Importance of Generalized Logistic Distribution in Extreme Value Modeling

Applied Mathematics, 2013, 4, 560-573 http://dx.doi.org/10.4236/am.2013.43080 Published Online March 2013 (http://www.scirp.org/journal/am)

K. Nidhin1, C. Chandran2 1Department of Mathematics, Indian Institute of Science, Bangalore, India 2Department of Statistics, University of Calicut, Calicut, India Email: [email protected], [email protected]

Received September 10, 2012; revised February 1, 2013; accepted February 8, 2013

ABSTRACT We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multivariate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it possess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc.

Keywords: Financial Risk Modeling; Stock Market Analysis; Generalized Logistic Distribution; Generalized Extreme Value Distribution; Tail Equivalence; Maximum Stability; Random Sample Size; Limit Distribution

1. Introduction different from normal and lognormal to model large values of finance data. For example t-distribution, alpha sta- An important problem from the field of stock market modeling is the determination of adequate model for ex- ble distribution, etc. In 1990’s the interest in modeling treme stock movement. In financial literature, the choice large values of finance data have been diverted to ex- of using an appropriate probability model for financial treme value theory where modeling maximum of the data, returns are clearly exemplified rather than selecting a generalized extreme value distribution for maxima (GEV conventional model (see for ex. [2]). The fitted tale dis- (max)) or generalized Pareto (GP) distribution are used tribution is crucially important in financial studies. For and for minimum of the data, generalized extreme value Value at Risk (VaR) estimation, one requires appropriate distribution for minima (GEV(min)) is used. These dis- probability distribution of extremes as input. A vast num- tributions enjoy strong theoretical support for analyzing ber of literature show the importance of best fitted dis- extreme movement of a data compared to other models. tribution in VaR analysis (see [3]). Another important Below we give an outline of these theoretical properties area of application of probability models of extremes is of the distributions. in hedging procedure. Hedging procedure is clearly based The theoretical representation for GEV(max) and on the probability of fitted distribution (see [4]). Also GEV(min) have been established by Jenkinson and Von measuring the risk attached to a share or portfolio will Mises (see von Mises (1954) and Jenkinson (1955)). We critically depends on the tale distribution. [5] largely il- define GEV(max) below. lustrates the importance of appropriate probability mod- Definition 1.1 A random variable X is said to follow els for extremes of financial returns data. the generalized extreme value distribution for maximum Initially, normal and lognormal distributions were used (GEV(max)) if its distribution function is given by, to model data which arise from financial sector. In the  1 k last five decades, different authors have been showing exp 1kx if k 0 Gx   that the distribution of extreme daily returns is far from k   exp expxk if 0 normal (see [5-11]. They used a number of distributions  

and the supports are Theorem 1.1 (see [12]): Let Xnn , 1 be a sequence of i.i.d.r.v.’s and M =maxXX , ,,X . If xk1 for k 0 nn12  there exist sequences of norming constants xk1 for k 0  abn 0,  n , and a non-degenerate d.f. G such that, xRfor k 0.  Mb w PxGnn x , (1.1) The related location-scale family can be introduced by an x   replacing the argument x above by for   R then G , in Equation (1.1), is the GEV(max) distribution  defined in Definition 1.4. and   0 . The parameter k  is known as the shape Since parameter of the distribution and different values of k min X12 ,XX , ,nn = max XXX1 , 2 , , , the leads to different well known distributions. That is, for asymptotic distribution of minima of i.i.d.r.v.’s can be k  0 , it is the Gumbel (Type I) distribution which is derived as 1 Gx  , where G is the GEV(max). interpreted as limk0 Gk, For k  0 , it is the reverse- Hence the asymptotic distribution for minima is the Weibull (Type III) distribution, for k  0 , it is Frechet GEV(min) distribution defined in Definition 1.2. These (Type II) distribution. Also GEV(min) can be defined as theories have been extended to multivariate case, where follows. multivariate generalized extreme value distributions play Definition 1.2 A random variable X is said to follow central role (see [13,14]). the generalized extreme value distribution for minimum Another alternative model for maxima is the gene- (GEV(min)), if its distribution function is given by, ralized Pareto distribution, has been suggested by two in- 1 k dependent works [15] and [16]. Below we define GP dis- 1exp1kx if k  0   tribution. Hx  k Definition 1.4 A random variable X is said to follow 1expexpxk if  0 the generalized Pareto (GP) distribution if its distribu- and the supports are tion function is given by, xk1 for k 0   1 k 1 11kx if k 0 xkfor k 0 Qx  xRfor k 0. 1exp xk if 0    The parameter k is the shape parameter and k  . and the supports are One can introduce the related location-scale family by xk1 for k 0 x    replacing the argument x above by for   R xk 1 for k 0   xk0for0. and   0 . For k  0 (which is interpreted as  limk0 Fk), the distribution is called reverse-Gumbel, for k  0 , it is called reverse-Frechet distribution and for The parameter k is the shape parameter and k  . k  0 , the distribution is the Weibull distribution. One can introduce the related location-scale family by x   The above two distributions possess the characterizing replacing the argument x above by for   R properties called max-stability and min-stability respec-  tively. In the following we define max-stability for non- and   0 . With respect to k  0 , k  0 and k  0 random sample size. we get the exponential distribution, beta distribution and Definition 1.3 A non-degenerate random variable X Pareto distribution. is said to be max-stable if, for each n  2,3, there are GP distribution possess characterizing property called Peak over Threshold (POT) stability w.r.t non-random constants an  0 and bn such that d sample size. In the following we define POT stability for maxX12 ,XXaXb , , nn n. non-random sample size. Definition 1.5 A non-degenerate random variable X where X ,,,XX are same copies of X . 12 n is said to be POT-stable if, for each u  0 there are Similarly, one can define min-stability. By the extre- constants a  0 and b such that mal types theorem, asymptotic distribution of maxima of u u d independent and identically distributed random variables X uX u aXb. (i.i.d.r.v.’s) under proper normalization can be approxi- uu mated by the GEV(max) distribution. Below we give an By Peak over threshold theorem, under proper norma- outline of the theorem. lization, the asymptotic distribution of the observations

Copyright © 2013 SciRes. AM 562 K. NIDHIN, C. CHANDRAN above exceedances of a level u has a generalized pareto variate extremal types theorem and multivariate random distribution iff asymptotic distribution of maximum of sum convergence theorem are special cases of this theo- the observations, under proper normalization, is the rem. In Section 5, we use the above general limit theo- GEV(max) distribution. We give an outline of the theorem to prove the convergence of random maxima and rem below. random minima to GL distribution. For an additional

Theorem 1.2 (see [15,16]) let Xnn ,1  be a se- support to our claim, we present a data analysis in Sec- quence of i.i.d.r.v.’s with common continuous distri- tion 6. We introduce a multivariate generalized logistic bution function F x . Suppose there exists a pair of se- distribution in Section 7, and also we prove a charac- quences an and bn with an  0 for all n and a teristic property of this multivariate distribution by using non-degenerate distribution function Gx such that the general multivariate theorem in Section 4. To prove the results we require some basic concepts lim PMnnn b a x n (1.2) which we discuss below. n Definition 1.6 The right end point and left end point of lim F axnn b G x n a d.f. F , denoted by xF and yF respectively, are for all x at which Gx is continuous. Let xxFxF  sup : 1 , 1F ux Px PX u xXu . u   yxFxinf : 0 . 1 Fu F    Definition 1.7 Two distributions F and G are equi- Then, under proper normalization, Pxu e can b approximated by generalized Pareto (GP) distribution, valent in their right tail if they have the same right end point, i.e. x  x , and when ux F (where xxF sup : Fx 1 ). F G This theory also extended to multivariate case, where 1 Fx multivariate generalized Pareto distribution plays central lim 1. xx F 1 Gx role (see [17]). Extremal types theorem (see [18]) and Peak over thre- Definition 1.8 Two distributions F and G are equi- shold theorem ([15,16]) facilitate a theoretical background valent in their left tail if they have the same left end point, to use generalized extreme value (GEV) distributions and i.e. yyF  G , and GP distribution in modeling extreme movement of stock Fx market indices. A number of researchers verified empiri- lim 1. cally that these models give sufficient fit to model ex- xy F Gx treme volatility in stock market, see for ex. [19,20]. Also For more details see [26]. Next, we introduce the max- many researches utilizes these models for measuring stability w.r.t. a discrete distribution. extremal behavior of stock markets, see for ex. [4,21,22]. Definition 1.9 (see [27]) Let F be a non-degenerate However, from 2004, [1] and many other researchers distribution function of a random variable X and N have been empirically showing that generalized logistic be a discrete random variable defined on set of positive (GL) distribution is a better model than GEV and GP in integers with probability mass function p . That is, modeling extremal behavior of different stock market n data, this includes US, UK, Germany, Japan, India, Athens, PN  n pn ,1,2,3, n  . African stock markets etc. See for eg. [1,23,24] etc. Also Then F is said to be max-stable w.r.t. pn (or r. v. [25] empirically showed that GL is better than GEV to X is said to be maximum stable w.r.t. a r. v. N ) if model extreme movements in the stock, commodities and there exist a  0 and b such that, bond markets. N N max XX , , , X  b d In this paper we show some theoretical motivation of 12 NN  X this claim, that is, we present a theoretical framework of aN the role of GL distribution in extreme value modeling. In where X , n 1 are same copies of X . Section 2, we define logistic distribution and some of the n Similarly, one can define min-stability w.r.t. a discrete known results of the logistic distribution which are im- distribution. portant in extreme value theory. We also define generalized logistic distribution in this section. Section 3 in- 2. Review of Logistic and Generalized troduces some of the notable properties of GL distribu- Logistic Distribution tion, namely, tale equivalence with GEV and GP distributions and stability property w.r.t geometric distribution. Logistic distribution is an important distribution used in We prove a general multivariate limit theorem in Section statistical modeling. It is used for modeling in a number 4, and show multivariate central limit theorem, multi- of research papers (see [28]). In literature, Logistic distri-

Copyright © 2013 SciRes. AM K. NIDHIN, C. CHANDRAN 563 bution plays some important role in modeling of ex- [29]). Among the various generalization of logistic dis- tremes of data. In this section, first, we define Logistic tribution, the one given by Hosking (see [30]) is called distribution and then discuss its importance in extreme the 5th generalized logistic distribution ([29]). This form value theory. of generalized logistic distribution is used for many real Definition 2.1 A random variable X is said to follow life applications, see for example [1,23,24,30,31]. Moti- standard logistic distribution if its d.f is given by vated by this we introduce the 5th generalized logistic 1 distribution and call this generalization of logistic dis- FxX x. (2.1) tribution as generalized logistic distribution. 1e x Definition 2.2 A random variable X is said to follow The following are some important results which con- the generalized logistic (GL) distribution if its distri- nects logistic distribution and extreme value theory. bution function is given by, Result 2.1 ([27]): Let Xn,1  be a sequence of n  1 i.i.d.r.v.s with symmetric distribution function F . . Let ifk  0    1 k N be an integer valued random variable independent of 11kx FxX   X n and 1  ifk  0 k 1  PN k p1,01,  p  p k 1. (2.2) 1exp()x and the supports are Let M N  maxXX12 , , , XN  and there exist con- 1 stants aN and bN such that xkfor k 0  Ma d 1 NN xk for k 0  X1 bN xR for k 0.  iff F is the logistic distribution function. The logistic distribution appears as a limiting distri- The parameter k is known as the shape parameter of butions as described in the following result. the distribution and k  . One can introduce the related location-scale family by replacing the argument x Result 2.2 (see [28]): Let Xnn ,1  be a sequence of independent and identically distributed random va- x   above by for   R and   0 . For k = 0, FX riables with distribution function F . . Let N be a in-  teger valued random variable which are not necessarily can be identified as the logistic distribution. For k  0 , and some scale transform, we get loglogistic distribution independent of X n. Assume F is in the domain of attraction of Gumbel distribution and as given in [27]. Similarly for k  0 , through some scale transform we get backward loglogistic distribution. The lim PN nz Az, z  0, N  statistical properties and estimation issues of the GL dis- where Az is a proper distribution function. Then the tribution are discussed in [29]. limit distribution of M N is logistic iff Az 1exp az, a 0. 3. Some Notable Properties of GL The next result brings the connection between logistic Distribution distribution and the extreme value theory through mid- In this section we prove some characters of GL distri- range. bution which are important to extreme value theory. We Result 2.3 (see [28]): Let Xnn ,1,2,  be a se- study the tail behavior of GL distribution compared to quence of independent and identically distributed other distributions. Also, as we see in GEV and GP dis- random variables with X1 follows F . . Let tributions, we prove GL distribution is also characterized

M nn maxXX12 , , , X and by a stable property. mXXn min12 , , , Xn. Assume F is symmetric distribution which belongs to the domain of attraction of 3.1. Tail Equivalence of GL, GEV and GP Gumbel distribution then under proper normalization the In this subsection, we compare tail distribution of GL midrange with GEV(max), GP and GEV(min). Figures 1 and 2 mM   nn gives the probability density functions and distribution n 2 functions of GL, GEV(max), GP and GEV(min) for k = 0 converges to the random variable Z which follows respectively. Figures 1 and 2 indicate that the right tails standard logistic distribution given in Definition 2.1. of the GEV(max), GP and GL distributions are similar In statistics literature, there are several ways of gene- and GEV(min) and GL are similar in left tails. This clearly ralizing the logistic distribution (for more details see indicates that these three distributions are asymptotically

sequence of i.i.d random variable follows GL distri bution

if and only if there exist real numbers aN  0 and bN such that, max XX , , , X  b d 12 NN  X aN where N is a discrete random variable follows g eometric distribution. Similarly, the result is true for minimum. Hence GL satisfies the stability property. In the next Section we prove a theorem which gives a direct relation between stability property and limit distribution of a general class of functions. We use this theorem to prove a limit theorem in extreme value theory where GL plays the central role.

4. A General Multivariate Limit Th eorem

Figure 1. Probability density functions of GEV(max), Here we prove a limit theorm of general functions of in- GEV(min), GP and GL distributions. dependent identically distributed continuous multivariate random variables (i.i.d.c.m.r.v) which posses property Q   (see Definition 4.1). Let g XX12,,, X be a Borel-   measurable function of X12,,,XX  , which is continues. We use the multivariate norming function as  ax  b a1, x 1,  b 1,,,, a 2, x 2,  b 2,  app,,,  x b p . Below we define a property Q of a Borel m easurable function.   Definition 4.1 Let g  XX12,,, X  be a Borel- measurable function then g satisfies property Q if,   gX 12,,, X X   ggX,,, X X  , gX  , X  ,,X ,,  1211 112 21 gX,,, X X  2111   21  12 21 

where ,  and  are integer valued rand om vari- Figure 2. Distribution functions of GEV(max), GEV(min), 1 2 GP and GL distributions. ables or integ ers such that P12 1. n X, maxXX , ,,X  , n X The functions  i1 in12 i1 i equivalent in their tails, which we prove in the following are some of the important examples of functions which theorems. satisfy Property Q. By maxX ,XX , ,  we mean Theorem 3.1 The distributions GEV(max), GP and GL 12 n the coordinate wise maximum. These functions with ran- are equivalent at their right tails. The distributions dom sample size also satisfy Property Q, for more details GEV(min) and GL are equivalent at their left tails. see [32].     3.2. Max-Stability and Min-Stability w.r.t Theorem 4.1 Let XXXnn 1,,, 2,np,X, n , n1 Geometric Distribution    be a sequence of i.i.d.c.m.r.v and g XX12,,,X be From Result 2.1, the logistic distribution is characterized a Boral-measurable function such that g satisfies by max-stability property. Here we prove the property  property Q , and  is independent of each X i . Let still remains with this generalization. Below theorem Y be a non-degenerate r.v. Then there exists sequences show that the GL distribution also characterized by max-  f and e  such that stability and min-stability w.r.t geometric distribution.    w Theorem 3.2 The generalized logistic distribution F   fx e  FY  x gX12,,, X X   given in Definition 2.2 characterizes max-stability w.r.t iff Y satisfies the following equation geometric distribution. That is, let XX12,, be a

FaxbFxx ,  and values of  Fabx   Fxx  k gY,,, Y Y   Y  n    X  12  i1 Xi 1 for some sequences a  0 and b , where Y are     i where X i are same copies of X , see [33 ,34]. Let same copies of Y .  Ynn ,1  be an i.i.d sequence of r.v’s and Z be a Many importan t limit theorem in probability can be   non-degenerate r.v. Then there exist l >0 and m  deduced as a special cases of Theorem 4.1, which in- such that cludes multivariate central limit theorem, multivariate  w  extremal types theorem and Random Sum Convergence F n  lx m  FZ  x i1Yi theorem, which we show in the following subsections. iff Z  X . 4.1. Multivariate Version of Generalized Central Limit Theorem as a Special Case 5. GL as a Limit Distribution of Random Maxima and Minima Let η = n and g XX,,, X  n X , then g satis- 12   i1 i The limit theory of random maxima and random minima fies property Q as described in Definition (4.1). Let X be has been studied by many authors (see for ex. [35]). In the class of random variables which follows multivariate this section we derive generalized logistic distribution as stable distributions. Then for all X  X there exist a se- the limit of random maxima and random minima when   quence an  0 and bn such that, f or each n, sample size follows geometric distribution using Theo- rem 3.2 and 4.1.    FaxbFxxn  nnX ,  i1 Xi 1 Theorem 5.1 Let Xnn , 1 be a sequence of i.i.d.c.r.v. Let M  maxXX , , , X ,   N  12 N  where X i are same copies of X , see [11]. Let   mXN  min 1 , XX2, , N  , and N is an inte ger valued Ynn ,1 be an i.i.d sequence of r.v ’s and Z be a   random variable which follows geometric distribution, non-degenerate r.v. Then there exist cn  0 and dn such that independent of X i  . Also let Y and Z be random variables with non-de generate distribution functions G  w  F n  cxnn d FZ  x and H respectively, a  0 , b , c  0 and d i1Yi N N N N such that,  iff Z  X . Ma NNdY b 4.2. Multivariate Extremal Types Theorem as a N Special Case and mc Let   n and NNd Z.     dN g XX12,,X,  max,,,XX12 Xn , then g satisfies property Q . Let X be the class of random variables Then follows mul tivariate extreme value distributions with cha-  1 ifk  0 racterizing property,  1 k  x  11k  n     F axnn b F x  xand n N Gx  XX11 1  ifk  0.     where X i are same copies of X . Let Ynn ,1 be a x   1exp sequence of i.i.d.c.m.r.v’s and Z be a non-degen erate   r.v. Then there exist c  0 and d such that n n with supports are w F   cxnnd F  x 1 maxYY12 , , ,Yn Z xk fork 0  1 iff Z  X . xkk for 0 xR for k 0. 4.3. Random Sum Convergence as a Special Case  also H has the same distributional form of G with Let   N , a r.v and g XX,,, X  N X , then 12   i1 i different parameters. g satisfie s property Q . Let X be the class of random Proof. The proof consist of two parts. First, f rom variables which follows multiva riate strictly geometric Theorem 3.2, it is verified that GL posses characterizing stable distribution. Then for all X  X there exist con- property of max-stability and min-stability w.r.t geo-   stants a  0 and b such that, fo r each n, metric distribution. By using Theorem 4.1, where p 1,

Copyright © 2013 SciRes. AM 566 K. NIDHIN, C. CHANDRAN the random maxima and random minimum, when sample and GS can be viewed as the weak limit distribution of size follows geometric distribution, converge to the GL sum of random number of random variables, whose ran- distribution, if it exists. dom number follows geometric distribution. Theorem 5 identifies the limit distribution of geometric random maxima and geometric random minima, if it 6. Goodness of Fit Test for the BSE Data exists, as the GL distribution given in Section 2.2. That is, In this section, we empirically study the performance of GL distribution can be used as an asymptotic model for GL and GEV in modeling the behavior of minimum re- maximum and minimum of a random number  N  of random variables, when N follows geometric distribution. turns of Bombay stock exchange data. We use Anderson- Notably, unlike of GEV(max), GEV(min), and GP, GL Darling test for comparing goodness of fit of GEV and distribution provides the theory for both maxima and GL distributions in the extremal behavior of the data. minima. This is an alternative model for maxima and Since the test is based on empirical distribution function minima. The theory is parallel to generalized extreme (EDF) and among all the well known tests based on EDF, value model and generalized Pareto model in the sense Anderson-Darling test has the highest power in testing that the three models includes three important distri- normality against a number of alternatives when the pa- butions which is decided by the shape parameter. Also rameters are unknown [36]. tail of these models are asymptotically equivalent and Table 1 shows a comparison between performance of three models have a characterizing stability property in GEV(min) and GL on minima returns. For fifteen inter- different sense. Moreover, the three models can be con- vals, GL provides an adequate fit in eleven intervals than sidered as the limit distribution of extremes in different GEV(min), which indicates that GL distribution provides situations. These results justify the theoretical importance a better fit than GEV(min). However, GEV(min) proof the empirical findings, that GL can be a suitable model vides an adequate fit in 11 out of 15 intervals and in which over GEV model for extremes of share market data, see most of the intervals p-values are large, which shows that for example [1,25,26]. Many papers also show that GL GEV(min) also provides a model for minimum returns. distribution provides an adequate distribution in hydro- Table 2 shows that GL provides a better fit in six in- logical application, for eg. [30,31] etc. terval compared to GEV(min). While GEV(min) pro- Remark 5.1 It is also be noted that this theorem is vides an adequate fit in nine out of ten intervals showing somewhat similar to geometric stable theorem in the that GEV(min) is also a model for monthly minima. theory of partial sum of random variables where geo- Comparing GL and GEV(min) in quarterly minimum metric stable (GS) distribution plays a central role. GS data, Table 3 indicates the same result, that is, GL pro- distribution is widely using in stock market modeling, vides a better fit than GEV(min). Further GEV(min) and

Table 1. Performance of GEVmin and GL distribution for weekly minimum data.

GEVmin GL Intervals n loca scale shape p loca scale shape p Best fit 1 78 0.006 0.006 0. 226 0.396 −0.009 0.005 0.324 0.465 GL 2 78 0.004 0.007 0.270 0.000 −0.007 0.005 0.356 0.000 − 3 78 0.014 0.011 −0.107 0.112 −0.019 0.007 0.103 0.869 GL 4 78 0.012 0.009 −0.207 0.672 −0.016 0.006 0.044 0.95 GL 5 78 0.010 0.012 0.204 0.395 −0.015 0.009 0.308 0.225 GEVmin 6 78 0.017 0.014 0.135 0.599 −0.023 0.010 0.26 0.938 GL 7 78 0.010 0.0082 −0.101 0.031 −0.014 0.005 0.106 0.152 GL 8 78 0.011 0.009 −0.235 0.176 −0.015 0.005 0.027 0.219 GL 9 78 0.010 0.011 0.071 0.701 −0.015 0.007 0.217 0.932 GL 10 78 0.013 0.011 0.012 0.052 −0.017 0.007 0.178 0.58 GL 11 78 0.015 0.014 0.0113 0.541 −0.021 0.009 0.177 0.254 GEVmin 12 78 0.008 0.007 0.090 0.009 −0.011 0.005 0.229 0.125 GL 13 78 0.008 0.008 0.183 0.107 −0.012 0.006 0.293 0.348 GL 14 78 0.006 0.007 −0.036 0.401 −0.019 0.006 0.293 0.494 GL 15 88 0.009 0.0117 0.117 0.003 −0.013 0.008 0.242 0.00 GEVmin

Table 2. Performance of GEVmin and GL distribution for monthly minimum d ata.

GEVmin GL Intervals n Loca Scale Shape p Loca Scale Shape p Best Fit 1 26 0.015 0.006 0. 323 0.324 −0.018 0.005 0.396 0.206 GEVm in 2 26 0.024 0.015 −0.060 0.708 −0.03 0.01 0.132 0.511 GEVmin 3 26 0.025 0.007 −0.153 0.255 −0.027 0.004 0.075 0.525 GL 4 26 0.034 0.020 0.137 0.551 −0.043 0.015 0.261 0.517 GEVmin 5 26 0.020 0.008 −0.054 0.419 −0.023 0.005 0.135 0.350 GEVmin 6 26 0.024 0.009 0.141 0.208 −0.027 0.006 0.264 0.339 GL 7 26 0.027 0.012 0.156 0.257 −0.032 0.009 0.274 0.258 GL 8 26 0.023 0.012 0.051 0.593 −0.028 0.008 0.204 0.625 GL 9 26 0.015 0.006 0.467 0.049 −0.018 0.005 0.507 0.122 GL 10 35 0.019 0.013 −0.081 0.695 −0.023 0.007 0.256 0.886 GL

Table 3. P erformance of GEV( min) and G L distributi on for qua rterly minim um data. *Indicates n either dist ribution fit the data.

GEVmin GL Intervals n Loca Scale Shape p Loca Scale Shape p Best Fit 1 22 0.029 0.015 −0.074 0.419 −0.035 0.01 0.123 0.529 GL 2 22 0.034 0.015 0.3072 0.103 −0.04 0.012 0.383 0.186 GL 3 22 0.039 0.018 −0.073 0.054 −0.046 0.011 0.124 0.052 GEVmin 4 24 0.0252 0.010 0.263 0.01 −0.029 0.008 0.350 0.028 GL*

GL giv es a good model in th ree out o f four inte rvals,     Definition 7.1 Let XXXnnnpn 1,,,, 2,  X, ,n 1 which shows both distribution can be used to model  quarterly data and comparing these distributions GL is a be a sequence of i.i.d.c.m.r.v and let better model. M  maxXX , , , X  , where N is random variable On analysis of half yearly data, Table 4 clearly shows N  12 N  follows geometric distribution. Let a  0 and b are GL provides a good fit comparing to GEV(min). Finally, n n p-dimensional real sequences such that Table 5 illustrates GL provides a better for yearly mini-  mum data compared to GEV(min). Both distribution fit axNN b for the data at 0.05 level. ax b, ax b,, a x  b The above analysis reveals that for analyzing the mini- 1,N 1,NNNN 1, 2, 2, 2, N pN,,, pN pN mum returns in stock market data, GL provides a better Suppose the following convergence occur, fit than GEV distributions, which shows minimum be- F ax b w G haves like a geometric minimum than the usual minimum MNN  N in stock market data (for more details see [37]). Then we called G as multivariate generalized logi-

stic (MGL) distribution and F   DG, where DG  7. Multivariate Generalized Logistic X1 Distribution and a Characterization is the domain of attraction of G . Property Notice that, the univariate margins of G must be a In Section 5, we identified that GL distribution is also a generalized logistic distribution, this includes all three suitable model for extremes, moreover it plays better types of the marginal distribution: k = 0, k > 0 and k < 0 model than GEV in some situations. We also proved that correspond respectively to the logistic, loglogis tic and the above theorem is parallel to the existing extremal backward loglogistic distributio ns. B elow we prove a types theorem and peak over threshold theorem. Naturally, characterizing property of MGL distribution. one can think a multivariate version of GL distribution as Theorem 7.1 The MGL distribution is characterized parallel to the theory of multivariate extreme value dis- by max-stability w.r.t geometric distribution. That is, let tribution. N be a random variable follows geometric distribution.

Table 4. Performance of GEV(min) and GL distribution for half yearly minimum data.

GEVmin GL

Intervals n Loca Scale Shape p Loca Scale Shape p Best Fit

1 23 0.038 0.019 0.130 0.452 −0.046 0.013 0.256 0.569 GL

2 23 0.040 0.019 −0.008 0.041 −0.047 0.013 0.165 0.195 GL

Table 5. P erformance of GEV(mi n) and GL distribution for yearly m inimum da ta.

GEVmin GL

Intervals n Loca Scale Shape p Loca Scale Shape p Best Fit

1 23 0.049 0.023 0.001 0.109 −0.058 0.015 0.171 0.131 GL

   ddx1 Gx  let Xnnnpn=X1,,X 2, ,,, X, n1 be a sequen ce of lim x ddx1 F x i.i.d.c.m.r.v which follows MGL distribution iff there exist 2 a  0 and b are p-dimensional real sequences such 1expx n n lim expxx exp exp  that x expx   2 axNN b lim exp expxx 1 exp 1. x  ax b,,, ax b a x  b 1,N 1, N 1, N 2, N 2, N 2, N pN,,, pN pN Case 2   0 . Here also xxGF, and, 1 1 1 k 1 Fx  11x  lim  lim F ax b F x  1  MNN  N X1 xx 1 Gx    1exp1x Proof. The result is directly from Theorem 4.1. 0 which is again form. So applying L-Hospital’s rule, 7.1. Proof of Theorem 3.1 0

Since the proof contains only simple calculation, we only ddx1 F  x lim  give proof for tail equivalence of GEV(max)and GP. We x  ddx1 G  x use notations G and F for distribution functions of GEV(max)and GL respectively.  1   1x 1x lim As per the assumed notation for the distributions of  1  11  x 111xx1exp1xx GEV(max) and GL, to prove the asymptotic equivalence    of their right tails , by Definition 1.7, it is enough to prove 2 that x  x and 1x 1   GF  1x    2 11 1 Gx 11xx1   1 1exxxp1 lim 1,  .   xx F 1 Fx 1  1x 1 We prove this i n the following three cases. 1lim12  x 121xx  1 exp 1 x 1  Case 1   0 . Here xxGF,  1 Gx 1expexpx  lim lim .  xx  1 1 1 1 Fx 1 1lim 1expx x 1 1  exp 1 x 1  1  21 x  1x 0  This is form. So applying L-Hospital’s rule we 1 0 11.  get, 20 

1  1 Case 3   0 . In this case xx, where b  and c  . For k  0 , Y follows back- GF k k ward loglogistic distribution. By [27], we know that the 1 Fx  0 lim  1 1. class containing logistic, loglogistic and backward log- x1  11 Gx  logistic has the characterizing property max-stability w.r.t to geometric distribution. The relation (B.1) is one 7.2. Proof of Theorem 3.2 to one implies the max-stability property w.r.t to geometric distrib ution characterize generalized logistic dis- To prove the max-stability of GL, we use the following tribution. Similarly we can prove that min-stability cha- lemma from [27]. racterizes GL distribution w.r.t geometric distribution. Lemma B.1 Let FX be a non-degenerate distribution Remark B.1 In the above proof, we give short steps function of a r.v. X which is maximum stable w.r.t mainly using Voorn’s idea (see [27]). One can directly pnn , 1 . If yF  , xF  , and for any real prove the theorem. constant c , and d.f’s H1 and H2 defined by

FX xHc 1 exp x and 7.3. Proof of Theorem 4.1 F xHcexp x  for all real x , then H X 2  1     and H are non-degenerate and maximum stable w.r.t Let XXXnnnpn 1,,,, 2,  X, ,1 n  be a sequence of 2  pn, 1 with y  c and x  c respectively. n H1 H2 For detailed discussion see [27]. B elow we prove independent a nd identically distributed continues mul- Theorem 3.2. tivariate random variables (i.i.d.c.m.r.v) defined on some probability space ,,P . We use symbols  ,  Proof. Let FX be the distribution function of generalized logistic di stribution with k  0 , and location and and  to denote both non random integer and in teger scale parameters  and  respectively. That i s, valued random variable, pa rticularly, we write n when we consider non random inte ger and N, depends on some 1  Fx,. x parame ter p , for integer valued random variable. When X  1 k p x   k   n , non-ran dom, then   may be interpreted 11k  as n . To prove the theorem we need the following  T function in multivariate case. Now take the transformation x  exp x , we get,  H1  Definition C.1 Let XX,,n n 1 be a sequence of 1 i.i.d.c.m.r.v with F represents distribution function of Fxexp x ,x  Y XH1  1 k Y and g be a given Borel function. Define, k 1exp x  DxyxyFx,,p  ,  0  X  which is the distributio n function of Logistic distribution and for every value of m , where m is a positive inte- (see [27]). That is, F x exp x is logistic distri- XH1  ger, we get bution function and by Le mma (B.1), the distribution p function F is max-stable w.r.t geometric distribution. DxzxzFm ,,   x . X  gX12,,, X Xm  Similarly, for k  0 , the same proof works. That is, G L distribut ion satisfies max-stability w.r.t geometric dis- Also the cardinality of D0 and Dm are same. Let tribution. T o prove that GL is the only distribution with M m be a sequence of class of functions s uch that, property of max-stability, let X be a random variable MffDDfmm :0 and is a bijection which follows GL distribution with shape parameter k .

Now take the tran sformation Then for every value of   m ,  a TM mm such that,   Xmif 0 Y   m (5) xF,,  x  T  xF  x ,  gX12, X,, X  X (C.1) Xmif  0  x . then for k  0 , Y follows loglogistic distribution with Below we give a simple example of the above T distribution function  function. 0ify    Example C.1 Let X,,Xn n 1 be a sequence of  1 Fy c i.i.d.c.m.r.v and Y  x       1i fy  g  XX12,,, X   max,, XX12 ,Xn  is given. Then b  the function Tn is,

n Tn xy,,. xy limdT xFax ,  b,, TxFax  b0, x     11  

  Remark C.1 Let an  0 and bn be given se-   the relation is reflexive. Also let FF1,ab , ,and 2, rs ,  quences and let D0 be the following form, M  and F  F . That is,   1,ab , 2,r ,s Dxyy0 ,,x FX  x , limdT xFax ,  b ,, TxF  rx s 0 which implies,   12    DxzxzFmm,,  axb  m. implies gX12,,,X  Xm   limd T xF , 2 rx s ,, T xF 1 ax b 0 also M m and T can define accordingly.     In the following we prove Theorem 4.1. M  Proof. Let G be a non-degenerate distribution func- or F  F so that the relation is symmetric. Now  2,rs , 1,ab , tion. Let Xnn ,1 be a sequence of i. i.d.c .m.r.v such  M  that X1 F1 (for notational conve ni ence w e use F1  let F1,a ,b, F2, rs , and F3,fe ,  , such that F1,ab , F 2,rs , instead of FX ) and define 1 M   p and FF . That is, TxFaxb, x  , 2,rs , 3,f , e 1   w limdT xFax ,  b,, TxF  rx s 0, Tx,,Faxb  xGx    12   1   be the class of sequences of distribu tion functions which and  convergence to x,Gx. We use the notation F ,,ab limd T xF , rx s, T xF, fx  e 0.   23   for a member TxF, axb  in  . Now, we  Then, define a Functio n spac e  , w.r.t g , as limdT xFax ,  b,, TxF  fxe 0.  TxFaxb, x k ,   13    1  that is M  is transitive.  Tx,,Faxb TxFaxb    11 So we have a partition on  by the equivalence relation M  . with a matric d as 1 Now we show that TxFx  , 1  always exists and is dT xH,ax b,, TxFa  x b a graph of a distribution function. Clearly for any n111111nnmmm 1  (C.2) xm  ,1 mp  then  sup y 1   x Txxx 12,,,pp ,0 xxx 12 ,,, ,0 and  where y is from T 1 ,,,  ,0   ,, ,  ,0. Also T is 0, y Tx, H a x b T xF , a x b     nnnmmm1111111  one to one and onto function and each of its variable the and nm11, are particular values of  . It is easy to  1 verify d follows metric conditions on  . We denote function is monotone implies T monotonically non decreasing for each of its variable and and right con- T for Tx, Faxb  . F ,,,ab  tinuous for each of its coordinates. Next we define a relation M  on  . Th at is, let Then we pr ove the following equation,    Xnn ,1 , Ynn ,1 and Znn ,1  are t hree se- TxFxTTxFx  ,,11      xand , . (C.4) quence of random variables each forms i.i.d and    X11 F , YF12 and Z13 F . Now let F1,ab , , Let Xn , 1 be i.i.d.c.m.r.v’s and    n  and F2,lm ,  . We say that F1,ab , and F2,lm , are M   g  XX12,,, X  be a Borel measurable function. Now M  related (denoted by FF1,ab , 2, lm , ) if for i 1, define    limd T xF , ax b,TxF , lx  m0. (C.3) UgXii ii11,,, X   12 X 12  

  Then below we show that the relation M is an equi- then Unn , 1 is also a sequence of i.i.d.c.m.r.v’s. Let       valence relation on  . Let F1,ab ,  . Since PX 11x  F x and PU 11 x H x . Then

x,,,,PgUU U  x d xH , x. xF,,. x T xF  fx  e (C.6)  12  1  tkt11  which is same as, So that F satisfies stability property. tfe1,, Now, we have a partition on  and we know that xP, g g X , X ,, X ,g X  , X  ,, X  ,,    12 1 2 2 every equi valent class contains a stable m ember. let  0 be one of the equivalent class introduced by the relation gX,, X, X  x d xHx ,.  11  12     1  M and F  is a stable memb er in  . That is, 1,ab , 0 0 But we know that   TxFax ,11  b  TxFx,1, xand. xH,, xxPU  x  11   Then for all F2,rs , 0 ,

x,,,,PgX X X  x dxF, xTxF,,rxs   TxFx,as. p  (C.7) 12 1  21   1

Hence, Since, F1,ab , , F2,rs , 0 implies,   xP, g g X , X ,, X , g X  , X ,, X ,,lim d T xF, 12 ax b,, T xF rx s 0.    12 1 2 2  

But g XX,,, Xxd dxFx, . 11  12      1   k T xFaxb,,11   Tx Fx1 x But x,,,,PgX X X   x d xFx ,.  12    1 So by property Q and values of . The above two equation implies, dxFx ,11ddxFxx  ,  and  , . (C.5) limdT xF , rx s,T xF ,x  0. Now we can show e xistence of a special property in   2  11 the function space. That is, there exist a specia l member, Therefore, we call this member as stable member, in each equivalent class. TxFrxs,,21  TxFx 1,   Given that in every equ ivalent class, there exist at le ast as  p andx . one F such that tab2 ,, Hence, in the function space, every equivalent class TxFaxb,, w xFx . t2t1 contains at least a stable member and all other members of that cla ss wi ll converge to that stable member. It is which implies easy to prove that the limiting memb er is unique in every TxFaxb, w xFx ,. class, if it exist. This complete the proof.  t2  t1 Remark C.2 In Equation (C.6), we introduce a for every k 1, 2,. By using Equation (C.5) we get property called mu ltivariate stability property for some of the members in  . This is a multivariate extension TT xF, a x b w xF , x .   tt21   of stability property in [32]. Below we give an exact definitions and some examples of this property. That is, Definition C.2 Let T  be a given one to one func- w 1  TxFaxb,,  TxFx . tions. We say that F1,ab,  satisfies stability property tt21  if it is a constant sequence in  . In other words, Putting ToF F we get F  satisfies stability property if for each  ,  t2  1,ab ,  w 1  k x,,Faxb TxFx . dT xFax,,,011 b TxF 1 x, x  .  t1       then by using multivariate version of Khintchine’s That is for each  , theorem, there exist sequence f  0 and e such   TxFaxb,,   TxFx , x k . that  11  1

1  TxFx,,xFfxe    x. A member F1,ab ,  is called a stable member if it tt11  is a constant sequence in  . Using Equation (C.1) the  Here we need the assumption of weak convergence. stability property can be rewri tten for sequence X n in Which implies terms of the function g as follows.

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