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Bias-corrected estimation for conditional Pareto-type distributions with random right censoring

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Abstract

We consider bias-reduced estimation of the extreme value index in conditional Pareto-type models with random covariates when the response variable is subject to random right censoring. The bias-correction is obtained by fitting the extended Pareto distribution locally to the relative excesses over a high threshold using the maximum likelihood method. Convergence in probability and asymptotic normality of the estimators are established under suitable assumptions. The finite sample behaviour is illustrated with a simulation experiment and the method is applied to two real datasets.

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Acknowledgements

This work was supported by a research grant (VKR023480) from VILLUM FONDEN. Computation/simulation for the work described in this paper was supported by the DeIC National HPC Centre, SDU. The authors sincerely thank the editor, associate editor and the referees for their helpful comments and suggestions that led to substantial improvement of the paper. The authors also take pleasure in thanking Gilles Stupfler for providing the code to compute the estimator proposed in Stupfler (2016).

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Correspondence to Yuri Goegebeur.

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Appendix

Appendix

1.1 Derivatives of the log-likelihood function

Let j(γY, δY; β) and jk(γY, δY; β), j, k = 1, 2, denote the first and second order derivatives of (γY, δY; β) with respect to γY and δY, respectively. Then

$$\begin{array}{@{}rcl@{}} \ell_{1}(\gamma_{Y},\delta_{Y};\beta) & = & \frac{1}{n}\sum\limits_{i = 1}^{n} K_{h_{n}}(x_{0}-X_{i}) \left[-\frac{1}{\gamma_{Y}} {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y_{i} \le C_{i} \rbrace} + \frac{1}{{\gamma_{Y}^{2}}} \ln \frac{T_{i}}{t_{n}}\right.\\ &&\qquad\qquad\qquad\qquad\left.+\frac{1}{{\gamma_{Y}^{2}}} \ln \left( 1+\delta_{Y}-\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)\right]{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T_{i} >t_{n} \rbrace}, \\ \ell_{2}(\gamma_{Y},\delta_{Y};\beta) & = & \frac{1}{n}\sum\limits_{i = 1}^{n} K_{h_{n}}(x_{0}-X_{i}) \left[ -\frac{1}{1+\delta_{Y}-\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta}} \left( 1- \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right) {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y_{i} \le C_{i} \rbrace} \right. \\ & & +\frac{1}{1+\delta_{Y}\left( 1-(1-\beta) \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)} \left( 1-(1-\beta) \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right) {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y_{i} \le C_{i} \rbrace} \\ & & \left. - \frac{1}{\gamma_{Y}} \frac{1}{1+\delta_{Y}-\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta}} \left( 1- \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right) \right]{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T_{i} >t_{n} \rbrace}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \ell_{11}(\gamma_{Y},\delta_{Y};\beta) & = & \frac{1}{n}\sum\limits_{i = 1}^{n} K_{h_{n}}(x_{0}-X_{i})\\ &&\times\left[\frac{1}{{\gamma^{2}_{Y}}} {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y_{i} \le C_{i} \rbrace} - \frac{2}{{\gamma_{Y}^{3}}} \ln \frac{T_{i}}{t_{n}} -\frac{2}{{\gamma_{Y}^{3}}} \ln \left( 1+\delta_{Y} -\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)\right]{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T_{i} >t_{n} \rbrace}, \\ \ell_{22}(\gamma_{Y},\delta_{Y};\beta) & = & \frac{1}{n}\sum\limits_{i = 1}^{n} K_{h_{n}}(x_{0}-X_{i}) \left[ \frac{1}{\left( 1+\delta_{Y}-\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)^{2}} \left( 1- \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)^{2} {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y_{i} \le C_{i} \rbrace} \right. \\ & & -\frac{1}{\left( 1+\delta_{Y}\left( 1-(1-\beta) \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right) \right)^{2}} \left( 1-(1-\beta) \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)^{2} {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y_{i} \le C_{i} \rbrace} \\ & & \left. + \frac{1}{\gamma_{Y}} \frac{1}{\left( 1+\delta_{Y}-\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta}\right)^{2}} \left( 1- \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right)^{2} \right]{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T_{i} >t_{n} \rbrace}, \\ \ell_{12}(\gamma_{Y},\delta_{Y};\beta) & = & \frac{1}{n}\sum\limits_{i = 1}^{n} K_{h_{n}}(x_{0}-X_{i}) \frac{1}{{\gamma_{Y}^{2}}} \frac{1}{1+\delta_{Y}-\delta_{Y} \left( \frac{T_{i}}{t_{n}} \right)^{-\beta}} \left( 1- \left( \frac{T_{i}}{t_{n}} \right)^{-\beta} \right){\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T_{i} >t_{n} \rbrace}. \end{array} $$

1.2 Proof of Theorem 1

We focus on deriving the asymptotic expansion for \(\mathbb E(T_{n}^{(2)}(K,s,s^{\prime }|x_{0}))\). Then, \(\mathbb E(T_{n}^{(1)}(K,s,s^{\prime }|x_{0}))\) can be handled similarly, combined with some ideas from Dierckx et al. (2014). Note that Dierckx et al. (2014) also considered the statistic \(T_{n}^{(1)}(K,s,s^{\prime }|x_{0})\), though it was analysed under their high level assumption called \((\mathcal M)\), which is avoided in the present paper, allowing us to obtain a more precise statement of the remainder terms using the Hölder exponents from Assumption \((\mathcal H)\).

We have

$$\begin{array}{@{}rcl@{}} \mathbb E(T_{n}^{(2)}(K,s,s^{\prime}|x_{0})) & = & \mathbb E \left[ K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s} \left( \ln_{+} \frac{T}{t_{n}}\right)^{s^{\prime}} {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y \le C, T >t_{n} \rbrace} \right] \\ & = &\mathbb E \left[K_{h_{n}}(x_{0}-X)\mathbb E\left( \left( \frac{T}{t_{n}} \right)^{s} \left( \ln_{+} \frac{T}{t_{n}}\right)^{s^{\prime}} {\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y \le C, T >t_{n} \rbrace} | X \right) \right] \\ & = & \mathbb E \left[ K_{h_{n}}(x_{0}-X){\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} f_{Y}(y|X)\overline F_{C}(y|X)dy \right] \\ & = & {\int}_{\mathbb R^{d}} K_{h_{n}}(x_{0}-u){\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}}f_{Y}(y|u)\overline F_{C}(y|u)dyf_{X}(u)du \\ & = & {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}}f_{Y}(y|x_{0}-h_{n}z)\\ &&\times\overline{F}_{C}(y|x_{0}-h_{n}z)dyf_{X}(x_{0}-h_{n}z)dz. \end{array} $$

In view of the various Hölder conditions, the latter is further decomposed as

$$\begin{array}{@{}rcl@{}} &&{\mathbb E(T_{n}^{(2)}(K,s,s^{\prime}|x_{0})) } \\ &= & f_{X}(x_{0}) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} f_{Y}(y|x_{0})\overline F_{C}(y|x_{0})dy \\ & & + {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} f_{Y}(y|x_{0})\overline F_{C}(y|x_{0})dy {\int}_{S_{K}} K(z) (f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ & & + f_{X}(x_{0}) {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} f_{Y}(y|x_{0})(\overline F_{C}(y|x_{0}-h_{n}z)-\overline F_{C}(y|x_{0}))dydz \\ & & + {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} f_{Y}(y|x_{0})(\overline F_{C}(y|x_{0}-h_{n}z) \\ & &-\overline F_{C}(y|x_{0}))dy(f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ & & + f_{X}(x_{0}) {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}}(f_{Y}(y|x_{0}-h_{n}z)-f_{Y}(y|x_{0}))\overline F_{C}(y|x_{0}))dydz \\ & & + {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} (f_{Y}(y|x_{0}-h_{n}z)\\ & &-f_{Y}(y|x_{0}))\overline F_{C}(y|x_{0})dy(f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ & & +f_{X}(x_{0}) {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} (f_{Y}(y|x_{0}-h_{n}z)\\ & &-f_{Y}(y|x_{0}))(\overline F_{C}(y|x_{0}-h_{n}z)-\overline F_{C}(y|x_{0}))dydz \\ & & + {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s} \left( \ln \frac{y}{t_{n}}\right)^{s^{\prime}} (f_{Y}(y|x_{0}-h_{n}z)\\ & &-f_{Y}(y|x_{0}))(\overline F_{C}(y|x_{0}-h_{n}z)-\overline F_{C}(y|x_{0}))dy \\ & & \hspace{3cm} (f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ &=: & T_{1}+\cdots+T_{8}. \end{array} $$

Concerning T1 we have

$$\begin{array}{@{}rcl@{}} T_{1} = t_{n} f_{Y}(t_{n}|x_{0}) \overline F_{C}(t_{n}|x_{0})f_{X}(x_{0}) {\int}_{1}^{\infty} z^{s}(\ln z)^{s^{\prime}} \frac{f_{Y}(t_{n}z|x_{0})}{f_{Y}(t_{n}|x_{0})} \frac{\overline F_{C}(t_{n}z|x_{0})}{\overline F_{C}(t_{n}|x_{0})}dz. \end{array} $$

A slight modification of Proposition 2.3 in Beirlant et al. (2009) gives

$$\begin{array}{@{}rcl@{}} \sup\limits_{z \ge 1} z^{1/\gamma_{\bullet}(x)}\left| \frac{\overline F_{\bullet}(t_{n}z|x_{0})}{\overline F_{\bullet}(t_{n}|x_{0})} - \overline G(z;\gamma_{\bullet}(x_{0}),\delta_{\bullet}(t_{n}|x_{0}),\beta_{\bullet}(x_{0})) \right| = o(|\delta_{\bullet}(t_{n}|x_{0})|), \hspace{1cm} t_{n} \to \infty. \end{array} $$

This leads to the decomposition

$$\begin{array}{@{}rcl@{}} T_{1} & = & t_{n} f_{Y}(t_{n}|x_{0}) \overline F_{C}(t_{n}|x_{0})f_{X}(x_{0}) \left[ {\int}_{1}^{\infty} z^{s}(\ln z)^{s^{\prime}} \frac{f_{Y}(t_{n}z|x_{0})}{f_{Y}(t_{n}|x_{0})} \overline G(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz \right. \\ & & + \left. {\int}_{1}^{\infty} z^{s}(\ln z)^{s^{\prime}} \frac{f_{Y}(t_{n}z|x_{0})}{f_{Y}(t_{n}|x_{0})} \left( \frac{\overline F_{C}(t_{n}z|x_{0})}{\overline F_{C}(t_{n}|x_{0})} - \overline G(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0})) \right)dz \right] \\ & =: & t_{n} f_{Y}(t_{n}|x_{0}) \overline F_{C}(t_{n}|x_{0})f_{X}(x_{0}) (T_{1,1}+T_{1,2}). \end{array} $$

From (3) we can write

$$\begin{array}{@{}rcl@{}} T_{1,1}& = & {\int}_{1}^{\infty} z^{s-1/\gamma_{Y}(x_{0})-1}(\ln z)^{s^{\prime}}\overline G(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz \\ & & + \frac{1}{1+\left( \frac{1}{\gamma_{Y}(x_{0})}-\varepsilon_{Y}(t_{n}|x_{0}) \right)\delta_{Y}(t_{n}|x_{0})} \left[\frac{}{} \right. \\ & & \frac{\delta_{Y}(t_{n}|x_{0})}{\gamma_{Y}(x_{0})}{\int}_{1}^{\infty} z^{s-1/\gamma_{Y}(x_{0})-1}(\ln z)^{s^{\prime}}(z^{-\beta_{Y}(x_{0})}-1)\overline G(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz \\ & & + \frac{\delta_{Y}(t_{n}|x_{0})}{\gamma_{Y}(x_{0})}{\int}_{1}^{\infty} z^{s-1/\gamma_{Y}(x_{0})-1}(\ln z)^{s^{\prime}}\left( \frac{\delta_{Y}(t_{n}z|x_{0})}{\delta_{Y}(t_{n}|x_{0})} - z^{-\beta_{Y}(x_{0})}\right)\\ & &G(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz \\ & & - \varepsilon_{Y}(t_{n}|x_{0})\delta_{Y}(t_{n}|x_{0}) {\int}_{1}^{\infty} z^{s-1/\gamma_{Y}(x_{0})-1}(\ln z)^{s^{\prime}}\left( \frac{\varepsilon_{Y}(t_{n}z|x_{0})}{\varepsilon_{Y}(t_{n}|x_{0})} - 1\right) \frac{\delta_{Y}(t_{n}z|x_{0})}{\delta_{Y}(t_{n}|x_{0})} \\ & &\frac{}{}\overline{G}(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz \\ & & \left. - \varepsilon_{Y}(t_{n}|x_{0})\delta_{Y}(t_{n}|x_{0}) {\int}_{1}^{\infty} z^{s-1/\gamma_{Y}(x_{0})-1}(\ln z)^{s^{\prime}}\left( \frac{\delta_{Y}(t_{n}z|x_{0})}{\delta_{Y}(t_{n}|x_{0})}- 1\right)\right.\\ &&\left.\frac{}{}\overline{G}(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz \right] \\ & =: & T_{1,1,1}+ \frac{1}{1+\left( \frac{1}{\gamma_{Y}(x_{0})}-\varepsilon_{Y}(t_{n}|x_{0}) \right)\delta_{Y}(t_{n}|x_{0})} (T_{1,1,2}+\cdots+T_{1,1,5}). \end{array} $$

In order to deal with these integrals, the following expansion of the extended Pareto distribution is useful

$$\begin{array}{@{}rcl@{}} \overline G(z;\gamma_{\bullet}(x_{0}),\delta_{\bullet}(t_{n}|x_{0}),\beta_{\bullet}(x_{0}))= z^{-1/\gamma_{\bullet}(x_{0})} \left( 1-\frac{\delta_{\bullet}(t_{n}|x_{0})}{\gamma_{\bullet}(x_{0})}(1-z^{-\beta_{\bullet}(x_{0})})+O(\delta_{\bullet}^{2}(t_{n}|x_{0}))\right), \end{array} $$

where \(O(\delta _{\bullet }^{2}(t_{n}|x_{0}))\) is uniform in z ≥ 1.

A straightforward calculation gives then

$$\begin{array}{@{}rcl@{}} T_{1,1,1} & = & \frac{{\Gamma}(s^{\prime}+ 1)\gamma_{T}^{s^{\prime}+ 1}(x_{0})}{(1-s\gamma_{T}(x_{0}))^{s^{\prime}+ 1}}+\delta_{C}(t_{n}|x_{0})\frac{{\Gamma}(s^{\prime}+ 1)\gamma_{T}^{s^{\prime}+ 1}(x_{0})}{\gamma_{C}(x_{0})} \\ & & \times\left[ \frac{1}{(1+\beta_{C}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right]+O({\delta_{C}^{2}}(t_{n}|x_{0})), \\ T_{1,1,2} & = & \delta_{Y}(t_{n}|x_{0})\frac{{\Gamma}(s^{\prime}+ 1)\gamma_{T}^{s^{\prime}+ 1}(x_{0})}{\gamma_{Y}(x_{0})} \left[ \frac{1}{(1+\beta_{Y}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}\right. \\ & &\qquad\qquad\qquad\qquad\quad\qquad\left.- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right] + O(\delta_{Y}(t_{n}|x_{0})\delta_{C}(t_{n}|x_{0})). \end{array} $$

For T1,1,3 we use Proposition B.1.10 in de Haan and Ferreira (2006), see also Drees (1998). Thus, for ε > 0 and 0 < δ < 1/γT(x0) + βY(x0) − s, arbitrary, and n sufficiently large, we have

$$\begin{array}{@{}rcl@{}} |T_{1,1,3}| & \le & \varepsilon \frac{|\delta_{Y}(t_{n}|x_{0})|}{\gamma_{Y}(x_{0})} {\int}_{1}^{\infty} z^{-(1/\gamma_{Y}(x_{0})+\beta_{Y}(x_{0})-s-\delta)-1}(\ln z)^{s^{\prime}}\overline G(z;\gamma_{C}(x_{0}),\delta_{C}(t_{n}|x_{0}),\beta_{C}(x_{0}))dz. \end{array} $$

Since ε is arbitrary and by using calculations for the integral that are similar to those above, one finds that T1,1,3 = o(δY(tn|x0)). In the same way T1,1,4 = o(δY(tn|x0)), and

$$\begin{array}{@{}rcl@{}} T_{1,1,5} & = & - \varepsilon_{Y}(t_{n}|x_{0})\delta_{Y}(t_{n}|x_{0}){\Gamma}(s^{\prime}+ 1)\gamma_{T}^{s^{\prime}+ 1}(x_{0})\left[ \frac{1}{(1+\beta_{Y}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}\right.\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\left.- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right]+o(\delta_{Y}(t_{n}|x_{0})). \end{array} $$

Analogously one can show that T1,2 = o(δC(tn|x0)).

Collecting the terms gives then

$$\begin{array}{@{}rcl@{}} T_{1}& = & t_{n} f_{Y}(t_{n}|x_{0}) \overline F_{C}(t_{n}|x_{0})f_{X}(x_{0}){\Gamma}(s^{\prime}+ 1)\gamma_{T}^{s^{\prime}+ 1}(x_{0}) \left\lbrace \frac{1}{(1-s\gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right. \\ & &+ \frac{\delta_{C}(t_{n}|x_{0})}{\gamma_{C}(x_{0})} \left[ \frac{1}{(1+\beta_{C}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) \\ & & \left. + \delta_{Y}(t_{n}|x_{0}) \left( \frac{1}{\gamma_{Y}(x_{0})}-\varepsilon_{Y}(t_{n}|x_{0}) \right)\left[ \frac{1}{(1+\beta_{Y}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}\right.\right.\\ &&\left.\left.- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) \right\}. \end{array} $$

Note that

$$\begin{array}{@{}rcl@{}} t_{n} f_{Y}(t_{n}|x_{0})\overline F_{C}(t_{n}|x_{0}) = \frac{\overline F_{T}(t_{n}|x_{0})}{\gamma_{Y}(x_{0})}\left( 1-\frac{\varepsilon_{Y}(t_{n}|x_{0})\delta_{Y}(t_{n}|x_{0}) }{1+\frac{\delta_{Y}(t_{n}|x_{0})}{\gamma_{Y}(x_{0})}} \right), \end{array} $$

whence

$$\begin{array}{@{}rcl@{}} T_{1} &=& \overline F_{T}(t_{n}|x_{0} )f_{X}(x_{0}) {\Gamma}(s^{\prime}+ 1)\frac{\gamma_{T}^{s^{\prime}+ 1}(x_{0})}{\gamma_{Y}(x_{0})}\left\lbrace \frac{1}{(1-s\gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right. \\ & & + \frac{\delta_{C}(t_{n}|x_{0})}{\gamma_{C}(x_{0})} \left[ \frac{1}{(1+\beta_{C}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) \\ & & +\ \delta_{Y}(t_{n}|x_{0}) \left( \frac{1}{\gamma_{Y}(x_{0})}-\varepsilon_{Y}(t_{n}|x_{0}) \right)\left[ \frac{1}{(1+\beta_{Y}(x_{0})\gamma_{T}(x_{0})-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}}\right.\\ &&\left.- \frac{1}{(1-s \gamma_{T}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) \\ & & \left. -\delta_{Y}(t_{n}|x_{0})\varepsilon_{Y}(t_{n}|x_{0}) \frac{1}{(1-s\gamma_{T}(x_{0}))^{s^{\prime}+ 1}} (1+o(1))\right\rbrace. \end{array} $$

For T2, we use the Hölder condition on fX and obtain \(T_{2} = O(h^{\eta _{f_{X}}}\overline F_{T}(t_{n}|x_{0}))\).

By rearranging terms we obtain the following bound for T3

$$\begin{array}{@{}rcl@{}} |T_{3}| & \le & f_{Y}(t_{n}|x_{0}) \overline F_{C}(t_{n}|x_{0})f_{X}(x_{0}) \times \\ & & {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s}\left( \ln \frac{y}{t_{n}} \right)^{s^{\prime}} \frac{f_{Y}(y|x_{0})}{f_{Y}(t_{n}|x_{0})}\frac{\overline F_{C}(y|x_{0})}{\overline F_{C}(t_{n}|x_{0})}\left|\frac{\overline F_{C}(y|x_{0}-h_{n}z)}{\overline F_{C}(y|x_{0})}-1 \right| dy dz, \qquad \end{array} $$
(9)

and from Assumption \((\mathcal H)\), for n large, and some constants M1, M2 and M3,

$$\begin{array}{@{}rcl@{}} \left|\frac{\overline F_{C}(y|x_{0}-h_{n}z)}{\overline F_{C}(y|x_{0})}-1 \right| &\le& M_{1} \left( h_{n}^{\eta_{A_{C}}}+y^{M_{2}h_{n}^{\eta_{\gamma_{C}}}}h_{n}^{\eta_{\gamma_{C}}}\ln y + |\delta_{C}(y|x_{0})|h_{n}^{\eta_{B_{C}}} \right. \\ & & + \left. |\delta_{C}(y|x_{0})|y^{M_{3} h_{n}^{\eta_{\varepsilon_{C}}}}h_{n}^{\eta_{\varepsilon_{C}}} \ln y \right). \end{array} $$

Plugging the above inequality into (9), and computing integrals similar to those encountered above yields

$$\begin{array}{@{}rcl@{}} T_{3} = O \left( \overline F_{T}(t_{n}|x_{0}) (h_{n}^{\eta_{A_{C}}}+h_{n}^{\eta_{\gamma_{C}}}\ln t_{n} + \delta_{C}(t_{n}|x_{0})h_{n}^{\eta_{B_{C}}}+\delta_{C}(t_{n}|x_{0})h_{n}^{\eta_{\varepsilon_{C}}}\ln t_{n} ) \right). \end{array} $$

Using the Hölder condition on fX one easily verifies that T4 is of smaller order than T3.

As for T5, we can write

$$\begin{array}{@{}rcl@{}} |T_{5}| & \le & f_{Y}(t_{n}|x_{0}) \overline F_{C}(t_{n}|x_{0})f_{X}(x_{0}) \times \\ & & {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} \left( \frac{y}{t_{n}} \right)^{s}\left( \ln \frac{y}{t_{n}} \right)^{s^{\prime}} \frac{f_{Y}(y|x_{0})}{f_{Y}(t_{n}|x_{0})}\frac{\overline F_{C}(y|x_{0})}{\overline F_{C}(t_{n}|x_{0})}\left|\frac{f_{Y}(y|x_{0}-h_{n}z)}{ f_{Y}(y|x_{0})}-1 \right| dy dz, \end{array} $$

which, combined with the inequality

$$\begin{array}{@{}rcl@{}} \left|\frac{f_{Y}(y|x_{0}-h_{n}z)}{f_{Y}(y|x_{0})}-1 \right| &\le& M_{1} \left( h_{n}^{\eta_{A_{Y}}}+y^{M_{2}h_{n}^{\eta_{\gamma_{Y}}}}h_{n}^{\eta_{\gamma_{Y}}}\ln y + |\delta_{Y}(y|x_{0})|h_{n}^{\eta_{B_{Y}}} \right. \\ & &+ \left. |\delta_{Y}(y|x_{0})|y^{M_{3} h_{n}^{\eta_{\varepsilon_{Y}}}}h_{n}^{\eta_{\varepsilon_{Y}}} \ln y \right), \end{array} $$

valid for n large, where M1, M2 and M3 are some constants, leads to

$$\begin{array}{@{}rcl@{}} T_{5} = O \left (\overline F_{T}(t_{n}|x_{0}) (h_{n}^{\eta_{A_{Y}}}+h_{n}^{\eta_{\gamma_{Y}}}\ln t_{n} + \delta_{Y}(t_{n}|x_{0})h_{n}^{\eta_{B_{Y}}}+\delta_{Y}(t_{n}|x_{0})h_{n}^{\eta_{\varepsilon_{Y}}}\ln t_{n} ) \right). \end{array} $$

After tedious calculations, but essentially involving integrals similar to the ones above, one can verify that T6, T7 and T8, are of smaller order than terms that were already encountered before.

Collecting the terms then establishes Theorem 1.

1.3 Proof of Corollary 1

We only give the details for \(T_{n}^{(1)}(K,s,s^{\prime }|x_{0})/(\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}))\). The convergence in probability of \(T_{n}^{(2)}(K,s,s^{\prime }|x_{0})/(\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}))\) can be established analogously.From Theorem 1 we have immediately

$$\begin{array}{@{}rcl@{}} \mathbb E \left( \frac{T_{n}^{(1)}(K,s,s^{\prime}|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) \to \frac{{\Gamma}(s^{\prime}+ 1)\gamma_{T}^{s^{\prime}}(x_{0})}{(1-s\gamma_{T}(x_{0}))^{s^{\prime}+ 1}}. \end{array} $$

By independence

$$\begin{array}{@{}rcl@{}} \mathbb{V}ar\left( \frac{T_{n}^{(1)}(K,s,s^{\prime}|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) = \frac{\mathbb{V}ar \left( K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace} \right)}{n \overline {F_{T}^{2}}(t_{n}|x_{0}){f_{X}^{2}}(x_{0})}. \end{array} $$

Using \((\mathcal K)\) and Theorem 1 we obtain

$$\begin{array}{@{}rcl@{}} \lefteqn{\mathbb{V}ar \left( K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace} \right)} \\ &= & \frac{\|K\|_{2}^{2}}{{h_{n}^{d}}} \mathbb E \left( \frac{1}{\|K\|_{2}^{2}{h_{n}^{d}}} K^{2}\left( \frac{x_{0}-X}{h_{n}} \right)\left( \frac{T}{t_{n}} \right)^{2s} \left( \ln_{+} \frac{T}{t_{n}} \right)^{2s^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace} \right)+O(\overline {F_{T}^{2}}(t_{n}|x_{0})) \\ &= & \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})\frac{\|K\|_{2}^{2}{\Gamma}(2s^{\prime}+ 1)\gamma_{T}^{2s^{\prime}}(x_{0})}{{h_{n}^{d}}(1-2s\gamma_{T}(x_{0}))^{2s^{\prime}+ 1}}(1+o(1)), \end{array} $$

and thus

$$\begin{array}{@{}rcl@{}} \mathbb{V}ar\left( \frac{T_{n}^{(1)}(K,s,s^{\prime}|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) = \frac{1}{n{h_{n}^{d}}\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \frac{\|K\|_{2}^{2}{\Gamma}(2s^{\prime}+ 1)\gamma_{T}^{2s^{\prime}}(x_{0})}{(1-2s\gamma_{T}(x_{0}))^{2s^{\prime}+ 1}}(1+o(1)), \end{array} $$

which tends to zero under the assumption \(n{h_{n}^{d}}\overline F_{T}(t_{n}|x_{0})\to \infty \). This establishes (6).

1.4 Proof of Theorem 2

To prove convergence in probability we adapt the argument of the proof of Theorem 5.1 in Chapter 6 of Lehmann and Casella (1998) to our context. Let \(\widetilde \ell (\gamma _{Y},\delta _{Y};\beta ) := \ell (\gamma _{Y},\delta _{Y};\beta )/(\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}))\), and similarly for \(\widetilde \ell _{j}\) and \(\widetilde \ell _{jk}\). We denote by jkl, j, k, l = 1, 2, the third order derivatives of , and by \(\widetilde \ell _{jkl}\) their rescaled version. As a first step we show that for any ν > 0 sufficiently small

$$\begin{array}{@{}rcl@{}} \mathbb P_{(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))}(\widetilde \ell(\gamma_{Y},\delta_{Y};\beta)&&< \widetilde \ell(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \text{ for all } (\gamma_{Y},\delta_{Y}) \\ & &\text{ on the surface of } Q_{ \nu,n} ) \to 1 \end{array} $$

as n, where Qν, n denotes the circle centered at \((\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))\) with radius ν, and Qν, n is assumed to belong to the parameter space. By Taylor’s theorem

$$\begin{array}{@{}rcl@{}} \lefteqn{\widetilde \ell(\gamma_{Y},\delta_{Y};\beta)- \widetilde \ell(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) } \\ & =& \widetilde \ell_{1}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0})) \\ & &+ \widetilde \ell_{2}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0})) \\ & & + \frac{1}{2} [\widetilde \ell_{11}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))^{2} \end{array} $$
$$\begin{array}{@{}rcl@{}} & &+\widetilde \ell_{22}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0}))^{2} \\ & & + 2 \widetilde \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0})) ] \\ & & + \frac{1}{6} [ \widetilde \ell_{111}(\widetilde \gamma_{Y},\widetilde \delta_{Y};\beta)(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))^{3}+ \widetilde \ell_{222}(\widetilde \gamma_{Y},\widetilde \delta_{Y};\beta)(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0}))^{3} \\ & &+ 3\widetilde \ell_{112}(\widetilde \gamma_{Y},\widetilde \delta_{Y};\beta)(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))^{2}(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0})) \\ & &+ 3\widetilde \ell_{122}(\widetilde \gamma_{Y},\widetilde \delta_{Y};\beta)(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0}))^{2}] \\ & =:& S_{1}+S_{2}+S_{3}, \end{array} $$

where \((\widetilde \gamma _{Y},\widetilde \delta _{Y})\) is a point on the line segment connecting (γY, δY) and \((\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))\).

After linearisation, \(\widetilde \ell _{1}\) and \(\widetilde \ell _{2}\) can be written in terms of \(T_{n}^{(1)}(K,s,s^{\prime }|x_{0})\) and \(T_{n}^{(2)}(K,s,s^{\prime }|x_{0})\) as follows

$$\begin{array}{@{}rcl@{}} \widetilde \ell_{1}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = & -\frac{1}{\gamma_{Y}^{(0)}(x_{0})} \frac{T_{n}^{(2)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} + \frac{1}{\gamma_{Y}^{(0)2}(x_{0})} \frac{T_{n}^{(1)}(K,0,1|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ & & + \delta_{Y}^{(0)}(t_{n}|x_{0})\frac{1}{\gamma_{Y}^{(0)2}(x_{0})} \left[\frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}- \frac{T_{n}^{(1)}(K,-\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}\right] \\ & & + O_{\mathbb P}(\delta_{Y}^{(0)2}(t_{n}|x_{0})), \end{array} $$
(10)
$$\begin{array}{@{}rcl@{}} \widetilde \ell_{2}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = & \beta \frac{T_{n}^{(2)}(K,-\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}- \frac{1}{\gamma_{Y}^{(0)}(x_{0})} \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ & &+ \frac{1}{\gamma_{Y}^{(0)}(x_{0})} \frac{T_{n}^{(1)}(K,-\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ & & + \delta_{Y}^{(0)}(t_{n}|x_{0}) \left[ (2\beta-\beta^{2}) \frac{T_{n}^{(2)}(K,-2\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & &\left.-2 \beta \frac{T_{n}^{(2)}(K,-\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ && \left. +\frac{1}{\gamma_{Y}^{(0)}(x_{0})} \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}+\frac{1}{\gamma_{Y}^{(0)}(x_{0})} \frac{T_{n}^{(1)}(K,-2\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & & \left.-\frac{2}{\gamma_{Y}^{(0)}(x_{0})} \frac{T_{n}^{(1)}(K,-\beta,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right] + O_{\mathbb P}(\delta_{Y}^{(0)2}(t_{n}|x_{0})). \end{array} $$
(11)

Using Corollary 1 we have

$$\begin{array}{@{}rcl@{}} \widetilde \ell_{1}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \overset{\mathbb P}{\to} 0 \text{ and } \widetilde \ell_{2}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \overset{\mathbb P}{\to} 0, \end{array} $$

so, for any ν > 0 we have that \(| \widetilde \ell _{j}(\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0});\beta ) | < \nu ^{2}\), j = 1, 2, with probability tending to 1, and hence on Qν, n, |S1| < 2ν3 with probability tending to 1.Using again linearisation and Corollary 1 one has

$$\begin{array}{@{}rcl@{}} \widetilde \ell_{11}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & \overset{\mathbb P}{\to} & \dot{\ell}_{11} := -\frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)3}(x_{0})}, \\ \widetilde \ell_{22}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & \overset{\mathbb P}{\to} & \dot{\ell}_{22} := -\frac{\beta^{2}\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1 + 2\beta \gamma_{T}^{(0)}(x_{0}))}, \\ \widetilde \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & \overset{\mathbb P}{\to} & \dot{\ell}_{12} := \frac{\beta\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)2}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0}))}. \end{array} $$

Now write

$$\begin{array}{@{}rcl@{}} 2S_{2} & = & \dot{\ell}_{11}(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))^{2}+\dot{\ell}_{22}(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0}))^{2}+ 2\dot{\ell}_{12}(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0})) \\ & & + (\widetilde \ell_{11}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)-\dot \ell_{11})(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))^{2} \\ & & + (\widetilde \ell_{22}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)-\dot \ell_{22})(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0}))^{2} \\ & & + 2(\widetilde \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta)-\dot \ell_{12})(\gamma_{Y}-\gamma_{Y}^{(0)}(x_{0}))(\delta_{Y}-\delta_{Y}^{(0)}(t_{n}|x_{0})). \end{array} $$

It can easily be verified that the first three terms are a deterministic negative definite quadratic form in \(\gamma _{Y}-\gamma _{Y}^{(0)}(x_{0})\) and \(\delta _{Y}-\delta _{Y}^{(0)}(t_{n}|x_{0})\). By the spectral decomposition this quadratic form can be rewritten as \(\lambda _{1}{\xi _{1}^{2}}+\lambda _{2} {\xi _{2}^{2}}\), where λ2λ1 < 0 are the eigenvalues and ξ1 and ξ2 are the orthogonal transformations of \(\gamma _{Y}-\gamma _{Y}^{(0)}(x_{0})\) and \(\delta _{Y}-\delta _{Y}^{(0)}(t_{n}|x_{0})\). Note that in the new coordinate system Qν, n becomes \({\xi _{1}^{2}}+{\xi _{2}^{2}}=\nu ^{2}\). Thus \(\lambda _{1}{\xi _{1}^{2}}+\lambda _{2} {\xi _{2}^{2}} \le \lambda _{1} \nu ^{2}\). For the random terms of S2 we use the above convergences in probability and conclude that the random part is in absolute value less than 4ν3 with probability tending to 1. Combined, we have that there exists c > 0 and ν0 > 0 such that for ν < ν0 one has S2 < −cν2 with probability tending to 1.

For the third order derivatives one has that \(|\widetilde \ell _{jkl}(\gamma _{Y},\delta _{Y};\beta )| \le M_{jkl}(\bold V)\) for (γY, δY) ∈ Qν, n, where \(\bold V := \lbrace (T_{i},X_{i},{\mathrm {1}\hskip -2.2pt\mathrm {l}}_{\lbrace Y_{i} \le C_{i} \rbrace }), i = 1,\ldots ,n \rbrace \), and \(M_{jkl} \overset {\mathbb P}{\to } m_{jkl}\), bounded, j, k, l = 1, 2. These derivations are straightforward and for brevity omitted from the paper. We have then that, with probability tending to one, \(|\widetilde \ell _{jkl}(\widetilde \gamma _{Y},\widetilde \delta _{Y};\beta )| < 2m_{jkl}\) and thus |S3| < eν3 on Qν, n, where \(e = 1/3 {\sum }_{j = 1}^{2}{\sum }_{k = 1}^{2}{\sum }_{l = 1}^{2}m_{jkl}\).

Combining the above, with probability tending to 1,

$$\begin{array}{@{}rcl@{}} \max_{Q_{\nu,n}} (S_{1}+S_{2}+S_{3}) < -c\nu^{2}+(2+e)\nu^{3}, \end{array} $$

which is negative if ν < c/(2 + e).

To establish existence and convergence in probability we adjust the line of argumentation in the proof of Theorem 3.7 in Chapter 6 of Lehmann and Casella (1998). For ν > 0, small enough such that Qν, n is a subset of the parameter space, consider

$$\begin{array}{@{}rcl@{}} S_{n}(\nu) := \lbrace \boldsymbol v : \widetilde \ell(\gamma_{Y},\delta_{Y};\beta) < \widetilde \ell(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \text{ for all } (\gamma_{Y},\delta_{Y}) \text{ on the surface of } Q_{\nu,n}\rbrace. \end{array} $$

From the above we have that \(\mathbb P_{(\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))}(S_{n}(\nu )) \to 1\) for any such ν, and hence there exists a sequence νn 0 such that \(\mathbb P_{(\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))}(S_{n}(\nu _{n})) \to 1\) as n. By the differentiability of \(\widetilde \ell (\gamma _{Y},\delta _{Y};\beta )\) we have that vSn(νn) implies that there exists a point \((\widehat \gamma _{Y,n}(\nu _{n}),\widehat \delta _{Y,n}(\nu _{n})) \in Q_{\nu ,n}\) for which \(\widetilde \ell (\gamma _{Y},\delta _{Y};\beta )\) attains a local maximum, and thus \(\widetilde \ell _{j}(\widehat \gamma _{Y,n}(\nu _{n}),\widehat \delta _{Y,n}(\nu _{n});\beta )= 0\), j = 1, 2. Now let \((\widehat \gamma _{Y,n}^{*}(x_{0}),\widehat \delta _{Y,n}^{*}(t_{n}|x_{0})) := (\widehat \gamma _{Y,n}(\nu _{n}),\widehat \delta _{Y,n}(\nu _{n}))\) for vSn(νn) and arbitrary otherwise. Clearly

$$\begin{array}{@{}rcl@{}} \lefteqn{\mathbb P_{(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))}(\widetilde \ell_{1}(\widehat \gamma_{Y,n}^{*}(x_{0}),\widehat \delta_{Y,n}^{*}(t_{n}|x_{0}));\beta)= 0,\widetilde \ell_{2}(\widehat \gamma_{Y,n}^{*}(x_{0}),\widehat \delta_{Y,n}^{*}(t_{n}|x_{0}));\beta)= 0)} \\ & &\hspace{6.5cm} \ge \mathbb P_{(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))}(S_{n}(\nu_{n})) \to 1, \end{array} $$

as n. Thus with probability tending to 1 there exists a sequence of solutions to the likelihood equations. Also, for any fixed ν > 0 and n sufficiently large, and denoting by d the usual Euclidean distance,

$$\begin{array}{@{}rcl@{}} \lefteqn{\mathbb P_{(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))}(d((\widehat \gamma_{Y,n}^{*}(x_{0}),\widehat \delta_{Y,n}^{*}(t_{n}|x_{0})),(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))) < \nu)} \\ & & \ge \mathbb P_{(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))}(d((\widehat \gamma_{Y,n}^{*}(x_{0}),\widehat \delta_{Y,n}^{*}(t_{n}|x_{0})),(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))) < \nu_{n}) \\ & & \ge \mathbb P_{(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0}))}(S_{n}(\nu_{n})) \to 1, \end{array} $$

which establishes the convergence in probability of the sequence \((\widehat \gamma _{Y,n}^{*}(x_{0}),\widehat \delta _{Y,n}^{*}(t_{n}|x_{0}))\).

1.5 Proof of Theorem 3

The result of the theorem will be established by using the Cramér-Wold device (see, e.g., van der Vaart 1998, p.16). Take \(\theta :=(\theta _{1},\ldots ,\theta _{J+L})^{T} \in \mathbb R^{J+L}\). Then

$$\begin{array}{@{}rcl@{}} \lefteqn{\theta^{T} r_{n} (\mathbb T_{n}-\mathbb E(\mathbb T_{n})) } \\ &=& \sum\limits_{i = 1}^{n} \left\lbrace \sum\limits_{j = 1}^{J} \theta_{j} \left( \frac{{h_{n}^{d}}}{n \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X_{i}) \left( \frac{T_{i}}{t_{n}} \right)^{s_{j}} \left( \ln_{+} \frac{T_{i}}{t_{n}} \right)^{s_{j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T_{i} > t_{n}\rbrace} \right. \\ & & + \sum\limits_{j = 1}^{L}\theta_{J+j}\left( \frac{{h_{n}^{d}}}{n \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2}K_{h_{n}}(x_{0}-X_{i})\left( \frac{T_{i}}{t_{n}} \right)^{s_{J+j}} \left( \ln_{+} \frac{T_{i}}{t_{n}} \right)^{s_{J+j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace{Y_{i} \le C_{i},T_{i}>t_{n}\rbrace}} \\ & & -\mathbb E \left[ \sum\limits_{j = 1}^{J} \theta_{j} \left( \frac{{h_{n}^{d}}}{n \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{j}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace} \right. \\ & & + \left. \left. \sum\limits_{j = 1}^{L}\theta_{J+j}\left( \frac{{h_{n}^{d}}}{n \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2}K_{h_{n}}(x_{0}-X)\left( \frac{T}{t_{n}} \right)^{s_{J+j}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{J+j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace{Y \le C,T>t_{n}\rbrace}} \right] \right\rbrace \\ & =:& \sum\limits_{i = 1}^{n} W_{i}. \end{array} $$

Note that \(\mathbb Var(W_{1}) = \theta ^{T}\mathbb A \theta /n\), where \(\mathbb A\) has elements

$$\begin{array}{@{}rcl@{}} \mathbb A_{jk} &:= & \mathbb Cov \left( \left( \frac{{h_{n}^{d}}}{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{j}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace}, \right. \\ & & \left. \left( \frac{{h_{n}^{d}}}{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{k}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{k}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace} \right), j,k \in \lbrace 1,\ldots,J \rbrace, \\ \mathbb A_{jk} &:= & \mathbb Cov \left( \left( \frac{{h_{n}^{d}}}{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{j}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y \le C, T > t_{n}\rbrace}, \right. \\ & & \left. \left( \frac{{h_{n}^{d}}}{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{k}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{k}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y \le C, T > t_{n}\rbrace} \right), \\ & & \hspace{7.4cm} j,k \in \lbrace J + 1,\ldots,J+L \rbrace, \\ \mathbb A_{jk} &:= & \mathbb Cov \left( \left( \frac{{h_{n}^{d}}}{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{j}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{j}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace T > t_{n}\rbrace}, \right. \\ & & \left. \left( \frac{{h_{n}^{d}}}{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right)^{1/2} K_{h_{n}}(x_{0}-X) \left( \frac{T}{t_{n}} \right)^{s_{k}} \left( \ln_{+} \frac{T}{t_{n}} \right)^{s_{k}^{\prime}}{\mathrm{1}\hskip-2.2pt\mathrm{l}}_{\lbrace Y \le C , T > t_{n}\rbrace} \right), \\ & & \hspace{5.4cm} j \in \lbrace 1,\ldots,J \rbrace, k \in \lbrace J + 1,\ldots,J+L \rbrace. \end{array} $$

Using Theorem 1 we obtain that \(\mathbb A_{jk} = {\Sigma }_{jk}(1+o(1))\), j, k ∈{1,…, J + L}, and thus \(\mathbb Var(\theta ^{T} r_{n} (\mathbb T_{n}-\mathbb E(\mathbb T_{n}))) = \theta ^{T}{\Sigma }\theta (1+o(1))\).

In order to establish the weak convergence to a Gaussian random variable we need to verify the Lyapounov condition (see, e.g., Billingsley 1995, p. 362), which simplifies in our setting to showing that \(\lim _{n \to \infty } n\mathbb E(|W_{1}|^{3})= 0\). To this aim, note that W1 is of the form \(V-\mathbb E(V)\), leading to the inequality

$$\begin{array}{@{}rcl@{}} \mathbb E (|W_{1}|^{3}) \le \mathbb E(|V|^{3})+ 3 \mathbb E(V^{2})\mathbb E(|V|)+ 4(\mathbb E(|V|))^{3}. \end{array} $$

Again using the result from Theorem 1, we obtain the following orders

$$\begin{array}{@{}rcl@{}} \mathbb E(|V|^{3}) & = & O \left( \frac{1}{n^{3/2}\sqrt{{h_{n}^{d}} \overline F_{T}(t_{n}|x_{0}) }} \right), \\ \mathbb E(V^{2}) \mathbb E(|V|) & = & O\left( \frac{\sqrt{{h_{n}^{d}}\overline F_{T}(t_{n}|x_{0})}}{n^{3/2}} \right), \\ (\mathbb E(|V|))^{3} & = & O \left( \left( \frac{{h_{n}^{d}} \overline F_{T}(t_{n}|x_{0})}{n} \right)^{3/2} \right), \end{array} $$

so that \( n\mathbb E(|W_{1}|^{3}) \to 0\) under our assumption rn.

1.6 Proof of Theorem 4

We apply a Taylor series expansion of \(\widetilde \ell _{j}(\widehat \gamma _{Y,n}(x_{0}),\widehat \delta _{Y,n}(t_{n}|x_{0});\beta )\), j = 1, 2, around \((\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))\) and obtain, after rearranging terms

$$\begin{array}{@{}rcl@{}} \lefteqn{-r_{n} \left[ \begin{array}{c} \widetilde \ell_{1}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \\ \widetilde \ell_{2}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \end{array} \right] } \\ & &{\kern8pt} =\left[ \begin{array}{cc} \check \ell_{11}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & \check \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \\ \check \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & \check \ell_{22}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \end{array} \right] \\ & &{\kern20pt} \times\left[ \begin{array}{c} r_{n}(\widehat \gamma_{Y,n}(x_{0})-\gamma_{Y}^{(0)}(x_{0})) \\ r_{n}(\widehat \delta_{Y,n}(t_{n}|x_{0})-\delta_{Y}^{(0)}(t_{n}|x_{0})) \end{array} \right]\qquad \end{array} $$
(12)

where

$$\begin{array}{@{}rcl@{}} \check \ell_{11}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = & \widetilde \ell_{11}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \\ & &+\frac{1}{2}[ \widetilde \ell_{111}(\widetilde \gamma_{Y,n},\widetilde \delta_{Y,n};\beta)(\widehat \gamma_{Y,n}(x_{0})-\gamma_{Y}^{(0)}(x_{0})) \\ & & + \widetilde \ell_{112}(\widetilde \gamma_{Y,n},\widetilde \delta_{Y,n};\beta)(\widehat \delta_{Y,n}(t_{n}|x_{0})-\delta_{Y}^{(0)}(t_{n}|x_{0}))], \\ \check \ell_{22}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = & \widetilde \ell_{22}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \\ & &+\frac{1}{2}[ \widetilde \ell_{122}(\widetilde \gamma_{Y,n},\widetilde \delta_{Y,n};\beta)(\widehat \gamma_{Y,n}(x_{0})-\gamma_{Y}^{(0)}(x_{0})) \\ & & + \widetilde \ell_{222}(\widetilde \gamma_{Y,n},\widetilde \delta_{Y,n};\beta)(\widehat \delta_{Y,n}(t_{n}|x_{0})-\delta_{Y}^{(0)}(t_{n}|x_{0}))], \\ \check \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = & \widetilde \ell_{12}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \\ & &+\frac{1}{2}[ \widetilde \ell_{112}(\widetilde \gamma_{Y,n},\widetilde \delta_{Y,n};\beta)(\widehat \gamma_{Y,n}(x_{0})-\gamma_{Y}^{(0)}(x_{0})) \\ & & + \widetilde \ell_{122}(\widetilde \gamma_{Y,n},\widetilde \delta_{Y,n};\beta)(\widehat \delta_{Y,n}(t_{n}|x_{0})-\delta_{Y}^{(0)}(t_{n}|x_{0}))], \end{array} $$

and \((\widetilde \gamma _{Y,n},\widetilde \delta _{Y,n})\) is a random value on the line segment between \((\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))\) and \((\widehat \gamma _{Y,n}(x_{0}),\widehat \delta _{Y,n}(t_{n}|x_{0}))\).

We start by analysing the left hand side of (12). As a first step we rewrite the expansion for \(\mathbb E(T_{n}^{(2)}(K,s,s^{\prime }|x_{0}))\) in terms of \(\delta _{T}^{(0)}(t_{n}|x_{0})\).

  • Case 1: \(\delta _{Y}^{(0)}(t_{n}|x_{0})/ \delta _{C}^{(0)}(t_{n}|x_{0}) \to \pm \infty \)

    $$\begin{array}{@{}rcl@{}} \lefteqn{\mathbb E(T_{n}^{(2)}(K,s,s^{\prime}|x_{0})) } \\ &=& \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}) {\Gamma}(s^{\prime}+ 1) \frac{\gamma_{T}^{(0)(s^{\prime}+ 1)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \left\lbrace \frac{1}{(1-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right. \\ & & +\delta_{T}^{(0)}(t_{n}|x_{0})\frac{\gamma_{Y}^{(0)}(x_{0})}{\gamma_{T}^{(0)}(x_{0})} \left[ \frac{1}{\gamma_{Y}^{(0)}(x_{0})} \left( \frac{1}{(1+\beta_{T}^{(0)}(x_{0}) \gamma_{T}^{(0)}(x_{0})-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right.\right. \\ & &\left.\left.-\frac{1}{(1-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right) \right. \\ & & \left. +\frac{\beta_{T}^{(0)}(x_{0})}{(1+\beta_{T}^{(0)}(x_{0}) \gamma_{T}^{(0)}(x_{0})-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) + \left. O(h_{n}^{\eta_{f_{X}}\wedge\eta_{A_{Y}}\wedge \eta_{A_{C}} }) \right. \\ & &\left.+ O(h_{n}^{\eta_{\gamma_{Y}}\wedge\eta_{\gamma_{C}}}\ln t_{n})\right\rbrace. \end{array} $$
  • Case 2: \( \delta _{Y}^{(0)}(t_{n}|x_{0})/ \delta _{C}^{(0)}(t_{n}|x_{0}) \to 0\)

    $$\begin{array}{@{}rcl@{}} \lefteqn{\mathbb E(T_{n}^{(2)}(K,s,s^{\prime}|x_{0})) } \\ &=& \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}) {\Gamma}(s^{\prime}+ 1) \frac{\gamma_{T}^{(0)(s^{\prime}+ 1)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \left\lbrace \frac{1}{(1-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right. \\ & & + \frac{\delta_{T}^{(0)}(t_{n}|x_{0})}{\gamma_{T}^{(0)}(x_{0})}\left [ \frac{1}{(1+\beta_{T}^{(0)}(x_{0}) \gamma_{T}^{(0)}(x_{0})-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right. \\ & &\left.-\frac{1}{(1-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) \\ & & + \left. O(h_{n}^{\eta_{f_{X}}\wedge\eta_{A_{Y}}\wedge \eta_{A_{C}} }) + O(h_{n}^{\eta_{\gamma_{Y}}\wedge\eta_{\gamma_{C}}}\ln t_{n})\right\rbrace. \end{array} $$
  • Case 3: \(\delta _{Y}^{(0)}(t_{n}|x_{0}) / \delta _{C}^{(0)}(t_{n}|x_{0}) \to a\), where 0 < |a| <

    $$\begin{array}{@{}rcl@{}} \lefteqn{\mathbb E(T_{n}^{(2)}(K,s,s^{\prime}|x_{0})) } \\ &=& \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}) {\Gamma}(s^{\prime}+ 1) \frac{\gamma_{T}^{(0)(s^{\prime}+ 1)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \left\lbrace \frac{1}{(1-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right. \\ & & +\delta_{T}^{(0)}(t_{n}|x_{0})\frac{\gamma_{Y}^{(0)}(x_{0})\gamma_{C}^{(0)}(x_{0})a}{\gamma_{T}^{(0)}(x_{0})(\gamma_{Y}^{(0)}(x_{0})+a\gamma_{C}^{(0)}(x_{0}))} \left[ \left( \frac{1}{\gamma_{Y}^{(0)}(x_{0})}+ \frac{1}{\gamma_{C}^{(0)}(x_{0})a} \right) \times \right. \\ & & \left( \frac{1}{(1+\beta_{T}^{(0)}(x_{0}) \gamma_{T}^{(0)}(x_{0})-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} -\frac{1}{(1-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right) \\ & & \left. +\frac{\beta_{T}^{(0)}(x_{0})}{(1+\beta_{T}^{(0)}(x_{0}) \gamma_{T}^{(0)}(x_{0})-s\gamma_{T}^{(0)}(x_{0}))^{s^{\prime}+ 1}} \right](1+o(1)) + \left. O(h_{n}^{\eta_{f_{X}}\wedge\eta_{A_{Y}}\wedge \eta_{A_{C}} }) \right. \\ & &\left.+ O(h_{n}^{\eta_{\gamma_{Y}}\wedge\eta_{\gamma_{C}}}\ln t_{n})\right\rbrace. \end{array} $$

Now let

$$\begin{array}{@{}rcl@{}} \mathbb U_{n} := \frac{1}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \left[ \begin{array}{c} T_{n}^{(1)}(K,0,0|x_{0}) \\ T_{n}^{(1)}(K,-\beta,0|x_{0}) \\ T_{n}^{(1)}(K,0,1|x_{0}) \\ T_{n}^{(2)}(K,0,0|x_{0}) \\ T_{n}^{(2)}(K,-\beta,0|x_{0}) \end{array} \right], \mu := \left[ \begin{array}{c} 1 \\ \frac{1}{1+\beta \gamma_{T}^{(0)}(x_{0})} \\ \gamma_{T}^{(0)}(x_{0}) \\ \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \\ \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta \gamma_{T}^{(0)}(x_{0}))} \end{array} \right]. \end{array} $$

From Theorem 3 we have

$$\begin{array}{@{}rcl@{}} r_{n}(\mathbb U_{n}-\mu) \leadsto N(\lambda \widetilde \mu,{\Sigma}) \end{array} $$

where Σ is a (5 × 5) symmetric matrix with elements

$$\begin{array}{llllllll} &\sigma_{11} := \|K\|_{2}^{2} && \sigma_{12} := \|K\|_{2}^{2} \frac{1}{1+\beta \gamma_{T}^{(0)}(x_{0})} \\ &\sigma_{22} := \|K\|_{2}^{2} \frac{1}{1 + 2\beta\gamma_{T}^{(0)}(x_{0})} && \sigma_{13} := \|K\|_{2}^{2} \gamma_{T}^{(0)}(x_{0}) \\ &\sigma_{33} := \|K\|_{2}^{2} 2 \gamma_{T}^{(0)2}(x_{0}) && \sigma_{14} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \\ &\sigma_{44} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} && \sigma_{15} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0}))} \\ &\sigma_{55} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1 + 2\beta\gamma_{T}^{(0)}(x_{0}))}&& \sigma_{34} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)2}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})}\\ &\sigma_{23} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{(1+\beta\gamma_{T}^{(0)}(x_{0}))^{2}} &&\sigma_{35} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)2}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0}))^{2}}\\ &\sigma_{24} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0}))} &&\sigma_{45} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0}))}\\ &\sigma_{25} := \|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1 + 2\beta\gamma_{T}^{(0)}(x_{0}))}&& \end{array}$$

and \(\widetilde \mu \) is a vector with elements

$$\begin{array}{@{}rcl@{}} \widetilde \mu_{1} & := & 0, \\ \widetilde \mu_{2} & := & \frac{\beta\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{(1+\beta \gamma_{T}^{(0)}(x_{0}))(1+\beta \gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0}))}, \\ \widetilde \mu_{3} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{1+\beta_{T}^{(0)}(x_{0}) \gamma_{T}^{(0)}(x_{0})}, \end{array} $$

and

  • Case 1: \(\delta _{Y}^{(0)}(t_{n}|x_{0})/ \delta _{C}^{(0)}(t_{n}|x_{0}) \to \pm \infty \)

    $$\begin{array}{@{}rcl@{}} \widetilde \mu_{4} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})) } + \frac{\beta_{T}^{(0)}(x_{0})}{1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0}) }, \\ \widetilde \mu_{5} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0})) (1+\beta\gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0}))} \\ & &+\frac{\beta_{T}^{(0)}(x_{0})}{1+\beta\gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0}) }, \end{array} $$
  • Case 2: \( \delta _{Y}^{(0)}(t_{n}|x_{0})/ \delta _{C}^{(0)}(t_{n}|x_{0}) \to 0\)

    $$\begin{array}{@{}rcl@{}} \widetilde \mu_{4} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})) } , \\ \widetilde \mu_{5} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0})) (1+\beta\gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0}))}, \end{array} $$
  • Case 3: \(\delta _{Y}^{(0)}(t_{n}|x_{0}) / \delta _{C}^{(0)}(t_{n}|x_{0}) \to a\), where 0 < |a| <

    $$\begin{array}{@{}rcl@{}} \widetilde \mu_{4} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})) }+ \frac{a\beta_{T}^{(0)}(x_{0})\gamma_{C}^{(0)}(x_{0})}{(\gamma_{Y}^{(0)}(x_{0})+a\gamma_{C}^{(0)}(x_{0}) )(1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})) } , \\ \widetilde \mu_{5} & := & -\frac{\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0})) (1+\beta\gamma_{T}^{(0)}(x_{0}) + \beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0}))} \\ & & + \frac{a\beta_{T}^{(0)}(x_{0})\gamma_{C}^{(0)}(x_{0})}{(\gamma_{Y}^{(0)}(x_{0})+a\gamma_{C}^{(0)}(x_{0}) )(1+\beta\gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})) } . \end{array} $$

Based on (10) and (11) we can write

$$\begin{array}{@{}rcl@{}} r_{n} \widetilde \ell_{1}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = &\xi_{1} r_{n}(\mathbb U_{n}-\mu)+r_{n} \delta_{Y}^{(0)}(t_{n}|x_{0})\frac{\beta \gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)2}(x_{0})(1+\beta \gamma_{T}^{(0)}(x_{0}))}(1+o_{\mathbb P}(1)), \\ r_{n} \widetilde \ell_{2}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) & = & \xi_{2} r_{n}(\mathbb U_{n}-\mu)-r_{n} \delta_{Y}^{(0)}(t_{n}|x_{0}) \frac{\beta^{2} \gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1 + 2\beta \gamma_{T}^{(0)}(x_{0}))}(1+o_{\mathbb P}(1)), \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \xi_{1} & := & \left[0,0,\frac{1}{\gamma_{Y}^{(0)2}(x_{0})},-\frac{1}{\gamma_{Y}^{(0)}(x_{0})},0 \right], \\ \xi_{2} & := & \left[-\frac{1}{\gamma_{Y}^{(0)}(x_{0})},\frac{1}{\gamma_{Y}^{(0)}(x_{0})},0,0,\beta \right], \end{array} $$

and thus

$$\begin{array}{@{}rcl@{}} r_{n} \left[ \begin{array}{c} \widetilde \ell_{1}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \\ \widetilde \ell_{2}(\gamma_{Y}^{(0)}(x_{0}),\delta_{Y}^{(0)}(t_{n}|x_{0});\beta) \end{array} \right] \leadsto N (\lambda \kappa, {\Delta}), \end{array} $$

where Δ is a symmetric (2 × 2) matrix with elements

$$\begin{array}{@{}rcl@{}} {\Delta}_{11} & := &\|K\|_{2}^{2} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)3}(x_{0})}, \\ {\Delta}_{22} & := &\|K\|_{2}^{2} \frac{\beta^{2}\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(1 + 2\beta\gamma_{T}^{(0)}(x_{0}))}, \\ {\Delta}_{12} & := & - \|K\|_{2}^{2}\frac{\beta \gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)2}(x_{0})(1+\beta\gamma_{T}^{(0)}(x_{0}))}, \end{array} $$

and κ has elements

  • Case 1: \(\delta _{Y}^{(0)}(t_{n}|x_{0})/ \delta _{C}^{(0)}(t_{n}|x_{0}) \to \pm \infty \)

    $$\begin{array}{@{}rcl@{}} \kappa_{1} & := & -\frac{1}{\gamma_{Y}^{(0)}(x_{0})}\left[\frac{\beta_{T}^{(0)}(x_{0})}{1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}-\frac{\beta}{1+\beta\gamma_{T}^{(0)}(x_{0})} \right], \\ \kappa_{2} & := & \frac{\beta \beta_{T}^{(0)}(x_{0})}{1+\beta\gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})} -\frac{\beta^{2}}{1 + 2\beta \gamma_{T}^{(0)}(x_{0})}, \end{array} $$
  • Case 2: \( \delta _{Y}^{(0)}(t_{n}|x_{0})/ \delta _{C}^{(0)}(t_{n}|x_{0}) \to 0\)

    $$\begin{array}{@{}rcl@{}} \kappa_{1} =\kappa_{2} = 0, \end{array} $$
  • Case 3: \(\delta _{Y}^{(0)}(t_{n}|x_{0}) / \delta _{C}^{(0)}(t_{n}|x_{0}) \to a\), where 0 < |a| <

    $$\begin{array}{@{}rcl@{}} \kappa_{1} & := & -\frac{a\gamma_{C}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})(\gamma_{Y}^{(0)}(x_{0})+a\gamma_{C}^{(0)}(x_{0}))}\left[\frac{\beta_{T}^{(0)}(x_{0})}{1+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}-\frac{\beta}{1+\beta\gamma_{T}^{(0)}(x_{0})} \right], \\ \kappa_{2} & := &\frac{a\beta\gamma_{C}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})+a\gamma_{C}^{(0)}(x_{0})}\left[\frac{\beta_{T}^{(0)}(x_{0})}{1+\beta\gamma_{T}^{(0)}(x_{0})+\beta_{T}^{(0)}(x_{0})\gamma_{T}^{(0)}(x_{0})}-\frac{\beta}{1 + 2\beta\gamma_{T}^{(0)}(x_{0})} \right]. \end{array} $$

Now we focus on the right hand side of (12). Concerning the terms \(\check \ell _{jk}(\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0});\beta )\), j, k = 1, 2, we have by the convergence in probability of \((\widehat \gamma _{Y,n}(x_{0}),\widehat \delta _{Y,n}(t_{n}|x_{0}))\) and because \(|\widetilde \ell _{jkl}(\gamma _{Y},\delta _{Y};\beta )| \le M_{jkl} (\boldsymbol V)\), in some open neighborhood of \((\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0}))\), with \( M_{jkl}(\boldsymbol V) =O_{\mathbb P}(1)\), j, k, l = 1, 2, that \(\check \ell _{jk}(\gamma _{Y}^{(0)}(x_{0}),\delta _{Y}^{(0)}(t_{n}|x_{0});\beta ) \overset {\mathbb P}{\to } \dot \ell _{jk}\), j, k = 1, 2. Let

$$\begin{array}{@{}rcl@{}} \mathbb C := \left[ \begin{array}{cc} \dot \ell_{11} & \dot \ell_{12} \\ \dot \ell_{12} & \dot \ell_{22} \end{array} \right]. \end{array} $$

From the proof of the convergence in probability, we know that ℂ is a negative definite matrix, and thus invertible. Then, according to Lemma 5.2 in Chapter 6 of Lehmann and Casella (1998), for the solution of the system of (12), we have the following convergence

$$\begin{array}{@{}rcl@{}} r_{n} \left[ \begin{array}{c} \widehat \gamma_{Y,n}(x_{0})-\gamma_{Y}^{(0)}(x_{0}) \\ \widehat \delta_{Y,n}(t_{n}|x_{0})-\delta_{Y}^{(0)}(t_{n}|x_{0}) \end{array} \right] \leadsto N(-\lambda \mathbb C^{-1}\kappa, \mathbb C^{-1} {\Delta} \mathbb C^{-1}). \end{array} $$

Working out the matrix products appearing in the mean and variance of the limiting distribution yields the result of the theorem.

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Goegebeur, Y., Guillou, A. & Qin, J. Bias-corrected estimation for conditional Pareto-type distributions with random right censoring. Extremes 22, 459–498 (2019). https://doi.org/10.1007/s10687-019-00341-7

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