Extreme value statistics for censored data with heavy tails under competing risks

Abstract

This paper addresses the problem of estimating, from randomly censored data subject to competing risks, the extreme value index of the (sub)-distribution function associated to one particular cause, in a heavy-tail framework. Asymptotic normality of the proposed estimator is established. This estimator has the form of an Aalen-Johansen integral and is the first estimator proposed in this context. Estimation of extreme quantiles of the cumulative incidence function is then addressed as a consequence. A small simulation study exhibits the performances for finite samples.

Appendix

This “Appendix” contains various results: some of them are used repeatedly in the proof of the main result (in particular Proposition 4, Lemmas 7, and 10, and to a lesser extent Lemmas 9 and 8), the other ones concern parts of the main proof which are postponed to the “Appendix” for better clarity of the main flow of the proof (Lemmas 11,  12 and 13). All the proofs can be found as a supplementary material file located in the first author’s webpage at http://lmv.math.cnrs.fr/annuaire/julien-worms/article/julien-worms-english.

Definition 1

An ultimately positive function f : $\mathbb {R}^+ \rightarrow \mathbb {R}$ is regularly varying (at infinity) with index $\alpha \in \mathbb {R}$, if

$$\begin{aligned} \lim _{t \rightarrow + \infty } \frac{f(tx)}{f(t)} = x^{\alpha } \quad (\forall x >0). \end{aligned}$$

This is noted $f \in RV_{\alpha }$. If $\alpha =0$, f is said to be slowly varying.

Proposition 4

(See Haan and Ferreira 2006 Proposition B.1.9)

Suppose $f \in RV_{\alpha }$. If $x < 1$ and $\epsilon >0$, then there exists $t_0=t_0(\epsilon )$ such that for every $t\ge t_0$,

$$\begin{aligned} (1-\epsilon ) x^{\alpha +\epsilon }< \frac{f(tx)}{f(t)} < (1+\epsilon ) x^{\alpha -\epsilon } \end{aligned}$$

and if $x \ge 1$ ,

$$\begin{aligned} (1-\epsilon ) x^{\alpha -\epsilon }< \frac{f(tx)}{f(t)} < (1+\epsilon ) x^{\alpha +\epsilon } . \end{aligned}$$

(43)

Lemma 7

Let $x \in \mathbb {R}_+^*$ , $\alpha \in \mathbb {R}_+$, $\beta >-1$, and for a and b real numbers, f and g are two regular varying functions at infinity, with index, respectively, a and b. Then, as $t \rightarrow + \infty$,

• (i) $\displaystyle J_{\beta }(x) = \int _1^{+\infty } \log ^{\beta } (y) \ y^{-x-1} dy = \frac{\varGamma (\beta +1)}{x^{\beta +1}}$.

• (ii) $\displaystyle I_{\alpha ,a,b} = \int _{1}^{+\infty } \log ^{\alpha } (y) \ \frac{f(yt)}{f(t)} \ \frac{dg(yt)}{g(t)} \rightarrow \frac{b \varGamma (\alpha +1)}{(-a-b)^{\alpha +1}}$, if $a+b <0$

• (iii) $\displaystyle J_{a,b} = \int _0^{1} \frac{f(yt)}{f(t)} \ \frac{dg(yt)}{g(t)} \rightarrow \frac{b}{a+b}$, if $a+b > 0$

Lemma 8

For any $\delta >0$, let $C_{\delta }$ denote the function

$$\begin{aligned} C_{\delta }(t)=\int _0^t \frac{dG(v)}{\overline{G}(v)\overline{H}^{\delta }(v)}. \end{aligned}$$

Under condition (1), this function is regularly varying of order $\delta /\gamma$ and $C_{\delta }(t)\sim (\gamma /\gamma _C)/(\delta \overline{H}^{\delta }(t))$, as $t\rightarrow +\,\infty$.

Remark 3

In the lemma above, $C_1$ is the important function C introduced at the beginning of Sect. 5, and thus $C(t)\sim (\gamma /\gamma _C)/\overline{H}(t) = (1-\gamma /\gamma _F)/\overline{H}(t)$, as $t \rightarrow +\, \infty$. Hence, C is regularly varying at infinity with index $1/\gamma$, a property which proves useful several times in the main proofs.

Lemma 9

Let $\psi (\phi _n,u)= \int _u^{+\infty } \phi _n(s) d F^{(k)}(x)$, for $u \ge 0$ and $\phi _n(u)= \frac{1}{\overline{F}^{(k)}(t_n)} \log (u/t_n) \mathbb {I}_{u>t_n}$. Under condition (1), we have

$$\begin{aligned} \psi (\phi _n,u)= & {} \gamma _{n,k}, \hbox { if } u \le t_n \\= & {} \log \left( \frac{u}{t_n} \right) \frac{\overline{F}^{(k)}(u)}{\overline{F}^{(k)}(t_n)} + \gamma _k \left( \frac{u}{t_n} \right) ^{-1/ \gamma _k} + \epsilon _n(u) \left( \frac{u}{t_n} \right) ^{-1/ \gamma _k + \delta }\quad \hbox {if } u > t_n, \end{aligned}$$

where $\epsilon _n(u)$ is a sequence tending to 0 uniformly in u, as $n\rightarrow \infty$, and $\delta$ a positive real number such that $-\frac{1}{\gamma _k} + \delta <0$.

Lemma 10

Recalling that H is a distribution function with infinite right endpoint, we have:

• (i) $\sup _{0\le x<Z^{(n)}} \overline{H}(x)/\overline{H}_n(x) = O_{\mathbb {P}}(1)$

• (ii) for any $a<1/2$,

  $$\begin{aligned} \sqrt{n} \sup _{t \ge 0} \frac{ |\overline{H}_n(t)-\overline{H}(t)|}{(\overline{H}(t))^a} = O_{\mathbb {P}}(1) \quad \hbox {and}\quad \sqrt{n} \sup _{t \ge 0 } \frac{ |\overline{H}_n^{(0)}(t)-\overline{H}^{(0)}(t)|}{(\overline{H}^{(0)}(t))^a} = O_{\mathbb {P}}(1) . \end{aligned}$$

Lemma 11

Under conditions (1) and (2), suppose that $\alpha \ge 0$ and $d \ge 1$ are real numbers. If $\gamma _k < \gamma _C$ and

$$\begin{aligned} X_{i,n}= \frac{ \sqrt{v_n} }{ n^{1+d}} \frac{ \phi _n(Z_i) }{ \overline{G}(Z_i) (\overline{H}^{(0)}(Z_i))^{d+\alpha } } \mathbb {I}_{\xi _i=k}, \end{aligned}$$

then we have $\sum _{i=1}^n X_{i,n}{\mathop {\longrightarrow }\limits ^{\mathbb {P}}} 0$, as n tends to infinity, if $\alpha$ is 0 or sufficiently close to it.

Lemma 12

Suppose that $V_1$ and $W_2$ are independent improper random variables of respective subdistribution functions $H^{(0)}$ and $H^{(1,k)}$, and $Z_3$ is independent of $V_1$ and $W_2$ and has distribution H. Consider h, $\underline{h}$, ${\mathcal {H}}$ and $\underline{\mathcal {H}}$ the functions defined in (30) and (39).

• (i) For any $d\ge 1$, there exist some positive constants c and $c'$ such that

  $$\begin{aligned} \mathbb {E}\,(\, |{\mathcal {H}}^d(V_1,W_2)|\,)\le & {} c \, \mathbb {E}\,(\, h^d(V_1,W_2)\,) \quad \hbox {and}\quad \mathbb {E}\,(\, |\underline{\mathcal {H}}^d(Z_3,V_1,W_2)|\,) \\\le & {} c' \, \mathbb {E}\,(\, \underline{h}^d(Z_3,V_1,W_2) \,) . \end{aligned}$$

• (ii) For any $d\in ]1,1+(1+2\gamma _k/\gamma _C)^{-1}[$, we have

  $$\begin{aligned} \textstyle \mathbb {E}\,(\, h^d(V_1,W_2)\,) = O\left( (\overline{F}^{(k)}(t_n)\overline{G}(t_n))^{2(1-d)} \right) . \end{aligned}$$

  In particular, if $\gamma _k < \gamma _C$, then $\mathbb {E}(h^{4/3}(V_1,W_2))$ is of the order of $(\overline{F}^{(k)}(t_n)\overline{G}(t_n))^{-2/3}$ and $\mathbb {E}(h^d(V_1,W_2))$ is finite whenever d is (greater than but) sufficiently close to 4 / 3.

• (iii) For any $d\in ]1,1+(1+3\gamma _k/\gamma _C)^{-1}[$, we have

  $$\begin{aligned} \textstyle \mathbb {E}\,(\, \underline{h}^d(Z_3,V_1,W_2)\,) = O\left( (\overline{F}^{(k)}(t_n)\overline{G}(t_n))^{3(1-d)} \right) . \end{aligned}$$

  In particular, if $\gamma _k < \gamma _C$, then $\mathbb {E}(\underline{h}^{6/5}(Z_3,V_1,W_2))$ is of the order of $(\overline{F}^{(k)}(t_n)\overline{G}(t_n))^{-3/5}$ and $\mathbb {E}(\underline{h}^d(Z_3,V_1,W_2))$ is finite whenever d is (greater than but) sufficiently close to 6 / 5.

• (iv) For any $d\in ]1/2,(2\gamma _C^{-1}+\gamma _F^{-1}+\gamma _k^{-1})/(3\gamma _C^{-1}+2\gamma _F^{-1})[$, we have $\mathbb {E}\left( h^d(V_1,W_2)/\overline{H}^d(V_1)\right) =O\left( (\overline{F}^{(k)}(t_n)\overline{G}(t_n))^{2-3d}\right)$. In particular, if $\gamma _k<\gamma _C$ then taking $\delta$ (greater than but) sufficiently close to 4 / 5 is permitted, otherwise it is 2 / 3 instead of 4 / 5.

• (v) The integral $\theta _n=\iint h(v,w)dH^{(0)}(v)dH^{(1,k)}(w)$ is equivalent, as $n\rightarrow \infty$, to $\gamma _k(-\log \overline{G}(t_n))$.

Lemma 13

In this lemma, various notations defined in Sects. 5.2.2–5.2.4 are used.

• (i) The variables ${{\mathcal {H}}}^{**}_I$ for $I\in \{(i,j) \, ; \, 1\le i<j\le n\}$ are centred and uncorrelated. This is also true for the variables ${\underline{\mathcal {H}}}^{**}_I$ for $I\in \{(i,j,l) \, ; \, 1\le i<j<l\le n\}$.

• (ii) We have $\mathbb {E}\left[ ({{\mathcal {H}}}^{**}(V_1,W_2))^2\right] \le 48 \mathbb {E}[{{\mathcal {H}}}_1^2\mathbb {I}_{|{{\mathcal {H}}}_1|\le M_n}]$.

• (iii) We have $\mathbb {E}\left( \, |{{\mathcal {H}}}_1-{\mathcal {H}}^*(V_1,W_2)+{{\mathcal {H}}}^*_{1\bullet }(V_1)+{{\mathcal {H}}}^*_{{\bullet }1}(W_2)| \,\right) \le 4\mathbb {E}\left( |{{\mathcal {H}}}_1|\mathbb {I}_{|{{\mathcal {H}}}_1|>M_n}\right)$