A class of Minimum Distance Estimators in Markovian Multiplicative Error Models

Abstract

This paper proposes a class of minimum distance estimators for the underlying parameters in a Markovian parametric multiplicative error time series model. This class of estimators is based on the integrals of the square of a certain marked residual process. The paper derives the asymptotic distributions of the proposed estimators. In a finite sample comparison, some members of the proposed class of estimators dominate a generalized method of moments estimator in terms of the finite sample bias at a variety of chosen error distributions while neither dominate each other in terms of the mean squared error at these error distributions. A real data example is considered to illustrate the proposed estimation procedures.

Appendix A: Main Proofs

1.1 A.1 Proofs of Theorems 4.1 and 4.2

Proof 1 (Proof of Theorem 4.1).

The proof of (4.4) is given later. Arguing as in the proof of Theorem 5.4.1 of Koul (2002), one verifies that (4.1), (C.3) and (4.4) imply Eq. 4.5 and (4.6)(a).

To prove (4.6)(b), by the positive definiteness of ${\mathcal G}_{1}$ we clearly have $\widetilde t_{n1}= {\mathcal G}_{1}^{-1}V_{n1}$. Moreover, V_n1 is a vector of the sums of martingale difference arrays satisfying (4.3). By the martingale central limit theorem (CLT), see Hall and Heyde (1980), $V_{n1}\to _{D} N(0, \sigma ^{2}{\Sigma }_{1})$ and $\widetilde t_{n1}={\mathcal G}_{1}^{-1}V_{n1}\to _{D} N(0, \sigma ^{2} {\mathcal G}_{1}^{-1}{\Sigma }_{1} {\mathcal G}_{1}^{-1}).$ This fact together with the Slutsky Theorem, (4.5) and (4.6)(a) imply (4.6)(b).

Proof of (4.4).

Let θ_nt := θ + n^− 1/2t, $t\in {\mathbb R}^{q}$. Define, for $z\in {\mathbb R}_{+}^{p}, t\in {\mathbb R}^{q}$,

$$ \begin{array}{@{}rcl@{}} W_{n}(z,t)&:= n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{\psi(Z_{i-1},\theta)}{\psi(Z_{i-1}, \theta_{nt})}- 1 \Big] \varepsilon_{i} I(Z_{i-1}\le z), \end{array} $$

$$ \begin{array}{@{}rcl@{}} S_{n}(z) \!\!\!\!&:=&\!\!\!\! n^{-1}\sum\limits_{i=1}^n \varphi(Z_{i-1}) (\varepsilon_{i} - 1) I(Z_{i-1}\!\le\! z), \quad T_{n11}(t)\! :=\! \int \big(W_{n}(z,t)+t^{\prime}{\Psi}(z)\big)^{2}dL(z), \\ T_{n12}(t) \!\!\!\!&:=&\!\!\!\! \int (W_{n}(z,t)+t^{\prime}{\Psi}(z)) (U_{n}(z) - t^{\prime}{\Psi}(z))dL(z). \end{array} $$

Use the model assumption ε_i = Y_i/ψ(Z_i− 1, θ) to obtain

$$ \begin{array}{@{}rcl@{}} U_{n}(z,\theta_{nt})\!\!\!&=&\!\!\!n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{Y_{i}}{\psi(Z_{i-1}, \theta_{nt})}-1\Big]I(Z_{i-1}\le z) \\ \!\!\!&=&\!\!\! n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{Y_{i}}{\psi(Z_{i-1}, \theta_{nt})}- \frac{Y_{i}}{\psi(Z_{i-1}, \theta)} +\frac{Y_{i}}{\psi(Z_{i-1}, \theta)} -1\Big]I(Z_{i-1}\le z) \\ \!\!\!&=&\!\!\! n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{\psi(Z_{i-1},\theta)}{\psi(Z_{i-1}, \theta_{nt})}- 1\Big] \varepsilon_{i} I(Z_{i-1}\le z) + n^{-1/2}\sum\limits_{i=1}^n (\varepsilon_{i}-1)I(Z_{i-1}\le z) \\ \!\!\!&=&\!\!\!W_{n}(z,t)+U_{n}(z), \qquad \forall z\in {\mathbb R}_{+}^{p}, t\in {\mathbb R}. \end{array} $$

(A.1)

Hence,

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!T_{n1}(t)\!\!\!&=&\!\!\!\int {U_{n}^{2}}(z,\theta_{nt}) dL(z) = \int \Big(W_{n}(z,t)+t^{\prime}{\Psi}(z)+U_{n}(z)-t^{\prime}{\Psi}(z)\Big)^{2} dL(z) \\ \!\!\!&=&\!\!\!T_{n11}(t)+2 T_{n12}(t) + Q_{n1}(t). \end{array} $$

(A.2)

We shall shortly prove the following facts. For every $0<b<\infty$,

$$ \begin{array}{@{}rcl@{}} \text{(a)} \quad \sup_{\|t\|\le b} T_{n11}(t)=o_p(1), \qquad \text{(b)} \quad E\Big(\sup_{\|t\|\le b} Q_{n1}(t)\Big)=O(1). \end{array} $$

(A.3)

Then by the Cauchy-Schwarz inequality, $\sup _{\|t\|\le b}\big |T_{n12}(t)\big |^{2}\le \sup _{\|t\|\le b} T_{n11}$ $(t) \sup _{\|t\|\le b} Q_{n1}(t)$ = o_p(1). Hence the claim Eq. 4.4.

Proof of (A.3)(a).

Rewrite

$$ \begin{array}{@{}rcl@{}} W_{n}(z,t)\!\!\!&:=&\!\!\! n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{\psi(Z_{i-1},\theta)}{\psi(Z_{i-1}, \theta_{nt})}-1 \Big] \varepsilon_{i} I(Z_{i-1}\le z) \\ \!\!\!&=&\!\!\! - n^{-1/2}\sum\limits_{i=1}^n \frac{1}{\psi(Z_{i-1}, \theta_{nt})} \Big[\psi(Z_{i-1}, \theta_{nt}) - \psi(Z_{i-1},\theta)-n^{-1/2}t^{\prime} \dot \psi(Z_{i-1},\theta)\Big] \\&&\!\!\!\varepsilon_{i} I(Z_{i-1}\le z) \\ && \!\!\!- t^{\prime} n^{-1}\sum\limits_{i=1}^n \Big[\frac{1}{\psi(Z_{i-1}, \theta_{nt})} - \frac{1}{\psi(Z_{i-1},\theta)} \Big] \dot \psi(Z_{i-1},\theta) \varepsilon_{i} I(Z_{i-1}\le z)\\ && \!\!\!- t^{\prime}n^{-1}\sum\limits_{i=1}^n \varphi(Z_{i-1})(\varepsilon_{i}-1) I(Z_{i-1}\le z) \\ && \!\!\!- t^{\prime}n^{-1}\sum\limits_{i=1}^n \big[ \varphi(Z_{i-1}) I(Z_{i-1}\le z) - E\big(\varphi(Z_{0})I(Z_{0}\le z)\big)\big] - t^{\prime}{\Psi}(z). \end{array} $$

Let d_it := ψ(Z_i− 1, θ_nt) − ψ(Z_i− 1, θ) and $\delta _{it}:= d_{it} -n^{-1/2}t^{\prime } \dot \psi (Z_{i-1}, \theta ).$ Then the above identity is equivalent to

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\!\!\!\!\!\!\!\!W_{n}(z,t) + t^{\prime} {\Psi}(z)\\ &&\!\!\!\!\!\!\!\!\!\!\!\!= - n^{-1/2}\sum\limits_{i=1}^n \frac{\delta_{it}}{\psi(Z_{i-1}, \theta_{nt})} \varepsilon_{i} I(Z_{i-1}\le z) + t^{\prime} n^{-1}\sum\limits_{i=1}^n \frac{d_{it}}{\psi(Z_{i-1}, \theta_{nt})} \varphi(Z_{i-1}) \varepsilon_{i} I(Z_{i-1}\le z) \\ && \!\!\!\!\!\!-t^{\prime}S_{n}(z)-t^{\prime}\big({\Psi}_{n}(z)-{\Psi}(z)\big) \\ &&\!\!\!\!\!\!\!\!\!\!\!\!= A_{1}(z,t)+ A_{2}(z,t)-t^{\prime}S_{n}(z)-t^{\prime}\big({\Psi}_{n}(z)-{\Psi}(z)\big), \qquad \text{say}. \end{array} $$

(A.4)

In the sequel, the range of the vectors z, 𝜗 in $\inf _{z,\vartheta }$ is over ${\mathbb R}_{+}^{p} \times {\Theta }$, unless mentioned otherwise. By (C.1), $\inf _{z, \vartheta }\psi (z,\vartheta )\ge C>0$ and for every $b<\infty$, $\sup _{1\le i\le n, \|t\|\le b}n^{1/2}\big |\delta _{it}\big |=o_{p}(1)$. Moreover, because E(ε₀) = 1, we have $n^{-1}{\sum }_{i=1}^n \varepsilon _{i}=O_{p}(1)$. Hence

$$ \begin{array}{@{}rcl@{}} \sup_{z\in {\mathbb R}_+^p, \|t\|\le b}|A_1(z,t)|\le C^{-1}\sup_{1\le i\le n, \|t\|\le b}n^{1/2}\big|\delta_{it}\big|n^{-1}\sum\limits_{i=1}^n \varepsilon_i=o_p(1). \end{array} $$

(A.5)

Next, recall that the stationarity of the process $Y_{i}, i\in {\mathbb Z}$ and $E\|\dot \psi (Z_{0},\theta )\|^{2}$ $<\infty$ imply that $n^{-1/2} \max \limits _{1\le i\le n}\|\dot \psi (Z_{i-1},\theta )\| =o_{p}(1)$, $E\big (n^{-1}{\sum }_{i=1}^{n} \|\varphi$ $(Z_{i-1}) \| \varepsilon _{i}\big )= E\|\varphi (Z_{0})\|<\infty$, and by the Markov inequality, $n^{-1}{\sum }_{i=1}^{n} \|\varphi$ (Z_i− 1)∥ε_i = O_p(1). Hence, (C.1) and (C.2) imply that

$$ \begin{array}{@{}rcl@{}} &&D_{n} := \sup_{1\le i\le n, \|t\|\le b} |d_{it}| \le \sup_{1\le i\le n, \|t\|\le b} \big|\delta_{it}\big| + b n^{-1/2} \max_{1\le i\le n}\|\dot \psi(Z_{i-1},\theta)\| =o_{p}(1),\\ &&\sup_{z\in {\mathbb R}_{+}^{p}, \|t\|\le b}|A_{2}(z,t)| \le b C^{-1} D_{n} n^{-1}\sum\limits_{i=1}^n \|\varphi(Z_{i-1})\| \varepsilon_{i} =o_{p}(1). \end{array} $$

(A.6)

Next, consider S_n(z). Observe that S_n(z) is a vector of weighted empirical processes with the summands of each component being stationary and ergodic and ES_n(z) ≡ 0. Using a Glivenko-Cantelli Lemma type argument one obtains that $\sup _{z\in {\mathbb R}_{+}^{p}}\|S_{n}(z)\|=o_{p}(1)$. For details see Stute (1976) and Koul (2019). Similarly, $\sup _{z\in {\mathbb R}_{+}^{p}}\big \|{\Psi }_{n}(z)-{\Psi }(z)\big \|^{2}=o_{p}(1)$. Upon combining these two facts with (A.4), (A.5) and (A.6) we obtain that for every $0<b<\infty$,

$$ \begin{array}{@{}rcl@{}} \sup_{z\in {\mathbb R}_+^p, \|t\|\le b}\big|W_n(z,t) + t^{\prime} {\Psi}(z)\big|=o_p(1). \end{array} $$

(A.7)

This fact combined with the definition of T_n11 and L being a d.f. readily yields (A.3)(a).

Next, to prove (A.3)(b), note that

$$ \begin{array}{@{}rcl@{}} Q_{n1}(t)&:=& \int \Big(U_{n}(z,\theta)-t^{\prime}{\Psi}(z)\Big)^{2} dL(z)\le 2 T_{n1}(0) + 2 t^{\prime}{\mathcal G}_{1} t, \\ E\Big(\sup_{\|t\|\le b}Q_{n1}(t)\Big)&\le& 2ET_{n1}(0)+ 2 b^{2} \int \|{\Psi}(z)\|^{2} dL(z)\\ &=& 2 \int G(z)dL(z)\big[1+ b^{2} E\|\varphi(Z_{0})\|^{2}\big]<\infty. \end{array} $$

This also completes the proof of Theorem 4.1. □

Proof 2 (Proof of Theorem 4.2).

The proof is similar to that of Theorem 4.1, with some differences. Let

$$ \begin{array}{@{}rcl@{}} && \tilde V_{n2}:= \int U_n(z){\Psi}(z)dG_n(z), \qquad {\mathcal G}_{n2}:=\int {\Psi}(z){\Psi}(z)' dG_n(z), \\ && \tilde Q_{n2}(t):= \int \big(U_n(z)-t^{\prime}{\Psi}(z)\big)^2 dG_n(z)= M_{n2}(\theta) -2t^{\prime}\tilde V_{n2} +t^{\prime} {\mathcal G}_{n2} t, \\ &&\tilde T_{n2}(t):=M_{n2}(\theta+n^{-1/2}t), \qquad \tilde T_{n21}(t) \!:=\! \int \big(W_n(z,t)+t^{\prime}{\Psi}(z)\big)^2dG_n(z), \\ && \tilde T_{n12}(t) := \int (W_n(z,t)+t^{\prime}{\Psi}(z)) (U_n(z) - t^{\prime}{\Psi}(z))dG_n(z). \end{array} $$

Then akin to (A.2),

$$ \begin{array}{@{}rcl@{}} \tilde T_{n2}(t)=\int {U_{n}^{2}}(z,\theta_{nt}) dG_{n}(z) =\tilde T_{n21}(t)+2 \tilde T_{n22}(t) + \tilde Q_{n2}(t). \end{array} $$

Now, $\sup _{\|t\|\le b} \tilde T_{n21}(t)\le \sup _{z\in {\mathbb R}_{+}^{p}, \|t\|\le b}\big (W_{n}(z,t)+t^{\prime }{\Psi }(z)\big )^{2}=o_{p}(1),$ by (A.7). We shall shortly prove that for every $0<b<\infty$,

$$ \begin{array}{@{}rcl@{}} \sup_{\|t\|\le b} \big|\tilde Q_{n2}(t) - Q_{n2}(t)\big|=o_p(1). \end{array} $$

(A.8)

Argue as for (A.3)(b) to conclude that $E\big (\sup _{\|t\|\le b} Q_{n2}(t)\big )=O(1).$ Hence $\sup _{\|t\|\le b} \tilde Q_{n2}(t)=O_{p}(1)$ and $\sup _{\|t\|\le b}\big |\tilde T_{n22}(t)\big |^{2}\le \sup _{\|t\|\le b} \tilde T_{n21}(t) \sup _{\|t\|\le b}$ $\tilde Q_{n2}(t)$ = o_p(1). These facts together yield the claim Eq. 4.7.

Next, to prove (A.8), note that the left hand side of (A.8) is bounded from the above by

$$ \begin{array}{@{}rcl@{}} \big|\int {U_{n}^{2}}\big[dG_{n} -dG\big]\big| +b \big\|\int U_{n} {\Psi}\big[dG_{n}- dG\big]\big\|+b^{2} \big\|\int {\Psi} {\Psi}^{\prime}\big[dG_{n}-dG\big] \big\|. \end{array} $$

By the Ergodic Theorem, $\big \|{\int \limits } {\Psi } {\Psi }^{\prime }\big [dG_{n}-dG\big ]\big \|\to 0$, a.s. while by Lemmas A.1 and A.2 below, the first two terms tend to zero in probability. This completes the proof of (4.7). The proofs of the other two claims of this theorem are similar to those of (4.5) and (4.6)(a), (b) of Theorem 4.1. □

1.2 A.2 Proof of Lemma 4.1

Proof 3 (Proof of the claim about (C.3)).

Let h be a positive function on ${\mathbb R}_{+}^{p}$ with $0<{\int \limits } h^{2} dL<\infty$ and let ${\mathbf {\infty }}$ (0) denote the vector of p infinities (zeros). Let $\varphi (z):= {\int \limits }_{x\le z} h(x) dL(x)$. Note that φ is nondecreasing in each coordinate and $\varphi ({\mathbf {\infty }})<\infty$. Moreover, $\gamma (z):={\int \limits } I(z\le x) h(x) dL(x)={\int \limits } I(z\le x) d\varphi (x)\ge 0$, for all $z\in {\mathbb R}_{+}^{p}$. Define

$$ \begin{array}{@{}rcl@{}} {\mathcal V}_{n1}(t)&:=& \int U_{n}(z,\theta_{nt}) h(z)dL(z)= \int U_{n}(z,\theta_{nt}) d\varphi(z) \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=&n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{Y_{i}}{\psi(Z_{i-1}, \theta+n^{-1/2}t)} -1\Big]\int I(Z_{i-1}\le z)d \varphi(z) \\ &=&n^{-1/2} \sum\limits_{i=1}^n \Big[\frac{Y_{i}}{\psi(Z_{i-1}, \theta+n^{-1/2}t)} -1\Big] \gamma(Z_{i-1}). \end{array} $$

By the Cauchy-Schwarz inequality,

$$ \begin{array}{@{}rcl@{}} M_{n1}(\theta+n^{-1/2}t)\ge \Big(\int U_n(z,\theta_{nt}) h(z)dL(z)\Big)^2 \Big /\int h^2 dL, \quad \forall t\in {\mathbb R}^q. \end{array} $$

Because γ(z) ≥ 0 for all z, by (4.8), $\forall e\in R^{q}, \|e\|=1, {\mathcal V}_{n1}(r e)$ is monotonic in r. Hence

$$ \begin{array}{@{}rcl@{}} \inf_{\|t\|>b}M_{n1}(\theta + n^{1/2}t) = \inf_{e \in {\mathbb R}^{q}, \|e\|=1, |r|>b}M_{n}(\theta + n^{-1/2}re) \!\ge\! \inf_{e \in {\mathbb R}^{q}, \|e\|=1, |r|=b}{\mathcal V}_{n1}^{2}(re)\Big / \int h^{2} dL. \end{array} $$

Let

$$ \begin{array}{@{}rcl@{}} \widehat {\mathcal V}_{n1}(t) \!\!\!&:=&\!\!\! \int \big(U_{n}(z)-t^{\prime}{\Psi}(z)\big) h(z)dL(z)= \int U_{n}(z) d\varphi(z) - t^{\prime} \int {\Psi}(z)d\varphi(z) \\ \!\!\!&=&\!\!\! {\mathcal V}_{n1}(0) - t^{\prime} \int {\Psi}(z)d\varphi(z). \end{array} $$

In view of (A.1),

$$ {\mathcal V}_{n1}(t) - \widehat {\mathcal V}_{n1}(t)= \int \big\{U_{n}(z,\theta_{nt}) - U_{n}(z)+t^{\prime}{\Psi}(z)\big\} d\varphi(z) = \int \big\{W_{n}(z,\theta_{nt})+t^{\prime}{\Psi}(z)\big\}d\varphi(z) . $$

Because $\varphi ({\mathbf {\infty }})={\int \limits } h dL<\infty$, by (A.7),

$$ \begin{array}{@{}rcl@{}} \sup_{\|t\|\le b} \Big|{\mathcal V}_{n1}(t) - \widehat {\mathcal V}_{n1}(t)\Big|\le \sup_{\|t\|\le b} \big|W_{n}(z,\theta_{nt})+t^{\prime}{\Psi}(z)\big| \varphi({\mathbf{\infty}}) =o_{p}(1). \end{array} $$

Therefore,

$$ \begin{array}{@{}rcl@{}} \Big|\inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b}{\mathcal V}_{n1}(re) - \inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b} \widehat {\mathcal V}_{n1}(re)\Big|=o_p(1). \end{array} $$

(A.9)

Let $\tau _{2}:={\int \limits } h^{2} dL$. Fix an 𝜖 > 0 and $0<\eta <\infty$. By (A.9), there exists N_{𝜖, η} such that

$$ \begin{array}{@{}rcl@{}} && P\Big(\inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b}\Big|{\mathcal V}_{n1}(re)\Big| \ge (\tau_2 \eta)^{1/2} \Big) \\ && \hskip .5in \ge P\Big(\inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b}\Big|\widehat {\mathcal V}_{n1}(re)\Big| \ge (\tau_2 \eta)^{1/2} \Big ) - \frac{\epsilon}{2}, \qquad \forall n>N_{\epsilon,\eta}. \end{array} $$

Moreover, by the Cauchy-Schwarz inequality, $E{\mathcal V}_{n1}^{2}(0)=\sigma ^{2} E\gamma ^{2}(Z_{0}) \le \sigma ^{2}{\int \limits } h^{2} dL<\infty .$ Hence by the Markov inequality, for every 𝜖 > 0 there is a b_𝜖 such that

$$ \begin{array}{@{}rcl@{}} P\big(|{\mathcal V}_{n1}(0)|\le b_{\epsilon}\big)\ge 1-(\epsilon/2), \quad \forall n\ge 1. \end{array} $$

(A.10)

Recall the elementary fact that for any real numbers a, b, ||a|−|b||≤|a ± b|. Also let $K:= {\int \limits } \|{\Gamma }\| d \varphi$. Let b_𝜖 be as in (A.10). Choose b ≥ (b_𝜖 + (v₂η)^1/2)K^− 1. Then, ∀n ≥ N_{𝜖, η},

$$ \begin{array}{@{}rcl@{}} P\Big(\inf_{|t|> b} M_{n1}(\theta+n^{-1/2}t)\ge \eta \Big ) &\ge& P\Big(\inf_{e \in {\mathbb R}^{q}, \|e\|=1, |r|=b}\Big|{\mathcal V}_{n1}(re)\Big| \ge (\tau_{2} \eta)^{1/2} \Big )\\ &=&P\Big(\Big|{\mathcal V}_{n1}(\pm be)\Big| \ge (\tau_{2} \eta)^{1/2}, \forall \|e\|=1 \Big ) \\ &\ge& P\Big(\Big|\widehat {\mathcal V}_{n1}(\pm be)\Big| \ge (\tau_{2} \eta)^{1/2}, \forall \|e\|=1\Big ) - \epsilon/2 \\ &\ge &P\Big(\ \Big | \big|{\mathcal V}_{n1}(0 )|- bK\big |\ \Big| \ge (\tau_{2}\eta)^{1/2} \Big ) - \epsilon/2 \\ &\ge &P\Big(\ |{\mathcal V}_{n1}(0)|\le bK - (\tau_{2}\eta)^{1/2} \Big ) - \epsilon/2 \\ &\ge &P\Big(\ |{\mathcal V}_{n1}(0)|\le b_{\epsilon} \Big ) - \epsilon/2 \ge 1-\epsilon, \end{array} $$

(A.11)

thereby proving the claim about (C.3).

Proof of the claim about (C.4)

The proof of this claim is similar to that of the previous claim, but with some differences. In particular we need the following two preliminary lemmas.

Lemma A.1.

Under the above set up, U_n converges weakly to a continuous Gaussian process ${\mathcal Z}(x), x\in {\mathbb R}_{+}^{p}$ in Skorokhod space $D([0,\infty ]^{p})$ and uniform metric.

The proof of this lemma is similar to that of the main theorem in Stute (1976).

Lemma A.2.

Let ${\mathcal U}$ be a relatively compact subset of $D[0,\infty ]^{p}$. Let μ_n, μ be a sequence of possibly random multivariate distribution functions on $[0,\infty )^{p}$ such that $\sup _{x\in {\mathbb R}_{+}^{p}} \big |\mu _{n}(x)-\mu (x)\big |\to 0$, a.s. Then $\sup _{y\in {\mathbb R}_{+}^{p}, \alpha \in {\mathcal U}}\Big | {\int \limits }_{x\le y} \alpha (x)$ [dμ_n(x) − dμ(x)]| →_p0.

The proof of this lemma is similar to that of Lemma 3.1 of Chang (1990) and Lemma 4.2 of Koul and Stute (1999). Details are left out for the sake of brevity.

Now, let h be a positive function on ${\mathbb R}_{+}^{p}$ with $0<{\int \limits } h^{2} dG<\infty$ and ${\int \limits } h dG=1$. Let

$$ \begin{array}{@{}rcl@{}} \beta_{nk}(z)&:=&{\int}_{x\le z} h^{k}(x) dG_{n}(x) =n^{-1}\sum\limits_{j=1}^n h^{k}(Z_{j-1})I(Z_{j-1}\le z), \\ \beta_{k}(z)&:=&{\int}_{x\le z} h^{k}(x) dG(x), \quad k=1, 2, \ z\in {\mathbb R}_{+}^{p}, \qquad B_{n2}:=\beta_{n2}({\mathbf{\infty}}), \quad B_{2}:=\beta_{2}({\mathbf{\infty}}). \end{array} $$

Note that β_nk(0) = 0 = β_k(0), $0< \beta _{k}(z)\le \beta _{k}(\mathbf {\infty })={\int \limits } h^{k} dG<\infty ,$ for all $z\in {\mathbb R}_{+}^{p}$ and Eβ_nk(z) ≡ β_k(z), k = 1, 2. By the Ergodic Theorem and a Glivenko-Cantelli type argument, see Stute (1976) and Koul (2019),

$$ \begin{array}{@{}rcl@{}} \sup_{z\in[0,\infty]^p} \big|\beta_{nk}(z)-\beta_k(z)\big|\to 0, \quad \text{a.s.}, k=1, 2; \quad |B_{n2}-B_2|\to 0, \text{ a.s.} \end{array} $$

(A.12)

Thus for all sufficiently large n, β_nk(z) > 0, for all $z\in (0,\infty ]^{p}, k=1, 2$, B_n2 > 0 (a.s.). The arguments below are carried out on the event $B_{n2}={\int \limits } h^{2} dG_{n}>0$.

Recall θ_nt = θ + n^− 1/2t. By the Cauchy-Schwarz inequality,

$$ \begin{array}{@{}rcl@{}} M_{n2}(\theta+n^{-1/2}t)\ge \Big(\int U_n(z,\theta_{nt}) h(z)dG_n(z)\Big)^2 \Big / B_{n2}, \quad \forall t\in {\mathbb R}^q. \end{array} $$

Let $\alpha (z):= {\int \limits }_{x\ge z} d\beta _{1}(x)$, $\alpha _{n}(z):= {\int \limits }_{x\ge z} d\beta _{n1}(x)$, $z\in {\mathbb R}_{+}^{p}$ and

$$ \begin{array}{@{}rcl@{}} {\mathcal V}_{n2}(t)&\!\!\!:=&\!\!\!\! \int U_{n}(z,\theta_{nt}) h(z)dG_{n}(z) =n^{-1/2}\sum\limits_{i=1}^n \Big[\frac{Y_{i}}{\psi(Z_{i-1}, \theta_{nt})} -1\Big]\int I(Z_{i-1}\le z)d\beta_{n1}(z)\\ &=&\!\!\!n^{-1/2} \sum\limits_{i=1}^n \Big[\frac{Y_{i}}{\psi(Z_{i-1}, \theta+n^{-1/2}t)} -1\Big] \alpha_{n}(Z_{i-1}). \end{array} $$

Because α_n(z) ≥ 0 for all $z\in {\mathbb R}_{+}^{p}, n\ge 1$, w.p.1, by (4.8), ${\mathcal V}_{n2}(r e)$ is monotonic in r for all e ∈ R^q, ∥e∥ = 1. Hence

$$ \begin{array}{@{}rcl@{}} \inf_{\|t\|>b}M_{n2}(\theta+n^{1/2}t) = \inf_{e \in {\mathbb R}^{q}, \|e\|=1, |r|>b}M_{n2}(\theta+n^{-1/2}re)\ge \inf_{e \in {\mathbb R}^{q}, \|e\|=1, |r|=b}{\mathcal V}_{n2}^{2}(re)\Big / B_{n2}. \end{array} $$

Let

$$ \begin{array}{@{}rcl@{}} \widehat {\mathcal V}_{n2}(t) &:=& \int \big(U_{n}(z)-t^{\prime}{\Psi}(z)\big) h(z)dG_{n}(z)= \int U_{n}(z) d\beta_{n1}(z) - t^{\prime} \int {\Psi}(z)d\beta_{n1}(z) \\ &=& {\mathcal V}_{n2}(0) - t^{\prime} \int {\Psi}(z)d\beta_{n1}(z), \\ {\mathcal V}_{n}^{*} &:=& \int U_{n}(z) d\beta_{1}(z)=n^{-1/2} \sum\limits_{i=1}^n (\varepsilon_{i}-1)\alpha(Z_{i-1}). \end{array} $$

By Lemma A.1, the process $U_{n}(z), z\in [0,\infty ]^{p}$ is relatively compact with respect to the uniform metric. Hence (A.12) and Lemma A.2 applied with $\mu _{n}=\beta _{n1}/\beta _{n1}({\mathbf {\infty }}), \mu =\beta _{1}$ yield

$$ \begin{array}{@{}rcl@{}} \big|{\mathcal V}_{n2}(0)-{\mathcal V}_n^{*}\big|=\Big|\int U_n(z)[ d\beta_{n1}(z)-d\beta_1(z)]\Big|=o_p(1). \end{array} $$

Here we have used the fact $\beta _{1}({\mathbf {\infty }})={\int \limits } h dG=1$ and $\beta _{n1}({\mathbf {\infty }})\to _{p} \beta _{1}({\mathbf {\infty }})=1$.

Let $\bar {\mathcal V}_{n2}(t):={\mathcal V}_{n}^{*} - t^{\prime }{\int \limits } {\Psi } d\beta _{1}$. Because by the Ergodic Theorem, ${\int \limits } {\Psi } d\beta _{n1}\to _{p} {\int \limits } {\Psi } d\beta _{1}$ and

$$ \begin{array}{@{}rcl@{}} \sup_{\|t\|\le b}\big|\widehat {\mathcal V}_{n2}(t) - \bar {\mathcal V}_{n2}(t)\big|=o_p(1), \end{array} $$

(A.13)

by (A.1), we obtain

$$ {\mathcal V}_{n2}(t) - \widehat {\mathcal V}_{n2}(t)= \int \big\{U_{n}(z,\theta_{nt}) - U_{n}(z)+t^{\prime}{\Psi}(z)\big\} d\beta_{n1}(z)\\ = \int \big[W_{n}(z,\theta_{nt})+t^{\prime}{\Psi}(z)\big] d\beta_{n1}(z) . $$

Therefore, by (A.7), (A.12) and (A.13)

$$ \begin{array}{@{}rcl@{}} \sup_{\|t\|\le b} \Big|{\mathcal V}_{n2}(t) - \bar {\mathcal V}_{n2}(t)\Big| \!\!\!&\le&\!\!\! \sup_{\|t\|\le b} \Big|{\mathcal V}_{n2}(t) - \widehat {\mathcal V}_{n2}(t)\Big| + \sup_{\|t\|\le b} \Big|\widehat {\mathcal V}_{n2}(t) - \bar {\mathcal V}_{n2}(t)\Big|\\ &\le &\!\!\! \sup_{\|t\|\le b} \big|W_n(z,\theta_{nt})+t^{\prime}{\Psi}(z)\big| \beta_{n1}({\mathbf{\infty}})+o_p(1)=o_p(1). \end{array} $$

Also note that $\sup _{\|t\|\le b} |\bar {\mathcal V}_{n2}(t)|=O_{p}(1)= \sup _{\|t\|\le b} | {\mathcal V}_{n2}(t)|$ and by (A.12),

$$ \begin{array}{@{}rcl@{}} \sup_{\|t\|\le b} \Big|\frac{{\mathcal V}_{n2}(t)}{B_{n2}^{1/2}} - \frac{\bar {\mathcal V}_{n2}(t)}{B_{2}^{1/2}}\Big| \le \sup_{\|t\|\le b}\big|{\mathcal V}_{n2}(t)\big| \Big| \frac{1}{B_{n2}^{1/2}} - \frac{1}{B_{2}^{1/2}}\Big| + \frac{1}{B_{2}^{1/2}} \sup_{\|t\|\le b} \big|{\mathcal V}_{n2}(t) - \bar {\mathcal V}_{n2}(t)\big| =o_{p}(1). \end{array} $$

This fact in turn implies that

$$ \begin{array}{@{}rcl@{}} \Big|\inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b}\frac{{\mathcal V}_{n2}(re)}{B_{n2}^{1/2}} - \inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b} \frac{\bar {\mathcal V}_{n2}(re)}{B_2^{1/2}}\Big|=o_p(1). \end{array} $$

(A.14)

Fix an 𝜖 > 0 and η > 0. By (A.14), there exists N_{𝜖, η} such that

$$ \begin{array}{@{}rcl@{}} && P\Big(\inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b}\frac{\big|{\mathcal V}_{n2}(re)\big|}{B_{n2}^{1/2}} \ge \eta^{1/2} \Big) \\ & \ge& P\Big(\inf_{e \in {\mathbb R}^q, \|e\|=1, |r|=b}\big|\bar {\mathcal V}_{n2}(re)\big| \ge (B_2 \eta)^{1/2} \Big ) - \frac{\epsilon}{2}, \qquad \forall n>N_{\epsilon,\eta}. \end{array} $$

Moreover, $E\big ({\mathcal V}_{n}^{*}\big )^{2}= \sigma ^{2} E\alpha ^{2}(Z_{0})\le \sigma ^{2} B_{2}<\infty$. Hence, by the Markov inequality, for any 𝜖 > 0 there exists b_𝜖 such that

$$ \begin{array}{@{}rcl@{}} P\big(|{\mathcal V}_n^{*}|\le b_{\epsilon}\big)\ge 1-(\epsilon/2), \qquad \forall n\ge 1. \end{array} $$

(A.15)

Let $K:= {\int \limits } \|{\Gamma }\| d\beta$ and b_𝜖 be as in (A.15). Choose b ≥ (b_𝜖 + (B₂η)^1/2)K^− 1 and argue as for Eq. A.11 to obtain that ∀n ≥ N_{𝜖, η},

$$ \begin{array}{@{}rcl@{}} P\Big(\inf_{|t|> b} M_{n2}(\theta_0+n^{-1/2}t)\ge \eta \Big ) &\ge& P\Big(\ |{\mathcal V}_n^{*}|\le bK - (B_2\eta)^{1/2} \Big ) - \epsilon/2 \\ &\ge& P\Big(\ |{\mathcal V}_{n1}^{*}|\le b_{\epsilon} \Big ) - \epsilon/2 \ge 1-\epsilon, \end{array} $$

thereby proving the claim about (C.4). This also completes the proof of Lemma 4.1. □