Metrika (2010) 71:117–123 DOI 10.1007/s00184-008-0206-3
On extremes of mixtures of distributions M. Sreehari · S. Ravi
Received: 15 May 2008 / Published online: 11 November 2008 © Springer-Verlag 2008
Abstract In this article we study the max domains of attraction of mixtures of distributions belonging to the max domain of attraction of max stable laws. Keywords Extreme value distributions · Max domains of attraction · Max stable laws · p-max stable laws · Linear normalization · Power normalization · Mixtures 1 Introduction Suppose that F1 , F2 , . . . , Fk are distribution functions (df’s). Set F = p1 F1 + · · · + pk Fk , where pi > 0, p1 +· · ·+ pk = 1. Then F is a df and we denote the left extremity by l(F) = inf {x : F(x) > 0} and the right extremity by r (F) = sup {x : F(x) < 1} . The main problems that we investigate are the following. • If F j is in the max domain of attraction of a general max stable law (definition and properties are discussed in Sect. 2), for each j, 1 ≤ j ≤ k, is F in the max domain of attraction of some max stable law H, and if yes, what is the structure of H ? • If the mixture F is in the max domain of attraction of a max stable law H, what can be said about F j , 1 ≤ j ≤ k?
Research of S. Ravi is supported by DSA Programme of UGC, New Delhi, India. M. Sreehari 6-B, Vrundavan Park, New Sama Road, Vadodara 390008, India e-mail:
[email protected] S. Ravi (B) Department of Studies in Statistics, University of Mysore, Manasagangotri, Mysore 570006, India e-mail:
[email protected]
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These problems are of interest in reliability and statistical analysis concerning mixed populations. Kale and Sebastian (1995) discussed the limit behaviour of the maximum of sample observations from the mixture distribution G = α F1 + (1 − α)F2 , where F1 is in the max domain of attraction of an extreme value distribution of Gumbel type or Fre´chet type and the support of F2 is (−δ, δ). They were investigating nonnormal symmetric distributions with kurtosis 3. AL-Hussaini and El-Adll (2004) also investigated the problems cited above and their results are somewhat ambiguous and partly wrong. Galambos (1975) considered a related problem. Our investigation gives an affirmative answer to the first question above under some assumptions (Sect. 3) while the second question has a negative answer. We give some interesting examples in this connection. In Sect. 4, we consider some generalizations. In Sect. 2, we present necessary definitions and known results. 2 General max stable laws Definition 1 A df H is said to be max-stable if for every positive integer n there exists a strictly monotone continuous transformation f n (x) such that H n ( f n (x)) = H (x) ∀x ∈ R.
(2.1)
Pancheva (1985, 1994) proved that the class of max-stable laws is given by the two parameter family H (x) = exp (− exp (−c.h(x))) , where 0 < c ∈ R and h is a strictly increasing invertible continuous function in S(H ), the support of H. A df H is max-stable if and only if H r is max-stable for r > 0. This class of distributions, henceforth called general max-stable laws, contains both the well known extreme value distributions (Fre´chet, Weibull and Gumbel types) as well as the six p-max stable laws derived by Pancheva (1985). The transformation f n (.) is given by h −1 (h(.) + log n) . Pancheva proved that the class of possible limit distributions of normalized maxima of independent identically distributed random variables coincides with the class of general max-stable laws. Definition 2 A df F is said to belong to the max domain of attraction of general maxstable law H if there exists a sequence of strictly monotone continuous transformations {gn (.)} such that w
F n (gn (x)) → H (x), −1 (g (x)) , with m < n, m n → λ, and where gn (.) is such that gλ (x) = limn→∞ gm n n n n gn (.) considered as a function of λ is solvable, i.e. gλ (x) = t has a unique solution λ = g(x, ¯ t). Sreehari (2008) proved a necessary and sufficient condition for a given df F to belong to the max domain of attraction of H (F ∈ D(H )) . It is noted that a general max-stable distribution does not have gaps in its support. It is easily seen that the product of two non-degenerate general max-stable laws H1 and H2 cannot be max-stable if their supports S(H1 ) and S(H2 ) are disjoint. On the other hand, if S(H1 ) ∩ S(H2 ) is not empty, then H = H1 H2 is max stable in view of Corollary 2 in
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On extremes of mixtures of distributions
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Pancheva (1985) because it is a strictly increasing continuous function on the interval (max (l(H1 ), l(H2 )) , max (r (H1 ), r (H2 ))). Further a df F may belong to the max domain of attraction of two or more general max-stable laws (see, Mohan and Ravi 1992) and the same sequence {gn (x)} may serve as norming sequence for two different df’s F1 and F2 to belong to the max domain of attraction of same or different general max-stable laws. See AL-Hussaini and El-Adll (2004) and Sreehari (2008). 3 Extremes of mixtures of df’s We first discuss a simple result which is of interest in reliability theory. Theorem 3.1 Suppose that G(x) = p.E 0 (x) + (1 − p).F(x) where 0 < p < 1, r (F) > 0, and E 0 (x) =
if x < 0, if 0 ≤ x.
0 1
Then, for some norming functions {gn (x)} , w
G n (gn (x)) → H 1− p (x), if and only if w
F n (gn (x)) → H (x). Proof Note that r (G) = r (F). Also for large n, n.{1 − G(gn (x))} = n.{1 − p.E 0 (gn (x)) − (1 − p)F(gn (x))} = n.{1 − p − (1 − p)F(gn (x))} so that G ∈ D H 1− p iff F ∈ D(H ). Remark 1 If r (F) < 0, then ⎧ ⎨ (1 − p).F(x) G(x) = 1 − p ⎩ 1
if x < r (F), if r (F) ≤ x < 0, if 0 ≤ x. w
Hence G n (gn (x)) < (1 − p)n for x < 0 and = 1 for x > 0, so that G n (gn (x)) → E 0 (x), x ∈ R.
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Theorem 3.2 Let F1 , . . . , Fk be df’s such that w
F jn (gn (x)) → H j (x), 1 ≤ j ≤ k,
(3.1)
where gkn (x) is a strictly monotone continuous function for each n. Let pi > 0, 1 ≤ i ≤ k, pi = 1. Set F(x) = p1 .F1 (x) + · · · + pk .Fk (x). Let S(Hi ) ∩ S(H j ) = φ and i=1 for 1 ≤ i < j ≤ k. Then as n → ∞,
w
F (gn (x)) → H (x) = n
0 k
if x ≤ max1≤i≤k l(Hi ) if x > max1≤i≤k l(Hi ).
p
i i=1 Hi (x)
(3.2)
Proof We have n.{1 − F(gn (x))} = n. 1 −
k
pi .Fi (gn (x))
i=1
=
k
pi .n.{1 − Fi (gn (x))}
i=1
→ − log
k
p
Hi i (x),
(3.3)
i=1
for x > max1≤i≤k l(Hi ). Hence the result follows from the observation that for x ≤ max1≤i≤k l(Hi ), n.{1 − Fi (gn (x))} → ∞ for at least one value of i ≤ k. Remark 2 The above result is essentially the sufficiency part of Theorem 1 in AL-Hussaini and El-Adll (2004). They also claimed the converse of the above result to be true. The following examples demonstrate that the converse of Theorem 3.2 is false in linear normalization setup and non-linear normalization setup. Remark 3 In case r (Hi ) < r (H j ) for some pair (i, j) then the corresponding Hi will become unity in the product term in (3.2). Example 1 Let, for α > 0, F1 (x) =
if x < 1, if 1 ≤ x,
if x < 1, if 1 ≤ x,
0 1 − x −α . 1 +
1 c
sin(log x)
0 1 − x −α . 1 −
1 c
sin(log x)
and F2 (x) = where c > 1 + α1 . Let F(x) =
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1 1 .F1 (x) + .F2 (x) = 2 2
0 1 − x −α
if x < 1, if 1 ≤ x.
pi
On extremes of mixtures of distributions
121
Then F1 and F2 do not belong to the max domain of attraction of any max-stable law under linear norming but w 0 n α1 if x < 0, F (n .x) → α (x) = if 0 ≤ x. exp − x −α Example 2 Suppose p1 = p2 = 21 . Let F j (x) =
0 1−
1 (1+x) j
if x < 0, if 0 ≤ x, j = 1, 2. w
nx 2 (x) Let F(x) = F1 (x)+F . Set gn (x) = 1+x , x>0. Then F n (gn (x)) → 2 where the max-stable law H j is given by 0 if x < 0, H j (x) = if 0 ≤ x. exp −x − j 2
2
1
j=1
H j2 (x),
However, F jn (gn (x)) does not converge weakly to H j (x), j = 1, 2. 1
Remark 4 In the above example, H (x) = (H1 (x).H2 (x)) 2 satisfies (2.1) with the transformation
nx 2 + n 2 x 4 + 4nx 2 (1 + x) f n (x) = 2(1 + x) and hence is a general max-stable law.
∞ pi .Fi (x), pi > 0, Remark 5 Theorem 3.2 goes through even when F(x) = i=1 pi ∞ ∞ i=1 pi = 1 under the additional condition on Hi ’s that i=1 1 − Hi (x) < ∞ for x > sup1≤i 0, which is in the max domain of attraction of a general max-stable law Hy (x) = exp − exp −h y (x) . Suppose further that there exists a sequence {gn (x)} of strictly monotone (in x) continuous functions independent of y such that w
F n (gn (x), y) → Hy (x).
(4.1)
Suppose ∞ F(x, y)dU (y)
F(x) = 0
where U is a probability distribution with S(U ) ⊂ (0, ∞).
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Suppose h y (x) satisfies the condition ∞
exp −h y (x) dU (y) < ∞.
(4.2)
0
Then the following result is proved on the same lines as Theorem 3.2. Theorem 4.1 Under the conditions (4.1) and (4.2),
w
F (gn (x)) → H (x) = n
0 ∞ exp − 0 exp −h y (x) dU (y)
if x ≤ sup y l(Hy (x)), if x > sup y l(Hy (x)).
We omit the details of the proof. The next result uses the ideas of Theorem 3.2 and Remark 5. Let {X n } be a sequence of independent random variables and let N be a positive integer valued random variable independent of {X n } . Suppose the d f Fk of X k is such that for some norming function gn (x), independent of k, w
Fkn (gn (x)) → Hk (x), ∀k.
(4.3)
Denote Hk (x) = exp (− exp (−h k (x))). Let Z be a random variable whose conditional distribution, given N = k, is given by P Z = X j | N = k = αk j , where for each k ≥ 1, holds.
k
j=1 αk j
j = 1, 2, . . . , k,
= 1. Suppose P(N = k) = θk . Then the following
Theorem 4.2 w
G nZ (gn (x)) →
⎧ ⎨0
if x ≤ maxk l(Hk (x)),
⎩e
if x > maxk l(Hk (x)).
k ∞ −(h j (x)) − k=1 θk j=1 αk j e
(4.4)
provided ∞
k=1
where G is the df of Z .
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θk
k
j=1
αk j exp −h j (x) < ∞,
(4.5)
On extremes of mixtures of distributions
123
Proof Note that n. (1 − G z (gn (x))) = n.P (Z > gn (x)) ∞
= n. θk P (Z > gn (x) | N = k) k=1
= n.
∞
θk
k=1
= n.
∞
∞
k=1
=
∞
k=1
θk
αk j P X j > gn (x)
j=1
θk
k=1
→−
k
k
αk j 1 − F j (gn (x))
j=1
θk
k
αk j log H j (x)
j=1 k
αk j exp(−h j (x)),
j=1
for x > maxk l(Hk (x)) since (4.5) holds. Hence (4.4) prevails. References AL-Hussaini EK, El-Adll ME (2004) Asymptotic distribution of normalized maximum under finite mixture models. Stat Probab Lett 70:109–117 Galambos J (1975) Limit laws for mixtures with applications to asymptotic theory of extremes. Probab Theory Relat Fields 32(3):197–207 Kale BK, Sebastian G (1995) On a class of symmetric nonnormal distributions with a kurtosis of three. In: Nagaraja HN et al (eds) Statistical Theory and Applications, in honour of David HA. Springer, New York, pp 55–63 Karlin S, Taylor HM (1975) A first course in stochastic processes. 2nd edn. Academic, New York Mohan NR, Ravi S (1992) Max domains of attraction of univariate and multivariate p-max stable laws. Teor Veroyatnost Primenen 37:709–721 [English transl. Theory Probab Appl 1993, 37(4):632–643] Pancheva EI (1985) Limit theorems for extreme order statistics under nonlinear normalization. In: Stability problems for stochastic models (Uzhgorod, 1984), lecture notes in mathematics, vol 1155. Springer, Berlin, pp 284–309 Pancheva EI (1994) Extreme value limit theory with nonlinear normalization. In: Galambos J et al (eds) Extreme value theory and applications. Kluwer, Dordrecht, pp 305–318 Sreehari M (2008) General max-stable laws (in press)
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