Local Robust Estimation of Pareto-Type Tails with Random Right Censoring

Abstract

We propose a nonparametric robust estimator for the tail index of a conditional Pareto-type distribution in the presence of censoring and random covariates. The censored distribution is also of Pareto-type and the index is estimated locally within a narrow neighbourhood of the point of interest in the covariate space using the minimum density power divergence method. The main asymptotic properties of our robust estimator are derived under mild regularity conditions and its finite sample performance is illustrated on a small simulation study. A real data example is included to illustrate the practical applicability of the estimator.

Appendices

Appendix

Proof of Theorem 1

We focus on deriving the asymptotic expansion for $\mathbb E(T_{n}^{(2)}(K|x_{0}))$. 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, and this allows us to obtain a more precise statement of the remainder terms using the Hölder exponents from condition $(\mathcal H)$.

We have

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

$$ \begin{array}{@{}rcl@{}} \lefteqn{\mathbb E(T_{n}^{(2)}(K|x_{0})) =} \\ & & f_{X}(x_{0}) {\int}_{t_{n}}^{\infty} f_{Y}(y|x_{0})\overline F_{C}(y|x_{0})dy \\ & & + {\int}_{t_{n}}^{\infty} f_{Y}(y|x_{0})\overline F_{C}(y|x_{0})dy {\int}_{S_{K}} K(z) (f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ & & + f_{X}(x_{0}) {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} f_{Y}(y|x_{0})(\overline F_{C}(y|x_{0}-h_{n}z)-\overline F_{C}(y|x_{0}))dydz \\ & & + {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} f_{Y}(y|x_{0})(\overline F_{C}(y|x_{0}-h_{n}z)-\overline F_{C}(y|x_{0}))dy(f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ & & + f_{X}(x_{0}) {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} (f_{Y}(y|x_{0}-h_{n}z)-f_{Y}(y|x_{0}))\overline F_{C}(y|x_{0}))dydz \\ & & + {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} (f_{Y}(y|x_{0}-h_{n}z)-f_{Y}(y|x_{0}))\overline F_{C}(y|x_{0})dy(f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\ & & +f_{X}(x_{0}) {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} (f_{Y}(y|x_{0}-h_{n}z)-f_{Y}(y|x_{0}))(\overline F_{C}(y|x_{0}-h_{n}z)-\overline F_{C}(y|x_{0}))dydz \\ & & + {\int}_{S_{K}} K(z) {\int}_{t_{n}}^{\infty} (f_{Y}(y|x_{0}-h_{n}z)-f_{Y}(y|x_{0}))(\overline F_{C}(y|x_{0}-h_{n}z)\\&&-\overline F_{C}(y|x_{0}))dy(f_{X}(x_{0}-h_{n}z)-f_{X}(x_{0}))dz \\&&\quad=: T_{1}+\cdots+T_{8}. \end{array} $$

Concerning T₁ we have

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

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

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

This leads to the decomposition

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

From Eq. 2.1 we can write

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

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

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

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

A straightforward calculation gives then

$$ \begin{array}{@{}rcl@{}} T_{1,1,1} & = & \gamma_{T}(x_{0})+\delta_{C}(t_{n}|x_{0})\frac{{\gamma_{T}^{2}}(x_{0})\rho_{C}(x_{0})}{\gamma_{C}(x_{0})(\gamma_{C}(x_{0})-\gamma_{T}(x_{0})\rho_{C}(x_{0}))} +O({\delta_{C}^{2}}(t_{n}|x_{0})), \\ T_{1,1,2} & = & \delta_{Y}(t_{n}|x_{0})\frac{{\gamma_{T}^{2}}(x_{0})\rho_{Y}(x_{0})}{\gamma_{Y}(x_{0})(\gamma_{Y}(x_{0})-\gamma_{T}(x_{0})\rho_{Y}(x_{0}))} + O(\delta_{Y}(t_{n}|x_{0})\delta_{C}(t_{n}|x_{0})). \end{array} $$

For T_1,1,3 we use Proposition B.1.10 in de de Haan and Ferreira (2006), see also Drees (1998). Thus, for ε > 0 and 0 < δ < 1/γ_T(x₀) − ρ_Y(x₀)/γ_Y(x₀) arbitrary, and n sufficiently large, we have

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

Since ε is arbitrary and by using calculations for the integral that are similar to those above, one finds that T_1,1,3 = o(δ_Y(t_n|x₀)). In the same way T_1,1,4 = o(δ_Y(t_n|x₀)), and

$$ \begin{array}{@{}rcl@{}} T_{1,1,5} & = & - \varepsilon_{Y}(t_{n}|x_{0})\delta_{Y}(t_{n}|x_{0}) \frac{{\gamma_{T}^{2}}(x_{0})\rho_{Y}(x_{0})}{\gamma_{Y}(x_{0})-\gamma_{T}(x_{0})\rho_{Y}(x_{0})}+o(\delta_{Y}(t_{n}|x_{0})). \end{array} $$

Analogously one can show that T_1,2 = o(δ_C(t_n|x₀)).

Collecting the terms gives then

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

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

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

For T₂, we use the Hölder condition on f_X 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

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

(3.4)

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

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

Plugging the above inequality into Eq. 3.4, and computing integrals similar to those encountered above yields

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

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

As for T₅, we can write

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

which, combined with the inequality

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

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

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

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.

Proof of Theorem 2

To prove the result we make use of the Cramér-Wold device (see, e.g., van der Vaart, 1998, p.16).

Take $\xi =(\xi _{1},\ldots ,\xi _{J+1})^{T} \in \mathbb R^{J+1}$. Then

We have

$$ \begin{array}{@{}rcl@{}} \mathbb{V}ar (W_{1}) = \frac{\xi^{T} \mathbb{C} \xi}{n}, \end{array} $$

where $\mathbb {C}$ has elements

As for $\mathbb {C}_{j,k}$, with j,k ∈{1,…,J}, we have, by a straightforward application of Theorem 1

$$ \begin{array}{@{}rcl@{}} \mathbb{C}_{j,k} &=& \frac{{h_{n}^{d}}}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}\!\! \left[\frac{ \overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})\|K\|_{2}^{2}\gamma_{T}^{s_{j}^{\prime}\!+s_{k}^{\prime}}(x_{0}){\Gamma}(s_{j}^{\prime} + s_{k}^{\prime} + 1)}{{h_{n}^{d}}(1-(s_{j} + s_{k})\gamma_{T}(x_{0}))^{s_{j}^{\prime}+s_{k}^{\prime}+1}}\right. \\&&\left.(1\! + o(1)) + O(\overline {F^{2}_{T}}(t_{n}|x_{0})) \vphantom{\frac{1}{F_{T}(t_{n}|x_{0})f_{X}(x_{0})\|K\|_{2}^{2}\gamma_{T}^{s_{j}^{\prime}\!+s_{k}^{\prime}}}}\right] \\ &= & {\Sigma}_{j,k}(1+o(1)). \end{array} $$

In the same way, by using Theorem 1, $\mathbb {C}_{J+1,J+1}={\Sigma }_{J+1,J+1}(1+o(1))$. For $\mathbb {C}_{J+1,j}$, j ∈{1,…,J}, we need to evaluate an expectation of the form

By arguments similar to those used in deriving the expansion for $\mathbb E(T_{n}^{(2)}(K|x_{0}))$ we obtain

(3.5)

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 and Eq. 3.5, 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 r_n →∞.

Proof of Theorem 3

Let

$$ \begin{array}{@{}rcl@{}} \widetilde{\mathbb{S}}_{n}^{(j)} & := & \left\lbrace r_{n} \left[ \frac{T_{n}^{(1)}(K,s,j|x_{0})}{\overline F(t_{n}|x_{0})f_{X}(x_{0})} - \mathbb E\left( \frac{T_{n}^{(1)}(K,s,j|x_{0})}{\overline F(t_{n}|x_{0})f_{X}(x_{0})}\right) \right]; s\in [S,0] \right\rbrace, \hspace{1cm} j \in \lbrace 0,1,2,3 \rbrace, \\ \widetilde{\mathbb{S}}_{n}^{(4)} & := & r_{n} \left[ \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}- \mathbb E \left( \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) \right]. \end{array} $$

The weak convergence of the individual processes $\widetilde {\mathbb {S}}_{n}^{(j)}$, j ∈{0,1,2,3}, to tight, zero centered, Gaussian processes, with a covariance structure as given in the statement of the theorem, can be established following the arguments in the proof of Theorem 1 in Dierckx et al. (2014), while the weak convergence of $\widetilde {\mathbb {S}}_{n}^{(4)}$ to a zero centered Gaussian random variable with variance as in the statement of the theorem follows from Theorem 2. Then joint tightness will follow from the individual tightness. The joint tightness combined with the finite dimensional convergence from Theorem 2 leads then to the joint weak convergence of $(\widetilde {\mathbb {S}}_{n}^{(0)},\widetilde {\mathbb {S}}_{n}^{(1)},\widetilde {\mathbb {S}}_{n}^{(2)},\widetilde {\mathbb {S}}_{n}^{(3)},\widetilde {\mathbb {S}}_{n}^{(4)})$. It remains to verify the expressions for the expected values of $\mathbb {S}^{(j)}$, j ∈{0,1,…,4}. To this aim we study

$$ \begin{array}{@{}rcl@{}} r_{n} \left[ \mathbb E\left( \frac{T_{n}^{(1)}(K,s,j|x_{0})}{\overline F(t_{n}|x_{0})f_{X}(x_{0})}\right) - \frac{j!{\gamma_{T}^{j}}(x_{0})}{(1-s\gamma_{T}(x_{0}))^{j+1}}\right], \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} r_{n} \left[ \mathbb E\left( \frac{T_{n}^{(2)}(K|x_{0})}{\overline F(t_{n}|x_{0})f_{X}(x_{0})}\right) - \frac{\gamma_{T}(x_{0})}{\gamma_{Y}(x_{0})}\right]. \end{array} $$

A straightforward application of Theorem 1, and taking the link between δ_T(t_n|x₀), δ_Y(t_n|x₀) and δ_C(t_n|x₀) into account, one easily obtains the expressions for the expected values of the limiting processes $\mathbb {S}^{(0)}$, $\mathbb {S}^{(1)}$, $\mathbb {S}^{(2)}$ and $\mathbb {S}^{(3)}$, and the random variable $\mathbb {S}^{(4)}$.

Proof of Theorem 4

The consistency of $\widehat \gamma _{T,n}(x_{0}|\tilde \rho )$ and $\widehat \delta _{T,n}(x_{0}|\tilde \rho )$ for $\gamma _{T}^{(0)}(x_{0})$ and 0, respectively, follows from Proposition 1 in Dierckx et al. (2014).

Concerning $\widehat p_{n}(x_{0})$ we have from Theorem 1

$$ \begin{array}{@{}rcl@{}} \mathbb E \left( \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) & = & 1+o(1), \\ \mathbb E \left( \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) & = & \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} +o(1), \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \mathbb{V}ar\left( \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) & = & O \left( \frac{1}{n{h_{n}^{d}} \overline F_{T}(t_{n}|x_{0})} \right), \\ \mathbb{V}ar\left( \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right) & = & O \left( \frac{1}{n{h_{n}^{d}} \overline F_{T}(t_{n}|x_{0})} \right). \end{array} $$

Thus

$$ \begin{array}{@{}rcl@{}} \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \stackrel{P}{\to} 1 \hspace{.5cm} \text{ and} \hspace{.5cm} \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \stackrel{P}{\to} \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})}, \end{array} $$

and hence, by the continuous mapping theorem, $\widehat p_{n}(x_{0}) \stackrel {P}{\to } \gamma _{T}^{(0)}(x_{0})/\gamma _{Y}^{(0)}(x_{0})$. Another application of the continuous mapping theorem gives then $\widehat \gamma _{Y,n}(x_{0}|\tilde \rho ) \stackrel {P}{\to } \gamma _{Y}^{(0)}(x_{0})$.

Proof of Theorem 5

Let $\widetilde {\Delta }_{\alpha }(\gamma ,\delta ;\tilde \rho ) := \widehat {\Delta }_{\alpha }(\gamma ,\delta ;\tilde \rho )/(\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0}))$, and let $\widetilde {\Delta }_{\alpha ,u}(\gamma ,\delta ;\tilde \rho )$, u = 1,2, denote the derivatives with respect to γ and δ, respectively, apart from the common scale factor 1 + α. Similarly, $\widetilde {\Delta }_{\alpha ,u,v}(\gamma ,\delta ;\tilde \rho )$ and $\widetilde {\Delta }_{\alpha ,u,v,w}(\gamma ,\delta ;\tilde \rho )$, u,v,w = 1,2, will denote second and third order derivatives (again apart from the common scaling by 1 + α).

We apply a Taylor series expansion of the estimating equations (??) and (??) around $(\gamma _{T}^{(0)}(x_{0}),0)$, and extend these with $T_{n}^{(1)}(K,0,0|x_{0})$ and $T_{n}^{(2)}(K|x_{0})$ to obtain

$$ \begin{array}{@{}rcl@{}} \lefteqn{r_{n} \left[ \begin{array}{c} -\widetilde {\Delta}_{\alpha,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) \\ -\widetilde {\Delta}_{\alpha,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) \\ \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -1 \\ \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -\frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \end{array} \right] =} \\ & & \left[ \begin{array}{cccc} \overline {\Delta}_{\alpha,1,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & \overline {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & 0 & 0\\ \overline {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & \overline {\Delta}_{\alpha,2,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\&&\times\kern.8pc\left[ \begin{array}{c} r_{n} (\widehat \gamma_{T,n}(x_{0}|\tilde \rho)-\gamma_{T}^{(0)}(x_{0})) \\ r_{n} \widehat \delta_{T,n}(x_{0}|\tilde \rho) \\ r_{n} \left( \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -1 \right) \\ r_{n} \left( \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -\frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})}\right) \end{array} \right] \end{array} $$

with

$$ \begin{array}{@{}rcl@{}} \overline {\Delta}_{\alpha,1,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & := & \widetilde {\Delta}_{\alpha,1,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)+\frac{1}{2} \left[ \widetilde {\Delta}_{\alpha,1,1,1}(\breve \gamma_{T,n}(x_{0}|\tilde \rho),\right. \\ & & \left. \breve \delta_{T,n}(x_{0}|\tilde \rho);\tilde \rho) (\widehat \gamma_{T,n}(x_{0}|\tilde \rho)-\gamma_{T}^{(0)}(x_{0})) \right. \\ & & \left. + \widetilde {\Delta}_{\alpha,1,1,2}(\breve \gamma_{T,n}(x_{0}|\tilde \rho),\breve \delta_{T,n}(x_{0}|\tilde \rho);\tilde \rho) \widehat \delta_{T,n}(x_{0}|\tilde \rho) \right], \\ \overline {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & : = & \widetilde {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)+\frac{1}{2} \left[ \widetilde {\Delta}_{\alpha,1,2,2}(\breve \gamma_{T,n}(x_{0}|\tilde \rho),\breve \delta_{T,n}(x_{0}|\tilde \rho);\right. \\ & & \left.\tilde \rho) \widehat \delta_{T,n}(x_{0}|\tilde \rho) \right. \\ & & \left. + \widetilde {\Delta}_{\alpha,1,1,2}(\breve \gamma_{T,n}(x_{0}|\tilde \rho),\breve \delta_{T,n}(x_{0}|\tilde \rho);\tilde \rho) (\widehat \gamma_{T,n}\right. \\ & & \left.(x_{0}|\tilde \rho)-\gamma_{T}^{(0)}(x_{0})) \right], \\ \overline {\Delta}_{\alpha,2,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & := & \widetilde {\Delta}_{\alpha,2,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) +\frac{1}{2} \left[\widetilde {\Delta}_{\alpha,2,2,2}(\breve \gamma_{T,n}(x_{0}|\tilde \rho),\breve \delta_{T,n}(x_{0}|\tilde \rho);\tilde \rho)\right. \\ & & \left. \widehat \delta_{T,n}(x_{0}|\tilde \rho) \right. \\ & & \left. + \widetilde {\Delta}_{\alpha,1,2,2}(\breve \gamma_{T,n}(x_{0}|\tilde \rho),\breve \delta_{T,n}(x_{0}|\tilde \rho);\tilde \rho) (\widehat \gamma_{T,n}(x_{0}|\tilde \rho)-\gamma_{T}^{(0)}(x_{0})) \right], \end{array} $$

and where $(\breve \gamma _{T,n}(x_{0}|\tilde \rho ),\breve \delta _{T,n}(x_{0}|\tilde \rho ))$ is a point on the line segment connecting $(\gamma _{T}^{(0)}(x_{0}),0)$ and $(\widehat \gamma _{T,n}(x_{0}|\tilde \rho ),\widehat \delta _{T,n}(x_{0}|\tilde \rho ))$.

After tedious, but straightforward, derivations one obtains

$$ \begin{array}{@{}rcl@{}} \lefteqn{\widetilde {\Delta}_{\alpha,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & = & (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \left[-\frac{\alpha \gamma_{T}^{(0)}(x_{0})(1+\gamma_{T}^{(0)}(x_{0}))}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}\frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & & \left. +\gamma_{T}^{(0)}(x_{0})\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}\right. \\ & & \left.-\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),1|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right], \end{array} $$

$$ \begin{array}{@{}rcl@{}} \lefteqn{\widetilde {\Delta}_{\alpha,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & = & (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-1} \left[-\frac{\alpha \tilde \rho(1+\gamma_{T}^{(0)}(x_{0}))}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))][1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}\frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & & +\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \left. \right. \\ & &\left.-(1-\tilde \rho)\frac{T_{n}^{(1)}(K,-(\alpha(1+\gamma_{T}^{(0)}(x_{0}))-\tilde \rho)/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})}\right], \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \lefteqn{ \widetilde {\Delta}_{\alpha,1,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & = & (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \left[ \left( \frac{\alpha+2}{1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))}-\frac{2\alpha+4}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}} \right.\right. \\ &&\hspace{5cm}\left.\left. + \frac{2\alpha+2}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{3}} \right) \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & & \left.-(\alpha+1)\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \left. \right. \\ & &\left.+\frac{2\alpha+2}{\gamma_{T}^{(0)}(x)}\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),1|x_{0})}{\overline F(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & & \left.-\frac{\alpha}{(\gamma_{T}^{(0)}(x_{0}))^{2}}\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),2|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right], \\ \\ \\ \lefteqn{\widetilde {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & = & (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \left[ \left( \frac{1+\alpha(2+\alpha)(1+\gamma_{T}^{(0)}(x_{0}))}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}} \right. \right. \\ & & \left. -\frac{(1-\tilde \rho)^{2}-\alpha[\tilde \rho(1-\tilde \rho)-2(1+\gamma_{T}^{(0)}(x_{0}))(1-\tilde \rho)]+\alpha^{2}(1+\gamma_{T}^{(0)}(x_{0}))(1-\tilde \rho)}{[1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}} \right) \\ & & \times \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -(1+\alpha)\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ & & + (\alpha+1)(1-\tilde \rho) \frac{T_{n}^{(1)}(K,-(\alpha(1+\gamma_{T}^{(0)}(x_{0}))-\tilde \rho)/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ & & \left. +\frac{\alpha}{\gamma_{T}^{(0)}(x_{0})} \frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),1|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right. \\ & & \left. - \frac{(\alpha-\tilde \rho)(1-\tilde \rho)}{\gamma_{T}^{(0)}(x_{0})}\frac{T_{n}^{(1)}(K,-(\alpha(1+\gamma_{T}^{(0)}(x_{0}))-\tilde \rho)/\gamma_{T}^{(0)}(x_{0}),1|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right], \end{array} $$

$$ \begin{array}{@{}rcl@{}} \lefteqn{\widetilde {\Delta}_{\alpha,2,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & = & (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \left[ \left( \frac{1+\alpha+\gamma_{T}^{(0)}(x_{0})}{1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))}-\frac{2(1-\tilde \rho)(1+\gamma_{T}^{(0)}(x_{0})+\alpha)}{1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))} \right. \right. \\ & & \left. +\frac{(1+\gamma_{T}^{(0)}(x_{0}))(1-2\tilde \rho)+\alpha(1-\tilde \rho)^{2}}{1-2\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))} \right) \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F(t_{n}|x_{0})f_{X}(x_{0})} \\ & & -(\alpha+\gamma_{T}^{(0)}(x_{0}))\frac{T_{n}^{(1)}(K,-\alpha(1+\gamma_{T}^{(0)}(x_{0}))/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ && +2(1-\tilde \rho)(\alpha+\gamma_{T}^{(0)}(x_{0}))\frac{T_{n}^{(1)}(K,-(\alpha(1+\gamma_{T}^{(0)}(x_{0}))-\tilde \rho)/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \\ & & \left. -[(1 + \gamma_{T}^{(0)}(x_{0}))(1 - 2\tilde \rho)+(\alpha - 1)(1 - \tilde \rho)^{2}]\frac{T_{n}^{(1)}(K,\!-(\alpha(1 + \gamma_{T}^{(0)}(x_{0})) - 2\tilde \rho)/\gamma_{T}^{(0)}(x_{0}),0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} \right] . \end{array} $$

For brevity, the expressions for the third order derivatives are omitted from the paper.

Now let

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

$$ \begin{array}{@{}rcl@{}} \overline{\mathbb U}(\tilde \rho) := \left[ \begin{array}{c} 1 \\ \frac{1}{1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))} \\ \frac{1}{1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))} \\ \frac{\gamma_{T}^{(0)}(x_{0})}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}} \\ \frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \end{array} \right]. \end{array} $$

Then by Theorem 3

$$ \begin{array}{@{}rcl@{}} \mathbb W_{n}(\tilde \rho) := r_{n}(\mathbb U_{n}(\tilde \rho)-\overline{\mathbb U}(\tilde \rho)) \leadsto N(\lambda \mu(\tilde \rho), {\Sigma}(\tilde \rho)), \end{array} $$

where $\mu (\tilde \rho )$ a (5 × 1) vector with elements

$$ \begin{array}{@{}rcl@{}} \mu_{1}(\tilde \rho) & = & 0, \\ \mu_{2}(\tilde \rho) & = & -\frac{\alpha \rho_{T}^{(0)}(x_{0})(1+\gamma_{T}^{(0)}(x_{0}))}{\gamma_{T}^{(0)}(x_{0})[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))][1-\rho_{T}^{(0)}(x_{0})+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}, \\ \mu_{3}(\tilde \rho) & = &-\frac{[\alpha(1+\gamma_{T}^{(0)}(x_{0}))-\tilde \rho]\rho_{T}^{(0)}(x_{0})}{\gamma_{T}^{(0)}(x_{0})[1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))][1-\rho_{T}^{(0)}(x_{0})-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}, \\ \mu_{4}(\tilde \rho) & = & \frac{\rho_{T}^{(0)}(x_{0})(1-\rho_{T}^{(0)}(x_{0}))-\alpha^{2}\rho_{T}^{(0)}(x_{0})(1+\gamma_{T}^{(0)}(x_{0}))^{2}}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}[1-\rho_{T}^{(0)}(x_{0})+\alpha(1+\gamma_{T}^{(0)}(x_{o}))]^{2}} , \\ \mu_{5}(\tilde \rho) & = & \mu, \end{array} $$

where μ is given by Eq. 3.1, apart from the factor λ, and ${\Sigma }(\tilde \rho )$ a (5 × 5) symmetric matrix with elements

$$ \begin{array}{@{}rcl@{}} {\Sigma}_{11}(\tilde \rho) & := & \|K\|_{2}^{2}, \\ {\Sigma}_{21}(\tilde \rho) & := & \frac{\|K\|_{2}^{2}}{1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))}, \\ {\Sigma}_{22}(\tilde \rho) & := & \frac{\|K\|_{2}^{2}}{1+2\alpha(1+\gamma_{T}^{(0)}(x_{0}))}, \\ {\Sigma}_{31}(\tilde \rho) & := & \frac{\|K\|_{2}^{2}}{1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))}, \\ {\Sigma}_{32}(\tilde \rho) & := & \frac{\|K\|_{2}^{2}}{1-\tilde \rho+2\alpha(1+\gamma_{T}^{(0)}(x_{0}))}, \\ {\Sigma}_{33}(\tilde \rho) & := & \frac{\|K\|_{2}^{2}}{1-2\tilde \rho+2\alpha(1+\gamma_{T}^{(0)}(x_{0}))}, \\ {\Sigma}_{41}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}, \\ {\Sigma}_{42}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{[1+2\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}, \\ {\Sigma}_{43}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{[1-\tilde \rho+2\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}, \\ {\Sigma}_{44}(\tilde \rho) & := & \frac{2(\gamma_{T}^{(0)}(x_{0}))^{2}\|K\|_{2}^{2}}{[1+2\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{3}}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} {\Sigma}_{51}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{\gamma_{Y}^{(0)}(x_{0})}, \\ {\Sigma}_{52}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{\gamma_{Y}^{(0)}(x_{0})[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}, \\ {\Sigma}_{53}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{\gamma_{Y}^{(0)}(x_{0})[1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}, \\ {\Sigma}_{54}(\tilde \rho) & := & \frac{(\gamma_{T}^{(0)}(x_{0}))^{2}\|K\|_{2}^{2}}{\gamma_{Y}^{(0)}(x_{0})[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}, \\ {\Sigma}_{55}(\tilde \rho) & := & \frac{\gamma_{T}^{(0)}(x_{0})\|K\|_{2}^{2}}{\gamma_{Y}^{(0)}(x_{0})}. \\ \end{array} $$

We have

$$ \begin{array}{@{}rcl@{}} r_{n} \left[ \begin{array}{c} -\widetilde {\Delta}_{\alpha,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) \\ -\widetilde {\Delta}_{\alpha,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) \\ \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -1 \\ \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -\frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \end{array} \right] = B(\tilde \rho) \mathbb W_{n}(\tilde \rho) \leadsto N(\lambda B(\tilde \rho)\mu(\tilde \rho),B(\tilde \rho){\Sigma}(\tilde \rho)B(\tilde \rho)^{T}), \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} B(\tilde \rho) := \left[ \begin{array}{ccccc} b_{11}(\tilde \rho) & -(\gamma_{T}^{(0)}(x_{0}))^{-\alpha-1} & 0 & (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} & 0 \\ b_{21}(\tilde \rho)& -(\gamma_{T}^{(0)}(x_{0}))^{-\alpha-1}& (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-1}(1-\tilde \rho) & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right], \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} b_{11}(\tilde \rho ) & := & \frac{\alpha(1+\gamma_{T}^{(0)}(x_{0}))}{(\gamma_{T}(x_{0}))^{\alpha+1}[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}, \\ b_{21}(\tilde \rho) & :=& \frac{\alpha \tilde \rho(1+\gamma_{T}^{(0)}(x_{0}))}{(\gamma_{T}(x_{0}))^{\alpha+1}[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))][1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}. \end{array} $$

As for $\overline {\Delta }_{\alpha ,u,v}(\gamma _{T}^{(0)}(x_{0}),0;\tilde \rho )$, u,v = 1,2, we have by Theorems 1, 3 and 4, and because $|\widetilde {\Delta }_{\alpha ,u,v,w}(\gamma ,\delta ;\tilde \rho )| \le M_{u,v,w}$, in some open neighborhood of $(\gamma _{T}^{(0)}(x_{0}),0)$ with $M_{u,v,w}=O_{\mathbb P}(1)$, u,v,w = 1,2, that $\overline {\Delta }_{\alpha ,u,v}(\gamma _{T}^{(0)}(x_{0}),0;\tilde \rho ) \stackrel { P}{ \to } {\Delta }_{\alpha ,u,v}(\gamma _{T}^{(0)}(x_{0}),0;\tilde \rho )$, where

$$ \begin{array}{@{}rcl@{}} \lefteqn{{\Delta}_{\alpha,1,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) := (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \frac{1+\alpha^{2}(1+\gamma_{T}^{(0)}(x_{0}))^{2}}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{3}},} \\ \lefteqn{{\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & & := (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \frac{\tilde \rho(1-\tilde \rho)[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))+\alpha^{2}(1+\gamma_{T}^{(0)}(x_{0}))^{2}]+\alpha^{3}\tilde \rho (1+\gamma_{T}^{(0)}(x_{0}))^{3}}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}[1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]^{2}}, \\ \lefteqn{{\Delta}_{\alpha,2,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho)} \\ & & := (\gamma_{T}^{(0)}(x_{0}))^{-\alpha-2} \frac{(1-\tilde \rho)\tilde \rho^{2}+\alpha \tilde \rho^{2}(1+\gamma_{T}^{(0)}(x_{0}))[\alpha(1+\gamma_{T}^{(0)}(x_{0}))-\tilde \rho]}{[1+\alpha(1+\gamma_{T}^{(0)}(x_{0}))][1-\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))][1-2\tilde \rho+\alpha(1+\gamma_{T}^{(0)}(x_{0}))]}. \end{array} $$

Let

$$ \begin{array}{@{}rcl@{}} {\Delta} (\tilde \rho):= \left[ \begin{array}{cc} {\Delta}_{\alpha,1,1}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) \\ {\Delta}_{\alpha,1,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) & {\Delta}_{\alpha,2,2}(\gamma_{T}^{(0)}(x_{0}),0;\tilde \rho) \end{array} \right] \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \breve {\Delta}(\tilde \rho) := \left[ \begin{array}{cc} {\Delta}(\tilde \rho) & 0 \\ 0 & I_{2} \end{array} \right], \end{array} $$

where I₂ is the (2 × 2) identity matrix. It can be verified that ${\Delta }(\tilde \rho )$ is positive definite and thus invertible. Then, by Lemma 5.2 in Chapter 6 of Lehmann and Casella (1998), we have

$$ \begin{array}{@{}rcl@{}} r_{n}\left[ \begin{array}{c} \widehat \gamma_{T,n}(x_{0}|\tilde \rho)-\gamma_{T}^{(0)}(x_{0}) \\ \widehat \delta_{T,n}(x_{0}|\tilde \rho) \\ \frac{T_{n}^{(1)}(K,0,0|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} - 1 \\ \frac{T_{n}^{(2)}(K|x_{0})}{\overline F_{T}(t_{n}|x_{0})f_{X}(x_{0})} -\frac{\gamma_{T}^{(0)}(x_{0})}{\gamma_{Y}^{(0)}(x_{0})} \end{array} \right] \leadsto N(\lambda \breve {\Delta}^{-1}(\tilde \rho)B(\tilde \rho)\mu(\tilde \rho),\breve {\Delta}^{-1}(\tilde \rho)B(\tilde \rho){\Sigma}(\tilde \rho)B(\tilde \rho)^{T} \breve {\Delta}^{-1}(\tilde \rho)). \end{array} $$

Finally, a straightforward application of the delta method gives

$$ \begin{array}{@{}rcl@{}} r_{n}(\widehat \gamma_{Y,n}(x_{0}|\tilde \rho)-\gamma_{Y}^{(0)}(x_{0})) \leadsto N(\lambda L^{T}\breve {\Delta}^{-1}(\tilde \rho)B(\tilde \rho)\mu(\tilde \rho),L^{T}\breve {\Delta}^{-1}(\tilde \rho)B(\tilde \rho){\Sigma}(\tilde \rho)B(\tilde \rho)^{T} \breve {\Delta}^{\!-1}(\tilde \rho)L), \end{array} $$

with $L^{T} := [\gamma _{Y}^{(0)}(x_{0})/\gamma _{T}^{(0)}(x_{0}),0,\gamma _{Y}^{(0)}(x_{0}),-(\gamma _{Y}^{(0)}(x_{0}))^{2}/\gamma _{T}^{(0)}(x_{0})]$.