .[2]Parameter
A parameter is a value, usually unknown (and therefore has to be estimated), used
to represent a certain population characteristic. For example, the population mean
µ is a parameter that is often used to indicate the average value of a quantity
10
Definition 1.2.3. .[2] Location Parameter:
let f(x) be any p.d.f .Then the family of p.d.f f(x − µ) indexed by the parameter
µ ,−∞ < µ < ∞ is called location family and µ is called Location Parameter .
Measures of location give information about the location of the central tendency
within a group of numbers.
Definition 1.2.4. .[2]Skewness
Not all distributions are bell shaped (or normal). In the normal distribution, there
are just as many observations to the right of the mean as there are to the left. The
median and mean are also equal. When this is not the case, we say the distribution
is skewed or asymmetrical. If the tail is drawn out to the left, then the curve is left
skewed If the tail is drawn out to the right, then the curve is right skewed.
Definition 1.2.5. .[2]Kurtosis
Another type of departure from normality is the kurtosis, or ”peakedness” of the
distribution. A leptokurtic curve has more values near the mean and at the tails, with
fewer observations at the intermediate regions relative to the normal distribution. A
platykurtic curve has fewer values at the mean and at the tails than the normal
curve, but more values in the intermediate regions. A bimodal (”double-peaked”)
distribution is an extreme example of a platykurtic distribution
11
The properties of the GLD distribution are studied in detail in Ramberg (1979)
at this time which called Ramberg -Schmeiser”RS distribution ” ) . In addition
to elaborating the richness of the fourth parameter of GLD to fit a wide variety
of frequency distributions, Ramberg (1979). A good summary of the shape GLD
distribution is well defined appears in King and MacGillivray (1999).
Recall that for the normal distribution there are also restrictions on (µ, σ2),
namely, σ > 0. The restrictions on λ1, λ2, λ3, λ4 that yield a valid, GLD distribu-
tion will be discussed . It is relatively easy to find the probability density function
from the percentile function of the GLD, as we now show.
Theorem 1.2.1. .[3]:
For the GLD(λ1, λ2, λ3, λ4), the probability density function is
f(x) =λ2
λ3yλ3−1 + λ4(1− y)λ4−1, at x = Q(y) (1.2.2)
where 0 ≤ y ≤ 1
Proof. :
Using the relationships: x = Q(y) and F (x) = y and differentiating with respect
to x, we get:
dy
dx= f(x) or f(Q(y)) =
dy
d(Q(y))=
1d(Q(y))
dy
(1.2.3)
so by differentiating (1.2.1) and outing it into (1.2.4) we find f(x).
dQ(y)
dy=
d
dy(λ1 +
yλ3 − (1− y)λ4
λ2
) =λ3y
λ3−1 + λ4(1− y)λ4−1
λ2
(1.2.4)
12
f(x) =λ2
λ3yλ3−1 + λ4(1− y)λ4−1, at x = Q(y).
In plotting the function f(x) for a density such as the normal, where f(x)is given as
a specific function of x, we calculate f(x) at x values, then plotting the pairs (x,f(x))
and connecting them with a smooth curve. For the GLD family, plotting f(x) proceeds
differently since (1.2.3) tells us the value of f(x) at x = Q(y). Thus, we take a grid of
y values (such as .01, .02, .03, , .99, that give us the ( points), find x at each of those
points from(1.2.1), and find f(x) at that x from (1.2.3). Then, we plot the pairs (x
,f(x)) and link them with a smooth curve.
Example
plot f(x) for a GLD, consider the GLD(λ1, λ2, λ3, λ4) with
The GLD(λ1, λ2, λ3, λ4) is valid in Region 1,2 ,3 and 4 , The GLD(λ1, λ2, λ3, λ4)
is not valid in Region 5 and 6. The situation is quite the same in Region 7,8 . A
point in Region 7, is valid if and only if
(1−λ3)(λ4−λ3)
(λ4 − 1)λ4−1 < λ3
λ4
see Karian and Dudewicz (2000). for an in depth study.
17
1.4 Shape Characteristics of the FMKL Parame-
terization
Freimer .(1988) devise a different parameterizations for the GLD, denoted FMKL,
which is given by
Q(y) = λ1 +1
λ2
[yλ3 − 1
λ3
− (1− y)λ4
λ4
] where 0 ≤ y ≤ 1 (1.4.1)
which is well defined over the entire (λ3, λ4)-plane. The variety of shapes offered by
this distribution classify the density shapes we need to know the role which each of
the parameters play within the GLD.From the defection FMKL we get
λ1 is the location parameter.
λ2 determines the scale.
λ3, λ4 determine the shape characteristics. For asymmetric distribution λ3 = λ4
Freimer (1988) classify the shapes returned by (1.4.1) as follows:
Class I
(λ3 < 1, λ4 < 1) : Unimodal densities with continuous tails. This class
can be subdivided with respect to the finite or infinite slopes of the densi-
ties at the end points.
Class Ia (λ3, λ4 < 12), Class Ib (1
2< λ3 < 1, λ4 ≤ 1
2) , and Class Ic (1
2< λ3 <
1, 12
< λ4 < 1) .
Class II
(λ3 > 1, λ4 < 1): Monotone p.d.f.s similar to those of the exponential or
χ2 distributions. The left tail is truncated.
Class III
18
(1 < λ3 < 2, 1 < λ4 < 2): U-shaped densities with both tails truncated.
Class IV
(λ3 > 2, 1 < λ4 < 2): Rarely occurring S-shaped p.d.fs with one mode and
one antimode. Both tails are truncated.
Class V
(λ3 > 2, λ4 > 2): Unimodal p.d.fs with both tails truncated.
Figures 1.3 to 1.10 show examples of each class of shapes.
see ”SusanneW.M.AuY enug”for an in depth study.
Figure 1.3: Class Ia p.d.fs including the normal distribution
Figure 1.4: Class Ib p.d.fs
19
Figure 1.5: Class Ic p.d.fs.
Figure 1.6: Class II p.d.fs includes the exponential distribution.
Figure 1.7: Class III U-shaped p.d.fs.
20
Figure 1.8: Class IV S-shaped p.d.fs.
Figure 1.9: Class V p.d.fs.
Chapter 2
The Moments of the GLD
Moment come in two formes.They are either raw moment or central moments.
Definition 2.0.1. .[2]The Kth raw moment of a probability density function f(x) of the random variableXis defined by :
E(Xk) =
∫ ∞
−∞Xkf(x)dx where k ≥ 1
.In particular the 1st moment E(X) = µ is the mean.
Definition 2.0.2. .[2]The Kth central moment is defined by
E(X − µ) =
∫ ∞
−∞(X − µ)kf(x)dx wherek > 1
2.1 Karian and Dudewicz (2000) approach :
The moments of the GLD(λ1, λ2, λ3, λ4) , parameterizations of the GLD can be de-
rived as follows : We start by setting λ1 = 0 to simplify this task; next, we obtain
the non-central moments of the GLD(λ1, λ2, λ3, λ4); and finally, we derive the central
GLD(λ1, λ2, λ3, λ4) moments.
Theorem 2.1.1.
If X is a GLD(λ1, λ2, λ3, λ4) random variable then Z = X−λ1 is GLD(0, λ2, λ3, λ4).
21
22
Proof. :
Since X is GLD(λ1, λ2, λ3, λ4)
QX(y) = λ1 +yλ3 − (1− y)λ4
λ2
andFX−λ1(x) = p (X − λ1 ≤ x) = p (X ≤ x + λ1) = FX(x + λ1) (2.1.1)
If we set FX(x + λ1) = y , we obtain
x + λ1 = QX(y) = λ1 +yλ3 − (1− y)λ4
λ2
(2.1.2)
From (2. 2.1) we also have FX−λ1(x) = y which with (2.2.2) yields
QX−λ1(y) = x =yλ3 − (1− y)λ4
λ2
Proving that X − λ1 is GLD(0, λ2, λ3, λ4).
Having established λ1 as a location parameter, we now determine the non-central
moment (when they exist) of the GLD(λ1, λ2, λ3, λ4)
Theorem 2.1.2.
If Z is GLD(0, λ2, λ3, λ4); then E(Zk), the expected value of ZK, is given by
E(Zk) =1
λk2
k∑i=0
[(ki )(−1)iβ(λ3(k − i) + 1 , λ4i + 1) (2.1.3)
where β(a, b) is the beta function defined by :
β(a, b) =
∫ 1
0
xa−1(1− x)b−1dx (2.1.4)
Proof.
E(Zk) =
∫ ∞
−∞Zkf(z)dz =
∫ 1
0
(Q(y))kdy
where
f(z) =dy
dz, Q(y) = z (2.1.5)
23
=
∫ 1
0
(yλ3 − (1− y)λ4
λ2
)kdy =1
λk2
∫ 1
0
(yλ3 − (1− y)λ4)kdy
By the binomial theorem,
(yλ3 − (1− y)λ4)k =k∑
i=0
[(ki )(y
λ3)(k−i)(−(1− y)λ4)i) (2.1.6)
using (2.2.6) in the last expression of (2.2.5), we get
E(ZK) =1
λk2
k∑i=1
[(ki )(−1)i
∫ 1
0
(yλ3)(k−i)(1− y)λ4idy]
=1
λk2
k∑i=1
[(ki )(−1)iβ(λ3(k − i) + 1, λ4i + 1)]
Completing the proof of the theorem.
Before continuing with our investigation of the GLD(λ1, λ2, λ3, λ4) moments,we
note the beta function that will be useful in our subsequent work where β(a, b) is the
beta function defined before.
The integral in (2.1.4) that defines the beta function will converge if and only if a
and b are positive (this can be verified by choosing c from the (0, 1) interval and
considering the integral over the subintervals (0, c) and (c, 1)).
Corollary 2.1.3.
The kth GLD(λ1, λ2, λ3, λ4) moment exists if and only if min (λ3, λ4) > −1/k.
Proof. .From Theorem (2.1.1) , E(Xk) will exist if and only if exists, which, by Theorem
(2.1.2), will exist if and only if E(Zk) = E((X − µ)K) exists, which, by Theorem
(2.1.2), will exist if and only if
24
λ3(k − i) + 1 > 0 and λ4i + 1 > 0, for i = 0, 1, ..., k.
Then λ3k − λ3i > −1 and λ3k > −1 + λ3i
Since λ3i is positive then
λ3k > −1, and λ3 > −1/k
Also we haveλ4i + 1 > 0 and λ4i > −1 for i = 0, 1, ., k
if i = k then λ4 > −1/k
This condition will done if
λ3 > −1/k and λ4 > −1/k.
Since, ultimately, we are going to be interested in the first four moments of the
GLD(λ1, λ2, λ3, λ4), we will need to impose the condition λ3 > −1/4 and λ4 > −1/4
throughout the remainder of this chapter.
The next theorem gives an explicit formulation of the first four centralized
GLD(λ1, λ2, λ3, λ4) moments.
Theorem 2.1.4.
If X is GLD(λ1, λ2, λ3, λ4) with λ3 > −1/4 and λ4 > −1/4, then its first fourmoments, α1 , α2 , α3 , α4 (mean , variance , skewness , and kurtosis, respectively),are given by
The proper interpretation of these table entries is (λ1(0, 1), λ2(0, 1)), λ3, λ4) = (-
0.07268,0.01603,0.008378,0.009564). With few exceptions, Table A-l provides values
of λ1, λ2, λ3, λ4 for which
Max |αi − αi| < 10−5
The exceptions occur when very small changes in λ3 or λ4 cause large variations in
α3 and α3. a situation that arises when λ3 or λ4 gets close to 0. When |λ3| < 10−2
or |λ4| < 10−2, we generally have
Max |αi − αi| < 10−3
68
69
In the rare instances where |λ3| < 10−4 or |λ4| < 10−4, we can only be assured of
Max |αi − αi| < 10−2
The entries of this part of Table A-l are from ”The Extended Generalized Lambda
Distribution (EGLD) System for Fittin g Distributions to Data with Moments, II:
Tables” by E.J. Dudewicz and Z.A. Kari-an, American Journal of Mathematical and
Management Sciences, V. 16, 3 and 4 (1996), pp. 287-307, copyright 1996 by Ameri-
can Sciences Press, Inc., 20 Cross Road, Syracuse, New York 13224-2104. Reprinted
with permission.
71
72
73
74
75
76
77
78
79
80
81
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