Gumbel Distribution: Generalizations and Applications

Transcription

1 CHAPTER 3 Gumbel Distribution: Generalizations and Applications 31 Introduction Extreme Value Theory is widely used by many researchers in applied sciences when faced with modeling extreme values of certain phenomena, such as ocean wave modeling, wind engineering, thermodynamics of earthquakes, risk assessment on financial markets etc The first results were developed considering independent observations but, more recently, models for extreme values have been constructed under the more realistic assumption of temporal dependence The importance of the Gumbel distribution in practice is due to its extreme value behavior It has been applied either as the parent distribution or as an asymptotic approximation, to describe extreme wind speeds, sea wave heights, floods, rainfall, age at death, minimum temperature, rainfall during droughts, electrical strength of materials, air pollution problems, geological problems, naval engineering etc Recently various authors have introduced several q-type distributions such as q-exponential, 49

2 q-weibull, q-logistic and various pathway models in the context of information theory, statistical mechanics, reliability modeling etc The q-exponential distribution can be viewed as a stretched model (Beck (2006), Beck and Cohen (2003)) for exponential distribution so that the exponential form can be reached as q 1 A theorem which is useful in stressstrength analysis is established by Jose and Naik (2009) They apply the model to data on remission times of cancer patients and compare it with the Weibull and the q-weibull models, which confirmed that the q-weibull model is a better fit Mathai (2005) introduced a pathway model connecting matrix variate gamma and normal densities A minification processes of the first order is given by X n = k min(x n, ε n ), n 1, (311) where {ε n, n 1} where k > 1 is a constant, and {ε n } is an innovation process of independently and identically distributed random variables chosen to ensure that {X n } is a stationary markov process with marginal distribution function F X (x) Because of the structure of (311), the process {X n } is called a minification processes (See Lewis and Mckenzie (1991)) Alice and Jose (2004) developed bivariate minification processes with semi Pareto marginal distribution Jose et al (2011) introduced a minification processes with Marshall Olkin bivariate Weibull distribution Minification processes with discrete marginals is discussed by Kalamkar (1995) Marshall and Olkin (1997) introduced a new family of survival functions which is constructed by adding a new parameter to an existing distribution The introduction of a new parameter will result in flexibility in the distribution For a random variable with a distribution function F (x) and survival function F (x), we can obtain a new family of distribution functions called univariate Marshall-Olkin family having cumulative distribution function G(x) given by G(x) = F (x) ; < x < ; 0 < α < α + (1 α)f (x) 50

3 Then the corresponding survival function is Ḡ(x) = α F (x) 1 (1 α) F ; < x < ; 0 < α < (312) (x) This new family involves an additional parameter α Jose and Alice (2001), Ghitany et al (2005) conducted detailed studies of Marshall-Olkin Weibull distribution Marshall-Olkin q-weibull Distribution and Max-Min Processes is discussed by Jose et al (2010) The symmetric q-laplace distribution and its applications in financial modeling is discussed by Jose et al (2008) In this chapter we deal with generalized Gumbel Maximum distribution in section 32 and generalized Gumbel Minimum distribution in section 33 q- Gumbel distribution is discussed in section 34 In section 35, we consider generalized Marshall-Olkin q-gumbel distribution Conclusions are given in section Generalized Gumbel distribution for maximum Now we can consider the generalized Gumbel distribution for maximum (GGUMX) The simplest form of the pdf of Gumbel distribution is f(x) = e x e e x, x R and the corresponding cumulative distribution function is F (x) = e e x The hazard rate function of the generalized Gumbel is h(x) = e x e e x 1 51

4 Then the survival function of the Marshall Olkin generalized Gumbel Maximum distribution(mo- GGUMX) is The corresponding pdf is Ḡ(x) = α(1 e e x), x R, α > 0 α + (1 α)e e x g(x) = αe x e e x (α + (1 α)e e x ) 2 The hazard rate function of the Marshall Olkin generalized Gumbel Maximum distribution is h g (x) = e x e e x (α + (1 α)e e x )(1 e e x ) Theorem 321 Marshall-Olkin generalized Gumbel maximum distribution is geometric extreme stable Proof Let X 1, X 2, be a sequence of independent identically distributed random variables Suppose N is independent of the X i s with a geometric(p) distribution and let U N = min(x 1, X 2,, X N ) and V N = max(x 1, X 2,, X N ) Then we obtain that Ḡ(x) = P (U N > x) = n=1 F n (x)(1 p) n p = p F (x) 1 (1 p) F (x) Suppose that F is the survival function of Marshall-Olkin generalised Gumbel maximum family Then Ḡ(x) = αp(1 e e x ) αp + (1 αp)e e x and it follows that U N is geometric minimum stable Let V N = max(x 1, X 2,, X N ) Then H(x) = P (V N < x) = F n (x)(1 p) n p = n=1 pf (x) 1 (1 p)f (x), so that H(x) = F (x) p+(1 p) F (x), x R If we suppose that F is the survival function of 52

5 MO-GGUMX distribution, then it follows that H(x) = α(1 e e x) α + (p α)e e x Hence V N is geometric maximum stable Thus the family of MO-GGUMX family of distributions is geometric extreme stable Theorem 322 Let {X i, i 1} be a sequence of independent and identically distributed random variables with common survival function F (x) and N be a geometric random variable with parameter p and P (N = n) = pq n ; n = 1, 2, ; 0 < p < 1, q = 1 p which is independent of {X i } for all i 1 Let U N = min 1 i N X i Then {U N } is distributed as MO-GGUMX if and only if {X i } is distributed as GGUMX Proof Consider H(x) = P (UN generalized Gumbel maximum family Then > x) Suppose that F is the survival function of the H(x) = p(1 e e x) p + (1 p)e e x which is the survival function of MO-GGUMX This proves the sufficiency part of the theorem Conversely suppose H(x) = Then we get p(1 e e x) p + (1 p)e e x F (x) = (1 e e x ) which is the survival function of generalized Gumbel maximum 321 An AR(1) model with MO-GGUMX marginal distribution Theorem 323 Consider an AR(1) structure given by ε n X n = min(x n, ε n ) with probability p with probability 1 p (321) 53

6 where 0 p 1, {ε n } is a sequence of independent and identically distributed random variables independent of {X n, X n 2, } Then {X n } is a stationary Markovian AR(1) process with MO-GGUMX marginals if and only if {ε n } is distributed as GGUMX distribution Proof From (321) it follows that F Xn (x) = p F εn (x) + (1 p) F Xn (x) F εn (x) (322) Under stationary equilibrium, this gives F X (x) = p F ε (x)/[1 (1 p) F ε (x)] If we take F ε (x) = 1 e e x then we obtain that F X is the survival function of MO-GGUMX Conversely, if we take F Xn (x) = p(1 e e x) p + (1 p)e e x then it is easy to show that F εn (x) is distributed as generalized Gumbel maximum and the process is stationary In order to establish stationarity we proceed as follows Assume X d n = MO-GGUMX and ε d n = GGUMX Then from (322) it follows that X n is distributed as MO-GGUMX Theorem 324 Consider an autoregressive minification process X n of order k with structure ε n with probability p 0 X n = min(x n, ε n ) with probability p 1 min(x n 2, ε n ) with probability p 2 min(x n k, ε n ) with probability p k (323) where 0 < p i < 1, p 1 + p p k = 1 p 0 Then {X n } has stationary marginal distribution as MO-GGUMX if and only if {ε n } is distributed as GGUMX and vice 54

7 versa 33 Generalised Gumbel distribution for minimum In the case of generalized Gumbel minimum (GGUMIN), the pdf is f(x) = e x e ex, x R and the corresponding cumulative distribution function is F (x) = 1 e ex The hazard rate function is h(x) = e x Survival function of the Marshall Olkin generalized Gumbel minimum distribution is and the corresponding pdf is Its hazard rate function is Ḡ(x) = αe ex 1 (1 α)e ex x R, α > 0 g(x) = h g (x) = αe x e ex (1 (1 α)e ex ) 2 e x 1 (1 α)e ex Theorem 331 Marshall-Olkin generalized Gumbel minimum (MO-GGUMIN)distribution is geometric extreme stable Proof is similar to the case of generalized Gumbel maximum Theorem 332 Let {X i, i 1} be a sequence of independent and identically distributed random variables with common survival function F (x) and N be a geometric 55

8 random variable with parameter p and P (N = n) = pq n ; n = 1, 2, ; 0 < p < 1, q = 1 p which is independent of {X i } for all i 1 Let U N = min 1 i N X i Then {U N } is distributed as MO-GGUMIN if and only if {X i } is distributed as GGUMIN Proof is similar to the case of generalized Gumbel maximum 331 An AR(1) model with MO-GGUMIN marginal distribution Theorem 333 Consider an AR(1) structure given by ε n X n = min(x n, ε n ) with probability p with probability 1 p (331) where 0 p 1, {ε n } is a sequence of independent and identically distributed random variables independent of {X n, X n 2, } Then {X n } is a stationary Markovian AR(1) process with MO-GGUMIN marginals if and only if {ε n } is distributed as GGU- MIN distribution Proof is similar to the case of generalised Gumbel maximum Theorem 334 Consider an autoregressive minification process X n of order k with structure ε n with probability p 0 X n = min(x n, ε n ) with probability p 1 min(x n 2, ε n ) with probability p 2 min(x n k, ε n ) with probability p k (332) where 0 < p i < 1, p 1 + p p k = 1 p 0 Then {X n } has stationary marginal distribution as MO-GGUMIN if and only if {ε n } is distributed as GGUMIN 34 Generalized q-gumbel distribution In this section we are studying various properties of generalized q-gumbel distribution under various conditions Graphs of density function and hazard rate function under various 56

9 conditions are also drawn Distribution function of q-gumbel maximum is as follows Case (i) when q > 1 and x > 0 we have, [1+(q)x] q F (x) = ke where and k = e e 1 [1+(q)x] q F (x) = 1 ke The corresponding pdf is [1+(q)x] q f(x) = ke [1 + (q 1)x] q q and the hazard rate function is h(x) = ke [1+(q)x] q [1 + (q 1)x] q q 1 ke [1+(q)x] q Graphs of the pdf and hazard rate function of generalized q-gumbel maximum distribution under case (i) when q > 1 and x > 0 are given in Fig 31a and in Fig 31b Case (ii) when q < 1 and 0 x 1, we have, F (x) = ke [1 ()x] 1 and F (x) = 1 ke [1 ()x] 1 57

10 Figure 31: Graphs of the pdf and hazard rate function of generalized q-gumbel maximum distribution under case (i) when q > 1 and x > 0 are given in Fig 31a and in Fig 31b Fig 31a pdf of case 1 when q = 15 Fig 31b hazard function of case1 when q = 5 The corresponding pdf is f(x) = ke [1 ()x] 1 [1 (1 q)x] q Its hazard rate function is where h(x) = ke [1 ()x] 1 [1 (1 q)x] q 1 ke [1 ()x] 1 k = e e 1 Graphs of the pdf and hazard rate function of generalized q-gumbel maximum distribution under case (ii) when q < 1 and 0 x 1, are given in Fig 32a and in Fig 32b Distribution function of q-gumbel minimum is obtained as follows 58

11 Figure 32: Graphs of the pdf and hazard rate function of generalized q- Gumbel maximum distribution under case (ii) when q < 1 and 0 x 1, are given in Fig 32a and in Fig 32b Fig 32apdf of case2 when q = 088 Fig 32bhazard function of case2 when q = 615 Case(i) when q > 1 and x > 0, we have, F (x) = 1 ke [1+(q)x] 1 q, corresponding pdf is f(x) = ke [1+(q)x] 1 q [1 + (q 1)x] q q Hazard rate function is where (Since h(x) = [1 + (q 1)x] 1 q 1 k 6734 {e z z 2 dz} = e E i (1, 1)) 59

12 Figure 33: Graphs of the pdf and hazard rate function of q Gumbel minimum distribution under case(i) when q > 1 and x > 0 are given in Fig 33a and in Fig 33b Fig 33apdf of case1 when q = 5 Fig 33bhazard function of case1 when q = 250 Graphs of the pdf and hazard rate function of q-gumbel minimum distribution under case(i) when q > 1 and x > 0 are given in Fig 33a and in Fig 33b Case(ii) when q < 1 and 0 < x < 1, we have, F [1 ()x] (x) = ke and [1 ()x]( F (x) = 1 ke ) The corresponding pdf is [1 ()x] f(x) = ke [1 (1 q)x] q Hazard rate function is h(x) = [1 (1 q)x] q, 60

13 Figure 34: Graphs of the pdf and hazard rate function of q-gumbel minimum distribution under case(ii)when q < 1 and 0 < x < 1, are given Fig 34a and in Fig 34b Fig 34a pdf of case2 when q = 05 Fig 34b hazard function of case2 when q = 002 where k = 6734 Graphs of the pdf and hazard rate function of q-gumbel minimum distribution under case(ii)when q < 1 and 0 < x < 1, are given Fig 34a and in Fig 34b 35 Marshall-Olkin generalized q-gumbel distribution When we apply the Marshall-Olkin method on generalized q-gumbel distributions we get generalized Marshall Olkin q-gumbel distribution as follows The survival function of the Marshall Olkin generalized q-gumbel maximum distribution; when q > 1 and x > 0 is Ḡ(x) = [1+(q)x] q α(1 ke ) 1 (1 α)(1 ke [1+(q)x]( q ) ) where k = e e 1 61

14 Figure 35: Graphs of the pdf and hazard rate function of Marshall Olkin generalized q-gumbel maximum distribution when q > 1 and x > 0are given in Fig 35a and in Fig 35b Fig 35a pdf of MO-GqGUMX1 when α = 1, q = 10 Fig 35b hazard function of MO-GqGUMX1 when α = 19, q = 65 and the corresponding pdf is g(x) = q αk[1 + (q 1)x] q e [1+(q)x] q (1 (1 α)(1 ke [1+(q)x] q )) 2 Its hazard rate is h g (x) = k[1 + (q 1)x] q q e [1+(q)x] q (1 (1 α)(1 ke [1+(q)x] q [1+(q)x]( q ))(1 ke ) ) Graphs of the pdf and hazard rate function of Marshall Olkin generalized q-gumbel maximum distribution(mo-gqgumx1) when q > 1 and x > 0are given in Fig 35a and in Fig 35b The survival function of the Marshall Olkin generalized q-gumbel maximum distribu- 62

15 tion; when q < 1 and 0 x 1 is where Ḡ(x) = and the corresponding pdf is α(1 ke [1 ()x] 1 ) 1 (1 α)(1 ke [1 ()x] 1 ) k = e e 1, q g(x) = α(ke [1+(q)x] [1 + (q 1)x] q q ) (1 (1 α)(1 ke [1 ()x] 1 )) 2 Its hazard rate is h g (x) = ke [1 ()x] 1 [1 (1 q)x] q (1 ke [1 ()x] 1 )(1 (1 α)1 ke [1 ()x] 1 ) Graphs of the pdf and hazard rate function of Marshall Olkin generalized q-gumbel maximum distribution(mo-gqgumx2) when q < 1 and 0 x 1 are given Fig 36a and in Fig 36b The survival function of the Marshall Olkin generalized q-gumbel minimum distribution; when q > 1 and x > 0 is Ḡ(x) = αke [1+(q)x] 1 q 1 (1 α)ke [1+(q)x] 1 q, and the corresponding pdf is 1 q g(x) = αke [1+(q)x] [1 + (q 1)x] q q (1 (1 α)ke [1+(q)x] 1 q ) 2 63

16 Figure 36: Graphs of the pdf and hazard rate function of Marshall Olkin generalized q-gumbel maximum distribution when q < 1 and 0 x 1 are given Fig 36a and in Fig 36b Fig 36a pdf of MO-GqGUMX2 when α = 3, q = 99 Fig 36b hazard function of MO-GqGUMX2 when α = 5, q = 5 Its hazard rate is h g (x) = ke [1+(q)x] 1 q [1 + (q 1)x] q q (ke [1+(q)x] 1 q )(1 (1 α)ke [1+(q)x] 1 q ), where k = 6734 Graphs of the pdf and hazard rate function of Marshall Olkin generalized q-gumbel minimum distribution (MO-GqGUMIN1) when q > 1 and x > 0 are given in Fig 37a and in Fig 37b The survival function of the Marshall Olkin generalized q-gumbel minimum distribution; when q < 1 and 0 x 1 is Ḡ(x) = [1 ()x] α(1 ke ) [1 ()x]( 1 (1 α)(1 ke ) ), 64

17 Figure 37: Graphs of the pdf and hazard rate function of Marshall Olkin generalized q-gumbel minimum distribution when q > 1 and x > 0 are given in Fig 37a and in Fig 37b Fig 37a pdf of MO-GqGUMIN1 when α = 2, q = 5 Fig 37b hazard function of MO-GqGUMIN1 when α = 06, q = 18 and the corresponding pdf is Its hazard rate is g(x) = αke [1 ()x] (1 (1 α)(1 ke [1 (1 q)x] q [1 ()x] )) 2 h g (x) = [1 ()x] ke [1 (1 q)x] q (1 ke [1 ()x] )(1 (1 α)(1 ke [1 ()x] )), where k = 6734 Graphs of the pdf and hazard rate function of the Marshall Olkin generalized q-gumbel minimum distribution (MO-GqGUMIN2) when q < 1 and 0 x 1 are given in Fig 38a and in Fig 38b 65

18 Figure 38: Graphs of the pdf and hazard rate function of the Marshall Olkin generalized q-gumbel minimum distribution when q < 1 and 0 x 1 are given in Fig 38a and in Fig 38b Fig 38apdfof MO-GqGUMIN2 when α = 99, q = 99 Fig 38bhazardfunctionof MOGqGUMIN2 when α = 15,q = 5 Theorem 351 Let {X i, i 1} be a sequence of independent and identically distributed random variables with common survival function F (x) and N be a geometric random variable with parameter p and P (N = n) = pq n ; n = 1, 2, ; 0 < p < 1, q = 1 p which is independent of {X i } for all i 1 Let U N = min 1 i N X i Then {U N } is distributed as MO-GqGMX if and only if {X i } is distributed as GqGUMX Proof is similar to the case of Generalized Gumbel maximum 351 An AR (1) model with MO-GqGUMX marginal distribution Theorem 352 Consider an AR (1) structure given by { εn X n = with probability p min(x n, ε n) with probability 1 p (351) where 0 p 1, {ε n} is a sequence of independent and identically distributed random variables independent of {X n, X n 2, } Then {X n} is a stationary Markovian AR (1) process with MO-GqGUMX marginals if and only if {ε n} is distributed as GqGUMX distribution 66

19 Figure 39: Simulated Sample Path of the Marshall Olkin q-gumbel minification processes for various values of the parameters are given in Fig 39a and in Fig 39b Fig 39aSample path when α = 15, q = 5,n=1000 Fig 39bSample path when α = 3, q = 3,n=1000 Proof is similar to the case of generalized Gumbel maximum Simulated Sample Path of the Marshall Olkin q-gumbel minification processes for various values of the parameters are given in Fig 39a and in Fig 39b Theorem 353 Consider an autoregressive minification process X n of order k with structure ε n with probability p 0 X n = min(x n, ε n ) with probability p 1 min(x n 2, ε n ) with probability p 2 min(x n k, ε n ) with probability p k (352) where 0 < p i < 1, p 1 + p p k = 1 p 0 Then {X n } has stationary marginal distribution as MO-GqGUMX if and only if {ε n } is distributed as GqGUMX 67

20 352 The max-min AR(1) processes Next we introduce a new model called the max-min process given by Jose et al (2008) which incorporates both maximum and minimum values of the process This has wide applications in atmospheric and oceanographic studies The structure is given as follows Theorem 354 Consider an AR(1) structure given by max(x n, ε n ) with probability p X n = min(x n, ε n ), with probability 1 p (353) X n, with probability 1 p 1 p 2 with the condition that 0 < p 1, p 2, p 3 < 1, p 2 < p 1 and 0 < p 1 +p 2 +p 3 < 1, where {ε n } is a sequence of iid random variables independently distributed of X n Then X n is a stationary Markovian AR(1) max-min process with stationary marginal distribution F X (x) if and only if {ε n } follows Marshall-Olkin distribution Proof From the given structure it follows that P (X n > x) = p 1 P (max(x n, ε n ) > x)+p 2 P (min(x n, ε n ) > x)+(1 p 1 p 2 )P (X n > x) on simplification we get P (X n > x) = p 1 {1 (1 F Xn (x))(1 F εn (x))} + p 2 FXn (x) F εn (x) + (1 p 1 p 2 ) F Xn (x) Under stationary equilibrium, F ε (x) = = p 2 FX (x) p 1 + (p 2 p 1 ) F X (x) p F X (x) 1 (1 p ) F X (x), where p = p 2 p 1 This has the same functional form of Marshall-Olkin survival function The converse can 68

21 be proved by mathematical induction, assuming that F Xn (x) = F X (x) 353 The max-min Process with Gumbel marginal distribution Now consider the generalized Gumbel maximum distribution with cumulative distribution function F (x) = e e x To obtain the generalized Gumbel max-min process, consider the max-min AR(1) structure and substitute the cdf we get, F ε (x) = p (1 e e x p + (1 p )(1 e e x ), which is the survival function of the Marshall-Olkin generalized Gumbel maximum distribution where p = p 2 p 1, p 2 < p 1 and p 1 + p 2 < The max-min Process with q-gumbel marginal distribution Now consider the Generalized q Gumbel maximum distribution with cumulative distribution function where (1+(q)x)( F (x) = ke q ) k = e e 1 To obtain the Generalized q-gumbel max-min process, consider the max-min AR(1) structure and substituting the cdf we get, F ε (x) = p (1+(q)x)( (1 ke q ) ) 1 (1 p )(1 ke (1+(q)x)( q ) ) q > 1 and x > 0 which is the survival function of the Marshall-Olkin q generalized Gumbel maximum distribution where k = e e and p = p 2 p 1, p 2 < p 1 and p 1 + p 2 < 1 Now consider a more general autoregressive structure which includes maximum, min- 69

22 imum as well as the innovations and the process values Theorem 355 Consider an AR(1) structure given by max(x n, ε n ) with probability p 1 min(x n, ε n ), with probability p 2 X n = ε n with probability p 3 X n, with probability 1 p 1 p 2 p 3 (354) with the condition that 0 < p 1, p 2, p 3 < 1, p 2 < p 1 and 0 < p 1 + p 2 + p 3 < 1, where {ε n } is a sequence of iid random variables independently distributed of X n Then X n is a stationary Markovian AR(1) max-min process with stationary marginal distribution F X (x) if and only if {ε n } follows Marshall-Olkin distribution 36 Conclusions As generalizations of Gumbel distribution, the Marshall-Olkin Gumbel maximum and minimum distribution as well as the Marshall-Olkin q-gumbel distribution are introduced and their properties are studied Minification processes with Marshall-Olkin q-gumbel marginal distribution are also developed and studied The max-min process with Gumbel and q- Gumbel marginal distributions are developed Graphical study of the pdf and hazard rate function of q-gumbel distribution under various cases are also considered References Alice, T, Jose, KK (2004) Bivariate semi Pareto minification processes, Metrika, 59, Beck, C (2006) Stretched exponentials from superstatistics, Physica A, 365, Beck, C, Cohen, ECD (2003) Superstatistics, Physica A, 322, Ghitany, ME, Al-Hussaini, EK, Al-Jarallah, R A (2005) Marshall-Olkin Extended Weibull Distribution and its Application to Censored Data, J Appl Stat, 32,

23 Jose, KK, Alice, T (2001) Marshall-Olkin generalized-weibull Distributions and Applications, STARS Int Journal, 2(1), 1-8 Jose, KK, Naik, SR, Ristic, MM (2010) Marshall-Olkin q-weibull Distribution and Max- Min Processes, Statistical Papers, 51, Jose, KK, Naik, SR, (2009) On the q-weibull Distribution and Its Applications, Communications in Statistics: Theory and Methods, 38(6), Jose, KK, Naik, SR (2008) The symmetric q-laplace distribution and its applications in financial modeling, Journal of System Science and Complexity Jose, KK, Ancy Joseph, Ristic, M M (2011) Marshall Olkin Weibull Distributions and Minification Processes, Statistical Papers, 52, Kalamkar, VA (1995) Minification processes with discrete marginals, J Appl Prob, 32, Lewis, PAW, Mckenzie, ED (1991) Minification Processes and their transformation, J Appl Prob 28, Marshall, AW, Olkin, I (1997) A new method for adding a parameter to a family of distributions with application to the exponential and weibull families, Biometrica, 84(3), Mathai, AM (2005) A pathway to matrix-variate gamma and normal densities, Linear Algebra and Its Applications, 396,