Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 42 / 2

Let $M_n={\max }_{1\le j\le n}{W}_j$ for independent and identically distributed random variables with distribution function $F$. Then $P(M_n\le u)=F(u{)}^n$. Membership in the maximum domain of attraction of a nondegenerate distribution function $G$ means that some deterministic $a_n>0$ and $b_n\in \mathbb{R}$ satisfyor equivalently $F(a_{n}x+b_n{)}^n\to G(x)$ at every continuity point of $G$.

Under the stated convergence, put $Z_n=(M_n-b_n)/a_n$, $r_n={\alpha }_n/a_n\to a>0$, and $s_n=({\beta }_n-b_n)/a_n\to b$. The Slutsky theorem says that combining a sequence with convergence in distribution with sequences converging in probability to constants preserves the corresponding sum, product and ratio limits, provided a limiting denominator is nonzero. Applying it here givesThe limiting distribution function is $P((Z-b)/a\le x)=G(ax+b)$, continuous for every real $x$ because $G$ is continuous and $a>0$. Consequently,

For large $t$, write the survival function asA positive function $\ell$ is slowly varying if $\ell (tu)/\ell (t)\to 1$ for every fixed $u>0$. Heresince $\log (tu)=\log t+\log u$ and $\log \log (tu)-\log \log t\to 0$. Thus $\overline{F}$ has regular variation of index $-2$.

The relevant extreme-value criterion is that an infinite right endpoint and a survival function regularly varying with index $-r$, $r>0$, give attraction to the standard Fréchet distribution of shape $r$. Its distribution function is $G_r(x)=e^{-x^{-r}}$ for $x>0$ and zero for $x\le 0$. Here is a direct verification, so no additional theorem is needed to establish the criterion in this example. Choose $a_n\to \infty$ satisfying $n\overline{F}(a_n)\to 1$ and set $b_n=0$. For $x>0$,If $u_n\to 0$ and $n{u}_n\to c$, then $n\log (1-u_n)=-n{u}_n+O(n{u}_n^2)\to -c$. Applying this with $u_n=\overline{F}(a_{n}x)$ gives $F(a_{n}x{)}^n\to e^{-x^{-2}}$. For $x\le 0$, the distribution function is zero because the observations are supported on $[10,\infty )$. ThereforeFor example, elementary asymptotic normalizers arefor sufficiently large $n$, with arbitrary positive definitions at smaller indices. Indeed $\log a_n\sim \frac{1}{2}\log n$ and $\log \log a_n\sim \log \log n$, so $a_n^2\log a_n\log \log a_n\sim Cn$. The precise valid choice of $C$ at the lower endpoint does not change the limiting shape.

The Gumbel distribution for maxima has distribution function $G(x)=\exp (-e^{-x})$ on the whole real line. For $n\ge 2$ setBoth the centering and scale are elementary, and the scale is positive. For any fixed real $x$, the normalized threshold ${\beta }_n+{\alpha }_{n}x$ is positive for all sufficiently large $n$. Writing $L=\log n$, a Taylor expansion givesConsequently,The logarithm calculation in part (i) now givesThus $F$ belongs to the maximum domain of attraction of the standard Gumbel distribution. The scale also has the Gumbel auxiliary function interpretation: the reciprocal hazard function is $2\sqrt{t}$, whose derivative $1/\sqrt{t}$ tends to zero at the infinite right endpoint. This agrees with the Von Mises conditions for extreme values, and at $t={\beta }_n$ gives exactly ${\alpha }_n=2\log n$.