Nonparametric estimation of the cross ratio function

Abstract

The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio, and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes. This dependence, however, disappears when the net income is large compared to the mean income.

Appendix: Proofs of Theorems 1–3

In this appendix, we present the proofs of Theorems 1–3 in the main text.

Proof of Theorem 1

$$\begin{aligned} \begin{array}{ll} &{}\widehat{\lambda }_m(t_1 \mid T_2 = t_2) - \lambda (t_1 \mid T_2 = t_2)\\ &{}\quad = \displaystyle \frac{1}{b_n} \displaystyle \int \left[ \widehat{\varLambda }_m (t_1 - b_n u\mid T_2 = t_2) - \varLambda (t_1 - b_n u\mid T_2 = t_2)\right] {\mathrm {d}}K_0(u)\\ &{}\qquad + \displaystyle \frac{1}{b_n} \displaystyle \int \varLambda (t_1 - b_nu \mid T_2 = t_2) {\mathrm {d}}K_0(u) - \lambda (t_1 \mid T_2 = t_2)\\ &{}\quad = (A) + (B). \end{array} \end{aligned}$$

The non-random term (B) equals

$$\begin{aligned} (B) = \displaystyle \frac{1}{2}b_n^2 \lambda '' (t_1 \mid T_2 = t_2) \mu _2(K_0) + o(b_n^2) \end{aligned}$$

(11)

where $\mu _2(K_0) = \displaystyle \smallint t^2 K_0(t) {\mathrm {d}}t$. This is because $\lambda (t_1 \mid T_2 = t_2)$ is twice continuously differentiable with respect to $t_1$.

For the integrand in (A), we first note that

$$\begin{aligned} \begin{array}{ll} &{}\widehat{\varLambda }_m(t_1 \mid T_2 = t_2) - \varLambda (t_1 \mid T_2 = t_2)\\ &{}\quad = \displaystyle \mathop \int \nolimits _0^{t_1} \displaystyle \frac{{\mathrm {d}}\widehat{F}_{t_2}(s)}{1-\widehat{F}_{t_2}(s)} - \displaystyle \mathop \int \nolimits _0^{t_1} \displaystyle \frac{{\mathrm {d}}F_{t_2}(s)}{1-F_{t_2}(s)} + O\left( \displaystyle \frac{m}{n}\right) \ \text{ a.s. } \end{array} \end{aligned}$$

Indeed, for n sufficiently large,

$$\begin{aligned} \left| \displaystyle \mathop \int \nolimits _0^{t_1} \displaystyle \frac{{\mathrm {d}}\widehat{F}_{t_2}(s)}{1-\widehat{F}_{t_2}(s-)} - \displaystyle \mathop \int \nolimits _0^{t_1} \displaystyle \frac{{\mathrm {d}}\widehat{F}_{t_2}(s)}{1- \widehat{F}_{t_2}(s)}\right| \le \displaystyle \frac{4}{[1-F_{t_2}(t_1)]^2} O\left( \displaystyle \frac{m}{n}\right) \end{aligned}$$

because the maximal jump of $\widehat{F}_{t_2}(\cdot )$ is $O\left( \displaystyle \frac{m}{n}\right)$ a.s. (see Janssen et al. 2016) and because $\widehat{F}_{t_2} (t_1)$ converges to $F_{t_2}(t_1)$. Hence, the term (A) can be written as

$$\begin{aligned} (A) = \displaystyle \frac{1}{b_n} \displaystyle \mathop \int \nolimits _{t_1-b_nL}^{t_1+b_nL} \ln \left[ \displaystyle \frac{1-\widehat{F}_{t_2}(s)}{1-F_{t_2}(s)}\right] {\mathrm {d}}K_0 \left( \displaystyle \frac{t_1-s}{b_n}\right) + O\left( \displaystyle \frac{m}{nb_n}\right) \ \text{ a.s. } \end{aligned}$$

By the mean value theorem, we obtain that

$$\begin{aligned} \ln \left[ \displaystyle \frac{1-\widehat{F}_{t_2} (s)}{1-F_{t_2}(s)}\right] = - \displaystyle \frac{\widehat{F}_{t_2} (s) - F_{t_2}(s)}{1-F_{t_2}(s)} - \displaystyle \frac{1}{2} \displaystyle \frac{[\widehat{F}_{t_2} (s) - F_{t_2}(s)]^2}{[1- \theta _n (s)]^2} \end{aligned}$$

for some $\theta _n(s)$ between $F_{t_2}(s)$ and $\widehat{F}_{t_2}(s)$. Hence,

$$\begin{aligned} (A)= & {} \displaystyle \frac{1}{b_n} \displaystyle \mathop \int \nolimits _{-L}^L \displaystyle \frac{\widehat{F}_{t_2} (t_1 - b_nu) - F_{t_2} (t_1 - b_n u)}{1-F_{t_2} (t_1 - b_n u)} {\mathrm {d}}K_0(u) \nonumber \\&+ \ R_n (t_1, t_2) + O\left( \displaystyle \frac{m}{nb_n}\right) \ \text{ a.s. }, \end{aligned}$$

(12)

where

$$\begin{aligned} R_n(t_1,t_2) = \displaystyle \frac{1}{2b_n} \displaystyle \mathop \int \nolimits _{-L}^L \displaystyle \frac{[\widehat{F}_{t_2} (t_1 - b_nu) - F_{t_2} (t_1 - b_n u)]^2}{[1-\theta _n (t_1 - b_nu)]^2} {\mathrm {d}}K_0(u). \end{aligned}$$

From Theorem 3 of Janssen et al. (2016) and the first part of the proof of Lemma 7 available in Electronic Supplementary Material provided by Janssen et al. (2016), we conclude that

$$\begin{aligned}&\sup \limits _{s} | \widehat{F}_{t_2}(s) - F_{t_2}(s) | = O\left( m^{1/4}n^{-1/2}(\ln n)^{1/2}\right) + O(n^{-1/2}(\ln n)^{1/2} + m^{-1} \nonumber \\&\qquad +\ m^{1/2}n^{-3/4}(\ln n)^{1/2}(\ln \ln n)^{1/4} + m^{13/12}n^{-1}(\ln n)^{1/2}(\ln \ln n)^{1/2})\ {\text{ a.s. }} \nonumber \\&\quad = O\left( m^{1/4}n^{-1/2}(\ln n)^{1/2}\right) \ \text{ a.s. }, \end{aligned}$$

(13)

by applying the assumptions in condition (d) of the theorem.

Since we assume that $K_0$ is a continuous density function of bounded variation, there exist two non-decreasing bounded and continuous functions $K_{01}$ and $K_{02}$ such that $K_0(u) = K_{01}(u) - K_{02}(u)$. Assume that $K_{01}$ and $K_{02}$ are supported on $[-L,L_1]$ and $[L_1, L]$, respectively, for some $-L \le L_1 \le L$. Hence, $K_{01}(-L) = K_{02}(-L) = 0 = K_{01}(L) = K_{02}(L)$ and $K_{01}(L_1) = -K_{02}(L_1)$. Therefore,

$$\begin{aligned} |R_n(t_1,t_2)|\le & {} \displaystyle \frac{1}{2b_n} \displaystyle \mathop \int \nolimits _{-L}^{L_1} \displaystyle \frac{[\widehat{F}_{t_2} (t_1 - b_n u) - F_{t_2} (t_1 - b_nu)]^2}{[1-\theta _n (t_1 - b_n u)]^2} {\mathrm {d}}K_{01}(u)\\&+ \displaystyle \frac{1}{2b_n} \displaystyle \mathop \int \nolimits _{L_1}^L \displaystyle \frac{[\widehat{F}_{t_2} (t_1 - b_n u) - F_{t_2} (t_1-b_n u)]^2}{[1-\theta _n (t_1 - b_n u)]^2} {\mathrm {d}}K_{02}(u). \end{aligned}$$

Furthermore, since $\sup \limits _s |\widehat{F}_{t_2}(s) - F_{t_2} (s)| \rightarrow 0$ a.s., we have that $\sup \limits _s |\theta _n(s) - F_{t_2}(s)| \rightarrow 0$ a.s.; hence, for some constant $C > 0$,

$$\begin{aligned} |R_n(t_1,t_2)|\le & {} \displaystyle \frac{2C}{b_n[1-F_{t_2} (t_1)]^2} \left[ \sup \limits _s |\widehat{F}_{t_2}(s) - F_{t_2} (s)|\right] ^2 K_{01} (L_1)\\= & {} O\left( \displaystyle \frac{m^{1/2}}{nb_n} \ln n \right) \ \text{ a.s. }, \end{aligned}$$

by using (13). Therefore, under the conditions in (d) we conclude that

$$\begin{aligned} (n m^{-1/2} b_n)^{1/2} R_n(t_1, t_2) \rightarrow 0\ \text{ a.s. } \end{aligned}$$

(14)

By the mean value theorem, the first term in the expression of (A) given in (12) becomes

$$\begin{aligned}&\frac{1}{b_n \left[ 1 - F_{t_2}(t_1)\right] }\mathop \int \nolimits _{-L}^{L} \left[ \widehat{F}_{t_2}(t_1 - b_n u) - F_{t_2}(t_1 - b_n u)\right] {\mathrm {d}}K_0(u)\nonumber \\&\qquad +\ \mathop \int \nolimits _{-L}^{L} \frac{u\left[ \widehat{F}_{t_2}(t_1 - b_n u) - F_{t_2}(t_1 - b_n u)\right] f_{t_2}\left[ \theta (u)\right] }{\left\{ 1 - F_{t_2}\left[ \theta (u)\right] \right\} ^{2}} {\mathrm {d}}K_0(u) \nonumber \\&\quad =: (A_{11}) + \tilde{R}_n(t_1, t_2), \end{aligned}$$

(15)

for some $\theta (u)$ between $t_1$ and $t_1 - b_n u$.

As above, we have that for some constant $C > 0$,

$$\begin{aligned} |\tilde{R}_n(t_1,t_2)|\le & {} \displaystyle \frac{Cf_{t_2}(t_1)}{[ 1- F_{t_2} (t_1)]^2} \left[ \sup \limits _s |\widehat{F}_{t_2}(s) - F_{t_2} (s)|\right] \mathop \int \nolimits _{-L}^{L} |u| {\mathrm {d}}\left[ K_{01}(u) + K_{02}(u)\right] \\= & {} O\left( \displaystyle m^{1/4} n^{-1/2} (\ln n)^{1/2} \right) \ \text{ a.s. } \end{aligned}$$

Under the conditions in (d), we have

$$\begin{aligned} (n m^{-1/2} b_n)^{1/2} \tilde{R}_n(t_1, t_2) \rightarrow 0\ \text{ a.s. } \end{aligned}$$

(16)

For $(A_{11})$ in Eq. (15), we write

$$\begin{aligned} (A_{11})= & {} \frac{1}{b_n \left[ 1 - F_{t_2}(t_1)\right] } \mathop \int \nolimits _{-L}^{L} \left\{ \widehat{F}_{t_2}(t_1 - b_n u) - E[\widehat{F}_{t_2}(t_1 - b_n u)]\right\} {\mathrm {d}}K_0(u)\\&+\ \frac{1}{b_n \left[ 1 - F_{t_2}(t_1)\right] } \mathop \int \nolimits _{-L}^{L} \left\{ E[\widehat{F}_{t_2}(t_1 - b_n u)] - F_{t_2}(t_1 - b_n u) \right\} {\mathrm {d}}K_0(u) \\=: & {} (A_{111}) + (A_{112}). \end{aligned}$$

For $(A_{112})$, which contributes to the bias, note that $E[\widehat{F}_{t_2}(t_1 - b_n u)] - F_{t_2}(t_1 - b_n u) = -\{E[\widehat{S}_{t_2}(t_1 - b_n u)] - S_{t_2}(t_1 - b_n u)\}$. In line with Remark 3 in Janssen et al. (2016), we have

$$\begin{aligned} E[\widehat{S}_{t_2}(t_1 - b_n u)] - S_{t_2}(t_1 - b_n u) = -\frac{1}{2}m^{-1}b\left[ S_1(t_1 - b_n u), S_2(t_2)\right] + o(m^{-1}), \end{aligned}$$

where

$$\begin{aligned} b(u,v) = (1 - 2v) C^{(2,2)}(u,v) + u(1-u)C^{(1,1,2)}(u,v) + v(1-v)C^{(2,2,2)}(u,v). \end{aligned}$$

Using partial integration, we obtain that

$$\begin{aligned} (A_{112}) = \frac{1}{2}m^{-1}\phi (t_1,t_2) + o(m^{-1}), \end{aligned}$$

(17)

where

$$\begin{aligned} \phi (t_1,t_2) = \frac{b^{(1)}\left[ S_{1}(t_1), S_{2}(t_2)\right] }{1 - F_{t_2}(t_1)}f_1(t_1), \end{aligned}$$

with $b^{(1)}(u,v) = \frac{\partial }{\partial u}b(u,v)$. For the first term we have, after partial integration,

$$\begin{aligned} (A_{111}) = \frac{1}{1 - F_{t_2}(t_1)}\left\{ \widehat{f}_{t_2}(t_1) - E[\widehat{f}_{t_2}(t_1)]\right\} , \end{aligned}$$

(18)

where $\widehat{f}_{t_2}(t_1)$ is precisely the Bernstein estimator for a conditional density function studied in Janssen et al. (2017).

The proof of the theorem follows directly from (11)–(18) and the Theorem in Janssen et al. (2017) by simply replacing Y by $T_1$ and X by $T_2$ in the aforementioned paper.

Also note that the term $\frac{1}{2}m^{-1}\phi (t_1,t_2)$ in the bias vanishes after multiplication with $(n m^{-1/2} b_n)^{1/2}$. This is because $(n m^{-1/2} b_n)^{1/2} m^{-1} \le n^{1/2} m^{-5/4} b_n^{-1/2} \rightarrow 0$ by the first relation in (d). This proves Theorem 1. $\square$

Proof of Theorem 2

Write

$$\begin{aligned}&\widehat{\lambda }_m(t_1 \mid T_2> t_2) - \lambda (t_1 \mid T_2> t_2)\\&\quad = \displaystyle \frac{1}{b_n} \displaystyle \int \left[ \widehat{\varLambda }_m(t_1 - b_n u \mid T_2> t_2) - \varLambda (t_1 - b_nu \mid T_2> t_2)\right] {\mathrm {d}}K_0(u)\\&\qquad + \frac{1}{b_n} \displaystyle \int \varLambda (t_1 - b_nu \mid T_2> t_2) dK_0(u) - \lambda (t_1 \mid T_2 > t_2)\\&\quad = (\widetilde{A}) + (\widetilde{B}). \end{aligned}$$

For the non-random term $(\widetilde{B})$ we have, similar to (11),

$$\begin{aligned} (\widetilde{B}) = O(b_n^2). \end{aligned}$$

(19)

For $(\widetilde{A})$, we perform analogous operations as we did in the proof of Theorem 1. This gives, in analogy with (12),

$$\begin{aligned} (\widetilde{A})= & {} \displaystyle \frac{1}{b_n} \displaystyle \mathop \int \nolimits _{-L}^L \displaystyle \frac{C_{m,n} [S_{1n} (t_1-b_nu), S_{2n}(t_2)] - C[S_1 (t_1 - b_nu), S_2(t_2)]}{C[S_1(t_1 - b_nu), S_2(t_2)]} {\mathrm {d}}K_0(u)\nonumber \\&+ \ \widetilde{R}_n (t_1, t_2) + O\left( \displaystyle \frac{m^{1/2}}{nb_n}\right) \ \text{ a.s. }, \end{aligned}$$

(20)

where

$$\begin{aligned} \widetilde{R}_n (t_1,t_2) = \displaystyle \frac{1}{2b_n} \displaystyle \mathop \int \nolimits _{-L}^L \displaystyle \frac{\{C_{m,n} [S_{1n} (t_1 - b_nu), S_{2n}(t_2)] - C[S_1(t_1-b_nu), S_2(t_2)]\}^2}{[1-\widetilde{\theta }_n(t_1-b_nu)]^2} {\mathrm {d}}K_0(u) \end{aligned}$$

for some $\widetilde{\theta }_n(t_1-b_nu)$ between $C_{m,n} [S_{1n} (t_1-b_nu), S_{2n}(t_2)]$ and $C[S_1 (t_1- b_nu), S_2(t_2)]$. The $O\left( \displaystyle \frac{m^{1/2}}{nb_n}\right)$ term in (20) comes from the replacement of $S_{1n}(s-)$ by $S_{1n}(s)$.

Indeed, for n sufficiently large, we have for some constant $M >0$:

$$\begin{aligned} \begin{array}{ll} &{}\left| \displaystyle \mathop \int \nolimits _0^{t_1} \left\{ \displaystyle \frac{1}{C_{m,n} [S_{1n} (s-), S_{2n}(t_2)]} - \displaystyle \frac{1}{C_{m,n} [S_{1n}(s), S_{2n}(t_2)]}\right\} d_s C_{m,n} [S_{1n}(s), S_{2n}(t_2)]\right| \\ \\ &{}\quad \le \displaystyle \frac{M}{C^2[S_1(t_1), S_2(t_2)]}\sup \limits _s \sum \limits _{k=1}^{m} \sum \limits _{\ell = 1}^{m} C_n \left( \displaystyle \frac{k}{m},\displaystyle \frac{\ell }{m}\right) \displaystyle \frac{P_{m,\ell } [S_{2n}(t_2)]}{n}\left| P'_{m,k} \left( S_{1n}(s) + \frac{1}{n}\right) \right| \\ \\ &{}\quad = O\left( \displaystyle \frac{m^{1/2}}{n}\right) \ \text{ a.s., } \text{ using } \text{ Lemma } \text{1 } \text{ in }\, Janssen et al. (2014). \end{array} \end{aligned}$$

Using that $C_{m,n}[S_{1n}(s), S_{2n}(t_2)] \rightarrow C[S_1(s), S_2(t_2)]$ a.s. and the fact that $K_0$ is of bounded variation we can make an argument completely analogous to the one used for $R_n(t_1,t_2)$ in (12). This gives the following bound for $\widetilde{R}_n(t_1,t_2)$:

$$\begin{aligned} \widetilde{R}_n(t_1,t_2) = O\left( \displaystyle \frac{1}{b_n}\left\{ \sup \limits _s \mid C_{m,n} [S_{1n} (s), S_{2n}(t_2)] - C[S_1(s), S_2(t_2)]\mid \right\} ^2\right) \ \text{ a.s. } \end{aligned}$$

Now,

$$\begin{aligned} \begin{array}{l} \mid C_{m,n} [S_{1n}(s)), S_{2n}(t_2)] - C[S_1(s), S_2(t_2)]\mid \\ \quad \le \,\displaystyle \mid C_{m,n} [S_{1n} (s), S_{2n}(t_2)] - C[S_{1n}(s), S_{2n}(t_2)]\mid \\ \qquad + \mid S_{1n}(s) - S_1(s)\mid + \mid S_{2n} (s) - S_2(s)\mid \end{array} \end{aligned}$$

by the Lipschitz continuity of C (see Nelsen 2006).

The supremum of the first term on the right-hand side is $O(n^{-1/2} (\ln \ln n)^{1/2} + m^{-1/2})$ a.s. (see the proof of Theorem 1 in Janssen et al. (2012)) and the supremum of the other two terms is $O(n^{-1/2} (\ln \ln n)^{1/2})$ a.s. So the bound for $\widetilde{R}_n(t_1,t_2)$ is

$$\begin{aligned} O\left( \displaystyle \frac{n^{-1} \ln \ln n}{b_n} + \displaystyle \frac{m^{-1}}{b_n}\right) \ \text{ a.s. } \end{aligned}$$

Combining this with (19) and (20), we obtain

$$\begin{aligned}&\widehat{\lambda }_m (t_1 \mid T_2> t_2) - \lambda (t_1 \mid T_2 > t_2)\\&\quad = \displaystyle \frac{1}{b_n} \displaystyle \mathop \int \nolimits _{-L}^L \displaystyle \frac{C_{m,n} [S_{1n} (t_1 - b_n u), S_{2n}(t_2)] - C[S_1 (t_1-b_nu), S_2(t_2)]}{C[S_1 (t_1 - b_nu), S_2(t_2)]} {\mathrm {d}}K_0(u)\\&\qquad + \ O \left( \displaystyle \frac{n^{-1}}{b_n} \ln \ln n + \displaystyle \frac{m^{-1}}{b_n} + \displaystyle \frac{m^{1/2}}{nb_n} + b_n^2\right) \ \text{ a.s. } \end{aligned}$$

For the first term in the right-hand side, we write

$$\begin{aligned}&C_{m,n} [S_{1n} (t_1), S_{2n}(t_2)] - C[S_1(t_1), S_2(t_2)]\\&\quad = \left\{ C_{m,n} [S_1 (t_1), S_2(t_2)] - C[S_1(t_1), S_2(t_2)]\right\} \\&\qquad + \ C_{m,n}^{(1)} [\theta _{1n} (t_1), \theta _{2n}(t_2)] \left[ S_{1n} (t_1) - S_1(t_1)\right] \\&\qquad + \ C_{m,n}^{(2)} [\theta _{1n} (t_1), \theta _{2n}(t_2)] \left[ S_{2n} (t_2) - S_2(t_2)\right] , \end{aligned}$$

with $(\theta _{1n} (t_1), \theta _{2n} (t_2))$ denoting an intermediate point between $(S_{1n} (t_1), S_{2n}(t_2))$ and $(S_1(t_1), S_2(t_2))$. Now using similar ideas as in Lemma 3 of Janssen et al. (2012) and the convergence rate of the Bernstein approximation given in (5) of the same paper, we obtain

$$\begin{aligned}&C_{m,n}[S_{1n}(t_1), S_{2n}(t_2)] - C[S_1(t_1), S_2(t_2)]\\&\quad = \displaystyle \frac{1}{n} \sum \limits _{i=1}^n Y_{mi} [S_1(t_1), S_2(t_2)] + O_P (m^{-1}) + o_P (n^{-1/2}) \end{aligned}$$

where the $Y_{m} (u_1,u_2)$ are independent zero mean random variables which are bounded. With this

$$\begin{aligned}&\widehat{\lambda }_{m}(t_1 \mid T_2> t_2) - \lambda (t_1 \mid T_2 > t_2)\\&\quad = \sum \limits _{i=1}^n W_{in} + O_P \left( \displaystyle \frac{n^{-1}}{b_n} \ln \ln n + \displaystyle \frac{m^{-1}}{b_n} + \displaystyle \frac{m}{nb_n} + b_n^2 + \displaystyle \frac{n^{-1/2}}{b_n}\right) , \end{aligned}$$

where

$$\begin{aligned} W_{in} = \displaystyle \frac{1}{nb_n} \displaystyle \int \displaystyle \frac{Y_{mi} [S_1(t_1-b_nu),S_2(t_2)]}{C[S_1(t_1-b_nu),S_2(t_2)]} {\mathrm {d}}K_0(u). \end{aligned}$$

Now

$$\begin{aligned} \begin{array}{ll} &{}\mathrm{Var}\left( \sum \limits _{i=1}^n W_{in}\right) = \\ \\ &{}\quad \displaystyle \int \displaystyle \int \displaystyle \frac{E\{Y_{mi} [S_1(t_1 - b_nu_1), S_2(t_2)] Y_{mi} [S_1(t_1-b_n u_2), S_2(t_2)]\}}{nb_n^2C[S_1 (t_1 - b_n u_1), S_2(t_2)] C[S_1(t_1-b_nu_2), S_2(t_2)]} {\mathrm {d}}K_0 (u_1) {\mathrm {d}}K_0(u_2)\\ \\ &{}\quad = O\left( \displaystyle \frac{1}{nb_n^2}\right) \end{array} \end{aligned}$$

by the boundedness of the $Y_{mi}$ and the fact that $K_0$ is of bounded variation.

Hence,

$$\begin{aligned} \sum \limits _{i=1}^n W_{in} = O_P\left( \displaystyle \frac{n^{-1/2}}{b_n}\right) \end{aligned}$$

and

$$\begin{aligned}&\widehat{\lambda }_m (t_1 \mid T_2> t_2) - \lambda (t_1 \mid T_2 > t_2)\\&\quad = O_P\left( \displaystyle \frac{n^{-1/2}}{b_n} + \displaystyle \frac{m^{-1}}{b_n} + \displaystyle \frac{m^{1/2}}{nb_n} + b_n^2\right) . \end{aligned}$$

The imposed conditions in (d) of Theorem 1 and the extra condition $m^{1/2} b_n \rightarrow \infty$ imply that all the terms in the right-hand side vanish after multiplication with $(nm^{-1/2} b_n)^{1/2}$. $\square$

Proof of Theorem 3

Linearization of the ratio gives that $\widehat{\theta }_m(t_1,t_2) - \theta (t_1, t_2)$ has the same limiting distribution as

$$\begin{aligned} \displaystyle \frac{1}{\lambda (t_1 \mid T_2> t_2)} [ \widehat{\lambda }_m(t_1 \mid T_2 = t_2) - \lambda (t_1 \mid T_2 = t_2)]\\\\ - \displaystyle \frac{\lambda (t_1 \mid T_2 = t_2)}{\lambda ^2(t_1 \mid T_2> t_2)} [\widehat{\lambda }_m (t_1 \mid T_2> t_2) - \lambda (t_1 \mid T_2 > t_2)]. \end{aligned}$$

Multiplication with $(nm^{-1/2} b_n)^{1/2}$ gives that the second term is $o_P(1)$ (by Theorem 2) and that the first term is asymptotically normal (by Theorem 1). $\square$