Let $Q_p$ be a class of probability distributions on non-negative reals parametrized by $p$, so that $$ Q_p([0,\infty)) = 1. $$ I wonder which known classes of distributions are closed under taking the maximum and, i.e. if $X_1\sim Q_{p_1}$ and $X_2\sim Q_{p_2}$ are independent then $\max(X_1,X_2)\sim Q_{p_3}$.
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Those would be the extreme value distributions. There are three of them, as they are usually presented, corresponding to three sets of conditions on the underlying distribution for which the limiting distribution of the maximum is being found. They are closed under finding the maximum, which is what you want. More-or-less copying from an old version of Methods of Statistical Analysis for Reliability and Life Data (Mann, Schafer, Singpurwalla), Type I: $F_{X(n)}(x) = \exp\left\{-\exp \left[-\frac{x-\gamma}{\alpha}\right] \right\},\space -\infty < x < \infty, \space \alpha > 0$ Type II: $F_{X(n)}(x) = \exp\left\{-\left(\frac{x-\gamma}{\alpha}\right)^{-\beta}\right\}, \space x \geq \gamma, \space \alpha,\beta > 0$ Type III: $F_{X(n)}(x) = \exp\left\{-\left[-\left(\frac{x-\gamma}{\alpha}\right)^\beta\right]\right\}. \space x \leq \gamma, \space \alpha,\beta > 0$ Edit: Read the comments, which extend this answer to make a greatly improved and more complete answer to this question! |
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It seems to me that proposing extreme value distributions really answers a different question. I will demonstrate that by addressing this question directly and showing it leads to distributions that are not among the extreme value types. Let's consider this from first principles. It is immediate, from the axioms of probability and definition of the CDF, that the distribution of the maximum of two independent random variables with CDFs $F_1$ and $F_2$ has $F_1 F_2$ for its CDF. Suppose a class of distributions $\Omega = \{F_{\theta}\}$ exists that is closed under pairwise maximum; that is, $$F_\theta \in \Omega, \ F_\phi \in \Omega \text{ implies } F_\theta F_\phi \in \Omega.$$ It is convenient to take logarithms, extending (as in Rudin's advanced analysis texts) the real numbers to include $-\infty$ as the log of $0$. Logs of CDFs of random variables essentially supported on $[0,\infty)$ are (i) mononotically nonincreasing, (ii) equal to $-\infty$ on $(-\infty,0)$, (iii) have right limits of $0$, and (iv) are cadlag. From this point of view, $\Omega$ must be a convex subset of a cone within the space of cadlag functions on $\mathbb{R}$. For it to be finitely parameterized, that cone must generate a finite-dimensional vector subspace. That still leaves a lot of possibilities. Some of these possibilities are well known. Consider, for example, the CDF of a uniform variable on $[0,1]$. Its CDF equals $0$ on $(-\infty,0]$, $x$ when $0 \le x \le 1$, and $1$ on $[1,\infty)$. The cone it generates is the set of CDFs of the form $$F_\theta(x) = \exp(\theta \log(x)) = x^\theta,\quad 0 \lt x \lt 1$$ parameterized by $\theta \gt 0$. Clearly the maximum of any two independent random variables with distributions in this family has a distribution also in this family (their parameters simply add). We may, if we wish, restrict to a convex subset of the form $\{F_\theta | \theta \ge \theta_0\}$ and still have a maximum-closed family. Notice, please, that no member of this family is an extreme-value distribution. This formulation includes discrete distributions (which obviously are not among the three types of extreme value distributions). For instance, consider the distributions supported on the natural numbers $0, 1, 2, \ldots, k, \ldots$ for which the probabilities are given by $${\Pr}_\theta(k) = \theta^{1/(k+1)} - \theta^{1/k}$$ (taking $\theta^{1/k}=0$ when $k=0$), parameterized by $0 \lt \theta \lt 1$. By construction, the CDF $F_\theta(k) = \theta^{1/(k+1)}$, whence it follows $$F_\theta(k) F_\phi(k) = \theta^{1/(k+1)}\phi^{1/(k+1)} = (\theta\phi)^{1/(k+1)},$$ and because the assumptions imply $0 \lt \theta\phi \lt 1$, this shows the family is closed under pairwise maxima. I hope that this analysis and these two examples show that, contrary to an opinion expressed in a comment, the approach of starting with a finite number of well-chosen CDFs and closing them with respect to the pairwise maximum (that is, forming their cones in an appropriate related vector space) not only is constructive but yields interesting and potentially useful classes of distributions. |
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jbowman beat me to the answer. A explanation for why they work is that Gnedenko's Theorem states that if $X_1,...,X_n$ is a sequence of n independent identically distributed random variables $M_n=\max(X_1, X_2,...,X_n)$ under certain conditions on the tail of the distribution converges to 1 of the three type that jbowman listed in his answer. Now since any type I, type II or type III distribtion can be expressed as the limit of the max of an iid sequence then if $G_1$ is say type I and is the limit distribution of $M_n=\max(X_1, X_2,...,X_n)$ as n tends to infinity and $G_2$ is also type I and is the limit of $N_n=\max(Y_1,Y_2,...,Y_n)$ then say $V_n=\max(M_n,N_n)$ and $G_3$ is the distribution of the limit as n approaches infinity for Vn then $G_3$ will be type I and be the distribution for the maximum of a rv with distribution $G_1$ and another with distribution $G_2$ and hence type I is closed under maximization. The same argument works for type II and type III. |
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