Robust estimation of the conditional stable tail dependence function

Abstract

We propose a robust estimator of the stable tail dependence function in the case where random covariates are recorded. Under suitable assumptions, we derive the finite-dimensional weak convergence of the estimator properly normalized. The performance of our estimator in terms of efficiency and robustness is illustrated through a simulation study. Our methodology is applied on a real dataset of sale prices of residential properties.

Appendix

The minimization of the empirical density power divergence ${{\widehat{\Delta }}}_{\alpha , 1-t}(\delta _{1-t}|x_0)$ is based on its derivative. Direct computations show that all the terms appearing in this derivative have the following form

$$\begin{aligned} S_{n,1-t}(s|x_0):={1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i)\left( {Z_{1-t,i} \over {{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)}\right) ^s \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}} \end{aligned}$$

for $s <0$.

Assuming $F_{Z_{1-t}}(y|x_0)$ is strictly increasing in y, we can rewrite this main statistic as follows:

$$\begin{aligned}&S_{n,1-t}(s|x_0) \nonumber \\&\quad = {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \left\{ 1+\int _{{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)}^{Z_{1-t,i}} {s\, u^{s-1} \over {{\widehat{U}}}^s_{Z_{1-t}}(n/k|x_0)} du \right\} \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}}\nonumber \\&\quad = {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}} \nonumber \\&\qquad + {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \left\{ \int _{{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)}^{Z_{1-t,i}} {s\, u^{s-1} \over {{\widehat{U}}}^s_{Z_{1-t}}(n/k|x_0)} du \right\} \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}}\nonumber \\&\quad = {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}} \nonumber \\&\qquad +\int _{{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)}^{\infty } {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) {s\, u^{s-1} \over {{\widehat{U}}}^s_{Z_{1-t}}(n/k|x_0)} \mathbbm {1}_{\{u<Z_{1-t,i}\}} du \nonumber \\&\quad = {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}} \nonumber \\&\qquad +\int _{{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)}^{\infty } {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) {s\, u^{s-1} \over {{\widehat{U}}}^s_{Z_{1-t}}(n/k|x_0)} \mathbbm {1}_{\{{{\overline{F}}}_{Z_{1-t}}(Z_{1-t,i}|x_0)< {k\over n}{n \over k} {{\overline{F}}}_{Z_{1-t}}(u|x_0)\}} du \nonumber \\&\quad = {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \mathbbm {1}_{\{Z_{1-t,i}>{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\}} \nonumber \\&\qquad + \int _{0}^{1} {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) s z^{-1-s} \mathbbm {1}_{\{{{\overline{F}}}_{Z_{1-t}}(Z_{1-t,i}|x_0) < {k\over n}{n \over k} {{\overline{F}}}_{Z_{1-t}}(z^{-1}{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) |x_0)\}} dz\nonumber \\&\quad =T_{n,1-t}\left( s_{n,1-t}(1|x_0)|x_0\right) + \int _0^1 T_{n,1-t}\left( s_{n,1-t}(z|x_0)|x_0\right) \, s\, z^{-1-s}\, dz, \end{aligned}$$

(10)

where

$$\begin{aligned} T_{n,1-t}(y|x_0)&:= {1\over k} \sum _{i=1}^n K_{h_n}(x_0-X_i) \mathbbm {1}_{\{{{\overline{F}}}_{Z_{1-t}}(Z_{1-t,i}|x_0) < {k\over n}y\}}, y\in (0, T],\\ s_{n,1-t}(z|x_0)&:= {n\over k} {{\overline{F}}}_{Z_{1-t}}\left( z^{-1} {{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)\Bigr |x_0\right) . \end{aligned}$$

Thus, we start this appendix with some auxiliary results allowing us to study the statistic $T_{n,1-t}(y|x_0)$ and subsequently in Section 7.2 we establish the weak convergence of $S_{n,1-t}(s|x_0)$. Finally, in Sect. 7.3, Theorem 1 will be established. The proof of Theorem 2 from Sect. 3 is deferred to the online Supplementary Material.

1.1 Auxiliary results in case of known margins

First, we establish the joint weak convergence of processes $W_{n,1-t_j} := \lbrace \sqrt{kh_n^d} [T_{n,1-t_j}(y|x_0)-yf_X(x_0)]; y \in (0,T] \rbrace$, $j=1,\ldots ,J$.

Lemma 1

Assume $({\mathcal {D}}_{1-t_j})$ and $({\mathcal {H}}_{1-t_j})$ for $j=1,\ldots ,J$, $({\mathcal {D}}_{0.5})$, $({\mathcal {H}}_{0.5})$, $({\mathcal {K}}_1)$, $x_0\in Int(S_X)$ with $f_X(x_0)>0$, and $y \mapsto F_{Z_{1-t_j}}(y|x_0)$, $j=1,\ldots ,J$, are strictly increasing. Consider sequences $k \rightarrow \infty$ and $h_n\rightarrow 0$ as $n \rightarrow \infty$ such that $k/n \rightarrow 0$, $kh_n^d \rightarrow \infty$, $h_n^{\eta _{\varepsilon _{1-t_1}}\wedge \cdots \wedge \eta _{\varepsilon _{1-t_J}}\wedge \eta _{\varepsilon _{0.5}}}\log \frac{n}{k} \rightarrow 0$, $\sqrt{kh_n^d}h_n^{\eta _{f_X}\wedge \eta _{G_{1-t_1}}\wedge \cdots \wedge \eta _{G_{1-t_J}}}\rightarrow 0$, and for $j=1,\ldots ,J$, $\sqrt{kh_n^d} |\delta _{1-t_j}(U_{Z_{1-t_j}}({n\over k}|x_0)|x_0)|h_n^{\eta _{C_{1-t_j}}}\rightarrow 0$ and $\sqrt{kh_n^d} |\delta _{1-t_j}(U_{Z_{1-t_j}}({n\over k}|x_0)|x_0)| h_n^{\eta _{\varepsilon _{1-t_j}}} \log {n\over k} \rightarrow 0$. Then, for $n \rightarrow \infty$, we have

$$\begin{aligned} (W_{n,1-t_1},\ldots ,W_{n,1-t_J}) \leadsto (W_{1-t_1},\ldots , W_{1-t_J}), \end{aligned}$$

in $\ell ^J((0,T])$, for any $T >0$.

Lemma 2

Under the assumptions of Lemma 1, for any sequence $u_n^{(j)}$ satisfying

$$\begin{aligned} \sqrt{kh_n^d} \left( \frac{{{\overline{F}}}_{Z_{1-t_j}}(U_{Z_{1-t_j}} (n/k|x_0)|x_0)}{{{\overline{F}}}_{Z_{1-t_j}}(u_n^{(j)}|x_0)} -1\right) \rightarrow c_j \in {\mathbb {R}}, \end{aligned}$$

as $n \rightarrow \infty$, $j=1, \ldots , J$, we have

$$\begin{aligned} \left( \begin{array}{c} \sqrt{n h_n^d {{{\overline{F}}}}_{Z_{1-t_1}}(u_n^{(1)}|x_0)} \left( {\widehat{{{\overline{F}}}}_{Z_{1-t_1}}(u_n^{(1)}|x_0) \over {{{\overline{F}}}}_{Z_{1-t_1}}(u_n^{(1)}|x_0)} - 1\right) \\ \vdots \\ \sqrt{n h_n^d {{{\overline{F}}}}_{Z_{1-t_J}}(u_n^{(J)}|x_0)} \left( {\widehat{{{\overline{F}}}}_{Z_{1-t_J}}(u_n^{(J)}|x_0) \over {{{\overline{F}}}}_{Z_{1-t_J}}(u_n^{(J)}|x_0)} - 1\right) \end{array} \right) \leadsto {1\over f_X(x_0)} \left( \begin{array}{c} W_{1-t_1}(1) \\ \vdots \\ W_{1-t_J}(1) \end{array} \right) . \end{aligned}$$

Lemma 3

Assume $({\mathcal {D}}_{1-t_j})$ and $({\mathcal {H}}_{1-t_j})$ for $j=1,\ldots ,J$ , $({\mathcal {D}}_{0.5})$ , $({\mathcal {H}}_{0.5})$ , $({\mathcal {K}}_1)$ , $x_0\in Int(S_X)$ with $f_X(x_0)>0$ , and $y \mapsto F_{Z_{1-t_j}}(y|x_0)$ , $j=1,\ldots ,J$ , are strictly increasing. Consider sequences $k \rightarrow \infty$ and $h_n\rightarrow 0$ as $n \rightarrow \infty$ such that $k/n \rightarrow 0$ , $kh_n^d \rightarrow \infty$ , $h_n^{\eta _{\varepsilon _{1-t_1}}\wedge \cdots \wedge \eta _{\varepsilon _{1-t_J}}\wedge \eta _{\varepsilon _{0.5}}}\log \frac{n}{k} \rightarrow 0$ , $\sqrt{kh_n^d}h_n^{\eta _{f_X}\wedge \eta _{G_{1-t_1}}\wedge \cdots \wedge \eta _{G_{1-t_J}}}\rightarrow 0$ , $\sqrt{kh_n^d} |\delta _{1-t_j}(U_{Z_{1-t_j}}({n\over k}|x_0)|x_0)|\rightarrow 0$ , $j=1,\dots ,J$ . Then, we have

$$\begin{aligned} \sqrt{k h_n^d} \left( \begin{array}{c} {{{\widehat{U}}}_{Z_{1-t_1}}\left( n/k|x_0\right) \over U_{Z_{1-t_1}}\left( n/k|x_0\right) }-1 \\ \vdots \\ {{{\widehat{U}}}_{Z_{1-t_J}}\left( n/k|x_0\right) \over U_{Z_{1-t_J}}\left( n/k|x_0\right) }-1 \end{array} \right) \leadsto {1\over f_X(x_0)} \left( \begin{array}{c} W_{1-t_1}(1) \\ \vdots \\ W_{1-t_J}(1) \end{array} \right) . \end{aligned}$$

1.2 Joint weak convergence of $S_{n,1-t_j}(s_j|x_0), j=1,\ldots , M$

We have now all the ingredients to state the joint weak convergence of $S_{n,1-t_j}(s_j|x_0)$, $j=1,\ldots ,M$. Note that we allow for the possibility that $t_j=t_{j'}$ for $j \ne j'$, but of course the statistics $S_{n,1-t_j}(s_j|x_0)$, $j=1,\ldots ,M$, must be different. This is due to the fact that, for a given value of t, the study of the MDPD estimator ${{\widehat{\delta }}}_{n,1-t}$ requires the joint convergence in distribution of several statistics $S_{n,1-t}(s|x_0)$, with different values of s.

Theorem 3

Under the conditions of Theorem 1, we have, for $s_1,\ldots ,s_M <0$,

$$\begin{aligned} \sqrt{kh_n^d} \left( \begin{array}{c} S_{n,1-t_1}(s_1|x_0) - {1\over 1-s_1} f_X(x_0) \\ \vdots \\ S_{n,1-t_M}(s_M|x_0) - {1\over 1-s_M} f_X(x_0) \end{array} \right) \leadsto \left( \begin{array}{c} s_1\, \int _0^1 \left[ {W_{1-t_1}(z)\over z} - W_{1-t_1}(1)\right] z^{-s_1} dz \\ \vdots \\ s_M\, \int _0^1 \left[ {W_{1-t_M}(z)\over z} - W_{1-t_M}(1)\right] z^{-s_M} dz \end{array} \right) . \end{aligned}$$

To prove this Theorem 3, we start to establish the weak convergence of an individual statistic $S_{n,1-t}(s|x_0)$, properly normalized. We have the following decomposition

$$\begin{aligned}&\sqrt{kh_n^d} \left( S_{n,1-t}(s|x_0) - {1\over 1-s} f_X(x_0)\right) \nonumber \\&\quad =\int _0^1 [W_{1-t}(z)-W_{1-t}(1)]\, s\, z^{-1-s} dz\nonumber \\&\qquad +\left\{ \sqrt{kh_n^d} \left[ T_{n,1-t}(s_{n,1-t}(1|x_0)|x_0)-s_{n,1-t}(1|x_0)f_X(x_0)\right] -W_{1-t}\left( s_{n,1-t}(1|x_0)\right) \right\} \nonumber \\&\qquad +\left\{ W_{1-t}\left( s_{n,1-t}(1|x_0)\right) -W_{1-t}(1)\right\} \nonumber \\&\qquad +\sqrt{kh_n^d} \left( s_{n,1-t}(1|x_0)-1\right) f_X(x_0)\nonumber \\&\qquad +\int _0^1 \left\{ \sqrt{kh_n^d} \left[ T_{n,1-t}(s_{n,1-t}(z|x_0)|x_0)-s_{n,1-t}(z|x_0)f_X(x_0)\right] -W_{1-t}\left( s_{n,1-t}(z|x_0)\right) \right\} \, s\, z^{-1-s} \, dz\nonumber \\&\qquad +\int _0^1 \left[ W_{1-t}\left( s_{n,1-t}(z|x_0)\right) -W_{1-t}(z)\right] \, s\, z^{-1-s} \, dz\end{aligned}$$

(11)

$$\begin{aligned}&+f_X(x_0)\, \sqrt{kh_n^d} \int _0^1 \left[ s_{n,1-t}(z|x_0)-z\right] \, s\, z^{-1-s} \, dz \nonumber \\&=: \int _0^1 [W_{1-t}(z)-W_{1-t}(1)]\, s\, z^{-1-s} dz + \sum _{i=1}^6 T_{i,k}. \end{aligned}$$

(12)

We study the terms separately. Clearly, using Lemma 5.2 from Goegebeur et al. (2021) we have that for n large, with arbitrary large probability,

$$\begin{aligned} |T_{1,k}|\le & {} \sup _{y\in (0, 2]} \left| \sqrt{kh_n^d} \left[ T_{n,1-t}(y|x_0)-y f_X(x_0)\right] -W_{1-t}\left( y\right) \right| , \end{aligned}$$

(13)

$$\begin{aligned} \text{ and } |T_{4,k}|\le & {} \sup _{y\in (0, 2]} \left| \sqrt{kh_n^d} \left[ T_{n,1-t}(y|x_0)-y f_X(x_0)\right] -W_{1-t}\left( y\right) \right| \left| \int _0^1 s\, z^{-1-s} dz\right| , \end{aligned}$$

(14)

and hence, by Lemma 1 combined with the Skorohod construction we obtain $T_{1,k}=o_{{\mathbb {P}}}(1)$ and $T_{4,k}=o_{{\mathbb {P}}}(1).$

Using again Lemma 5.2 in Goegebeur et al. (2021) with continuity, we have

$$\begin{aligned} |T_{2,k}|= & {} o_{\mathbb {P}}(1). \end{aligned}$$

(15)

Concerning $T_{3,k}$, we can use the following decomposition:

$$\begin{aligned} T_{3,k}&= \sqrt{kh_n^d} \left[ {{{\overline{F}}}_{Z_{1-t}}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over {{\overline{F}}}_{Z_{1-t}}(U_{Z_{1-t}}(n/k|x_0)|x_0)}-1\right] \, f_X(x_0)\\&= \sqrt{kh_n^d} \left[ \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-1} - 1 \right] \, {1+\delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+\delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)} \, f_X(x_0)\\&\quad + \sqrt{kh_n^d} \left[ {1+\delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+\delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)} - 1\right] \, f_X(x_0)\\&= \sqrt{kh_n^d} \left[ \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-1} - 1 \right] \, {1+\delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+\delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)} \, f_X(x_0)\\&\quad + \sqrt{kh_n^d}\, { \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+\delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)}\\&\quad \times \left\{ \left[ {\delta _{1-t} ({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over \delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)} - \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-\beta (x_0)}\right] \right. \\&\quad \left. +\left[ \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-\beta (x_0)} - 1\right] \right\} .\\&=: -\sqrt{kh_n^d} \left[ {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)} - 1 \right] \, f_X(x_0) (1+o_{\mathbb {P}}(1))\\&\quad + \sqrt{kh_n^d}\, { \delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+\delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)} T^{(1)}_{3,k}. \end{aligned}$$

By Proposition B.1.10 in de Haan and Ferreira (2006), for n large, with arbitrary large probability, we have for $\varepsilon , \xi >0$

$$\begin{aligned} |T^{(1)}_{3,k}|\le \varepsilon \, \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-\beta (x_0)\pm \xi } + \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-\beta (x_0)} + 1. \end{aligned}$$

(16)

In the above, the notation $a^{\pm \bullet }$ means $a^{\bullet }$ if $a\ge 1$ and $a^{-\bullet }$ if $a<1$. This implies by Lemma 3 and our conditions that

$$\begin{aligned} T_{3,k} \leadsto -W_{1-t}(1). \end{aligned}$$

(17)

Concerning now $T_{5,k}$, we have for any $\delta \in (0,1)$ small

$$\begin{aligned} |T_{5,k}|\le & {} \int _0^{\delta } \left| W_{1-t}\left( s_{n,1-t}(z|x_0)\right) -W_{1-t}(z)\right| \, |s|\, z^{-1-s} \, dz\nonumber \\&+\int _{\delta }^1 \left| W_{1-t}\left( s_{n,1-t}(z|x_0)\right) -W_{1-t}(z)\right| \, |s|\, z^{-1-s} \, dz\nonumber \\\le & {} |s|\left\{ \sup _{z\in (0,\delta ]} \left| W_{1-t}\left( s_{n,1-t}(z|x_0)\right) \right| +\sup _{z\in (0, \delta ]} |W_{1-t}(z)|\right\} \int _0^\delta z^{-1-s} \, dz\nonumber \\&+ |s| \sup _{z\in (\delta , 1]} \left| W_{1-t}\left( s_{n,1-t}(z|x_0)\right) -W_{1-t}(z)\right| \int _\delta ^1 z^{-1-s} \, dz\nonumber \\= & {} o_{{\mathbb {P}}}(1). \end{aligned}$$

(18)

Finally, concerning $T_{6,k}$, we have

$$\begin{aligned} T_{6,k}&= f_X(x_0)\, \sqrt{kh_n^d} \int _0^1 \left[ {{{\overline{F}}}_{Z_{1-t}}(z^{-1} {{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) |x_0) \over {{\overline{F}}}_{Z_{1-t}}(U_{Z_{1-t}}(n/k|x_0)|x_0)} -z\right] \, s\, z^{-1-s} \, dz\\&= f_X(x_0)\, \sqrt{kh_n^d} \left\{ \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-1} - 1\right\} \, s \, \int _0^1 z^{-s} dz\\&\quad + f_X(x_0)\, \sqrt{kh_n^d} \, \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-1} \\&\quad \times \int _0^1 \left( {1+\delta _{1-t}(z^{-1} {{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) |x_0) \over 1+\delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)} - 1\right) s \,z^{-s}dz\\&=: - f_X(x_0)\, {s \over 1-s} \sqrt{kh_n^d} \, \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}-1\right) (1+o_{\mathbb {P}}(1))\\&\quad + f_X(x_0)\, \sqrt{kh_n^d} \, \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-1} {\delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)\over 1+\delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)} \, T^{(1)}_{6,k} \end{aligned}$$

with

$$\begin{aligned} |T^{(1)}_{6,k}|\le & {} |s| \int _0^1 \left| {\delta _{1-t} (z^{-1} {{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over \delta _{1-t} (U_{Z_{1-t}}(n/k|x_0)|x_0)} - \left( z^{-1}{{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-\beta (x_0)}\right| \,z^{-s}dz \\&+|s|\int _0^1 \left| \left( z^{-1}{{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)}\right) ^{-\beta (x_0)} - 1\right| \,z^{-s}dz\\= & {} O_{\mathbb {P}}(1), \end{aligned}$$

using arguments similar to those for $T^{(1)}_{3,k}$. Consequently, using again Lemma 3, we deduce that

$$\begin{aligned} T_{6,k} \leadsto - {s \over 1-s}\, W_{1-t}(1). \end{aligned}$$

(19)

Combining decomposition (12) with (13)–(19), the proof of the marginal weak convergence of $S_{n,1-t}(s|x_0)$, properly normalized, is achieved.

The joint weak convergence of $( \sqrt{kh_n^d}[S_{n,1-t_j}(s_j|x_0) - f_X(x_0)/( 1-s_j) ], j=1,\ldots ,M )$ follows from Lemmas 1 and 3, respectively. $\square$

1.3 Proof of Theorem 1

Again we first consider the case of a single estimator ${{\widehat{L}}}_k(y_1,y_2|x_0)$. From (3), (4) and (5), we deduce that

$$\begin{aligned}&\displaystyle \sqrt{kh_n^d} \left( {{\widehat{L}}}_k(y_1, y_2|x_0)-L(y_1, y_2|x_0)\right) \\&\quad =-y_1 \sqrt{kh_n^d} \left( {{\widehat{G}}}_{1-t,k}(x_0)-G_{1-t}(x_0)\right) \\&\quad = - y_1 \sqrt{kh_n^d} \left( {k \over n} {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over 1+{{\widehat{\delta }}}_{n,1-t}} -G_{1-t}(x_0)\right) \\&\quad = - y_1 G_{1-t}(x_0) \sqrt{kh_n^d} \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)} {1+ \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+{{\widehat{\delta }}}_{n,1-t}} - 1\right) \\&\quad = - y_1 G_{1-t}(x_0) \sqrt{kh_n^d} \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)} -1 \right) \\&\qquad + y_1 G_{1-t}(x_0) \sqrt{kh_n^d} \left( {{\widehat{\delta }}}_{n,1-t} - \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)\right) {1 \over 1+{{\widehat{\delta }}}_{n,1-t}}\\&\qquad + y_1 G_{1-t}(x_0) {{{\widehat{\delta }}}_{n,1-t} - \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0) \over 1+{{\widehat{\delta }}}_{n,1-t}} \sqrt{kh_n^d} \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)} - 1\right) . \end{aligned}$$

Now remark that

$$\begin{aligned}&\sqrt{kh_n^d} \left| \delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) - \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)\right| \\&\quad =\sqrt{kh_n^d} \left| \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)\right| \left| {\delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \over \delta _{1-t}(U_{Z_{1-t}}(n/k|x_0)|x_0)} - 1 \right| \\&\quad =o_{\mathbb {P}}(1), \end{aligned}$$

by (16). This implies that

$$\begin{aligned}&\sqrt{kh_n^d} \left( {{\widehat{L}}}_k(y_1, y_2|x_0)-L(y_1, y_2|x_0)\right) \\&\quad = - y_1 G_{1-t}(x_0) \sqrt{kh_n^d} \left( {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)} -1 \right) \\&\qquad + y_1 G_{1-t}(x_0) \sqrt{kh_n^d} \left( {{\widehat{\delta }}}_{n,1-t} - \delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0)\right) +o_{\mathbb {P}}(1). \end{aligned}$$

Using the fact that

$$\begin{aligned}&\sqrt{kh_n^d} \left( \begin{array}{c} {{{\widehat{U}}}_{Z_{1-t}}(n/k|x_0) \over U_{Z_{1-t}}(n/k|x_0)} -1 \\ {{\widehat{\delta }}}_{n,1-t}- \delta _{1-t}({{\widehat{U}}}_{Z_{1-t}}(n/k|x_0)|x_0) \end{array} \right) \\&\quad \leadsto \left( \begin{array}{c} {W_{1-t}(1)\over f_X(x_0)} \\ c\left( 2\alpha \int _0^1 \left[ {W_{1-t}(z)\over z} - W_{1-t}(1)\right] z^{2\alpha } \, dz-(1+\beta )(2\alpha +\beta ) \int _0^1 \left[ {W_{1-t}(z)\over z} - W_{1-t}(1)\right] z^{2\alpha +\beta } \, dz \right) \end{array} \right) , \end{aligned}$$

we can deduce that

$$\begin{aligned}&\sqrt{kh_n^d} \left( {{\widehat{L}}}_k(y_1, y_2|x_0)-L(y_1, y_2|x_0)\right) \\&\quad \leadsto - y_1 G_{1-t}(x_0) {W_{1-t}(1)\over f_X(x_0)} + y_1 G_{1-t}(x_0) c \left\{ 2\alpha \int _0^1 \left[ {W_{1-t}(z)\over z} - W_{1-t}(1)\right] z^{2\alpha } \, dz \right. \\&\qquad \left. -(1+\beta )(2\alpha +\beta ) \int _0^1 \left[ {W_{1-t}(z)\over z} - W_{1-t}(1)\right] z^{2\alpha +\beta } \, dz\right\} . \end{aligned}$$

Now, concerning the finite-dimensional convergence, it follows from Lemma 3 combined with the following theorem which states the joint behavior of the MDPD estimator ${{\widehat{\delta }}}_{n,1-t_j}, j=1,\ldots , J$, and whose proof is deferred to the online Supplementary Material:

Theorem 4

Under the conditions of Theorem 1, with probability tending to one, there exists sequences of solutions $({{\widehat{\delta }}}_{n,1-t_j})_{n \ge 1}$, $j=1,\ldots ,J,$ to the MDPD estimating equations such that

$$\begin{aligned} \left( \begin{array}{c} {{\widehat{\delta }}}_{n,1-t_1} - \delta _{1-t_1}({{\widehat{U}}}_{Z_{1-t_1}}(n/k|x_0)|x_0) \\ \vdots \\ {{\widehat{\delta }}}_{n,1-t_J} - \delta _{1-t_J}({{\widehat{U}}}_{Z_{1-t_J}}(n/k|x_0)|x_0) \end{array} \right) {\mathop {\rightarrow }\limits ^{{\mathbb {P}}}} {\varvec{0}}. \end{aligned}$$

Moreover, for the consistent solution sequences one has that

$$\begin{aligned}&\sqrt{kh_n^d} \left( \begin{array}{c} {{\widehat{\delta }}}_{n,1-t_1} - \delta _{1-t_1}({{\widehat{U}}}_{Z_{1-t_1}}(n/k|x_0)|x_0) \\ \vdots \\ {{\widehat{\delta }}}_{n,1-t_J} - \delta _{1-t_J}({{\widehat{U}}}_{Z_{1-t_J}}(n/k|x_0)|x_0) \end{array} \right) \\&\quad \leadsto c \left( \begin{array}{c} 2\alpha \int _0^1 \left[ {W_{1-t_1}(z)\over z} - W_{1-t_1}(1)\right] z^{2\alpha } \, dz-(1+\beta )(2\alpha +\beta ) \int _0^1 \left[ {W_{1-t_1}(z)\over z} - W_{1-t_1}(1)\right] z^{2\alpha +\beta } \, dz \\ \vdots \\ 2\alpha \int _0^1 \left[ {W_{1-t_J}(z)\over z} - W_{1-t_J}(1)\right] z^{2\alpha } \, dz-(1+\beta )(2\alpha +\beta ) \int _0^1 \left[ {W_{1-t_J}(z)\over z} - W_{1-t_J}(1)\right] z^{2\alpha +\beta } \, dz \end{array} \right) , \end{aligned}$$

where c is defined in Theorem 1. $\square$