The S-U algorithm for missing data problems

Summary

We present a new Monte-Carlo method for finding the solution of an estimating equation that can be expressed as the expected value of a ‘full data’ estimating equation in which the expected value is with respect to the distribution of the missing data given the observed data. Equations such as these arise whenever the E-M algorithm can be used. The algorithm alternates between two steps: an S-step, in which the missing data are simulated, either from the conditional distribution described above or from a more convenient importance sampling distribution, and a U-step, in which parameters are updated using a closed-form expression that does not require a numerical maximization. We present two numerical examples to illustrate the method. Theoretical results are obtained establishing consistency and asymptotic normality of the approximate solution obtained by our method.

Appendix

Regularity assumptions for Theorem 1.

Let $\Omega = [\hat \theta - \eta ,\hat \theta + \eta ]$, η > 0 be a compact neighborhood of ${\hat \theta }$, let log⁺(x) = max[0, log(x)], x > 0, and let ∥ ∥ denote the Euclidian norm, and let ${\rm{G}}_\theta ^{ - 1{\rm{i}}}({\rm{u}}\vert{{\rm{y}}_{\rm{i}}}) = \mathop {\inf }\limits_{\rm{x}} \,\{ {\rm{G}}_\theta ^{\rm{i}}({\rm{x}}\vert{{\rm{y}}_{\rm{i}}}) \ge {\rm{u}}\}$. Then, the following conditions are assumed to hold for each 1 ≤ i ≤ N.

C1. There exist non-negative functions A_i and h_i with $\int\limits_0^1 {{{\rm{A}}_{\rm{i}}}({\rm{u}})}$ log⁺[A_i(u)] du < ∞, and h_i(x) ↓ 0 as x ↓ 0, such that

$$\begin{array}{c}{\left\Vert\mathrm{w}_{\theta}^{\mathrm{i}}[\mathrm{G}_{\theta}^{-1 \mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}]-\mathrm{w}_{\theta^{\prime}}^{\mathrm{i}}[\mathrm{G}_{\theta^{\prime}}^{-1 \mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}]\right\Vert} \\ { \leq \mathrm{A}_{\mathrm{i}}(\mathrm{u}) h_{\mathrm{i}}(\left\Vert\theta-\theta^{\prime}\right\Vert)}\end{array}$$

∀ θ, θ′ ∈ Ω, u ∈ (0,1), and

$${E_{{{\rm{G}}_\theta }({\rm{X}}\vert{{\rm{y}}_{\rm{i}}})}}[{{\rm{w}}_\theta }({\rm{X}},{{\rm{y}}_{\rm{i}}}){\log ^ + }{{\rm{w}}_\theta }({\rm{X}},{{\rm{y}}_{\rm{i}}})] < \infty $$

C2a. There exist non-negative functions B_i and k_i with $\int\limits_0^1 {{{\rm{B}}_{\rm{i}}}({\rm{u}})}$ log⁺[B_i(u)] du < ∞, and k_i(x) ↓ 0 as x ↓ 0, such that

$$\begin{array}{l}{\left\Vert \mathrm{S}_{\mathrm{i}}[\mathrm{G}_{\theta}^{-1\mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}] \mathrm{w}_{\theta}^{\mathrm{i}}[G_{\theta}^{-1\mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}]\right.} \\ {\quad-\mathrm{S}_{\mathrm{i}}[G_{\theta^{\prime}}^{-1\mathrm{i}}(\mathrm{u}\vert\mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}] \mathrm{w}_{\theta^{\prime}}^{\mathrm{i}}[\mathrm{G}_{\theta^{\prime}}^{-1\mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}]\left\Vert \leq \mathrm{B}_{\mathrm{i}}(\mathrm{u})k_{\mathrm{i}}(\left\Vert\theta-\theta^{\prime}\right\Vert)\right.}\end{array}$$

∀θ, θ′ ∈ Ω, u ∈ (0,1), and

$$E_{\mathrm{G}_{\theta}^{\mathrm{i}}(\mathrm{X} \vert \mathrm{y}_{\mathrm{i}})}[\left\Vert\mathrm{S}_{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\Vert \mathrm{w}_{\theta}^{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}}) \log ^{+}\{\left\Vert\mathrm{S}_{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\Vert \mathrm{w}_{\theta}^{\mathrm{i}}\left(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\}]<\infty$$

C2b. There exist non-negative functions ${\widetilde{\rm{B}}_{\rm{i}}}$ and ${\widetilde{k}_{\rm{i}}}$ with $\int\limits_0^1 {{{\widetilde{\rm{B}}}_{\rm{i}}}({\rm{u}}){{\log }^ + }[{{\widetilde{\rm{B}}}_{\rm{i}}}({\rm{u}})]\,{\rm{du}} < \infty }$, and ${\widetilde{k}_{\rm{i}}}({\rm{x}}) \downarrow 0$ as x ↓ 0, such that

$$\begin{array}{l}{ \left\Vert \tilde{\mathrm{S}}_{\mathrm{i}}[\mathrm{G}_{\theta}^{-1 \mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}] \mathrm{w}_{\theta}^{\mathrm{i}}[\mathrm{G}_{\theta}^{-1 \mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}]\right.} \\ {\ \ \left.-\tilde{\mathrm{S}}_{\mathrm{i}}[\mathrm{G}_{\theta^{\prime}}^{-1 \mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}] \mathrm{w}_{\theta^{\prime}}^{\mathrm{i}}[\mathrm{G}_{\theta^{\prime}}^{-1 \mathrm{i}}(\mathrm{u} \vert \mathrm{y}_{\mathrm{i}}), \mathrm{y}_{\mathrm{i}}]\right\Vert \leq \tilde{\mathrm{B}}_{\mathrm{i}}(\mathrm{u}) \tilde{k}_{\mathrm{i}}(\left\Vert\theta-\theta^{\prime}\right\Vert)}\end{array}$$

$$E_{\mathrm{G}_{\theta}^{\mathrm{i}}(\mathrm{X} \vert \mathrm{y}_{\mathrm{i}})}[\left\Vert\tilde{\mathrm{S}}_{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\Vert \mathrm{w}_{\theta}^{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}}) \log ^{+}\{\left\Vert\tilde{\mathrm{S}}_{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\Vert \mathrm{w}_{\theta}^{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\}]<\infty$$

C3. There exist non-negative functions C_i and l_i with $\int\limits_0^1 {{{\rm{C}}_{\rm{i}}}({\rm{u}})}$ log⁺[C_i(u)] du < ∞, and l_i(x) ↓ 0 as x ↓ 0, such that

$$\begin{array}{l}\left\Vert \mathcal{H}_{\mathrm{i}}[\mathrm{G}_{\theta}^{-1\mathrm{i}}(\mathrm{u}\vert \mathrm{y}_{\mathrm{i}}),\ \mathrm{y}_{\mathrm{i}}]\mathrm{w}_{\theta}^{\mathrm{i}}[\mathrm{G}_{\theta}^{-1\mathrm{i}}(\mathrm{u}\vert \mathrm{y}_{\mathrm{i}}),\ \mathrm{y}_{\mathrm{i}}]\right.\\ \left.-\mathcal{H}_{\mathrm{i}}[\mathrm{G}_{\theta^{\prime}}^{-1\mathrm{i}}(\mathrm{u}\vert \mathrm{y}_{\mathrm{i}}),\ \mathrm{y}_{\mathrm{i}}]\mathrm{w}_{\theta^{\prime}}^{\mathrm{i}}[\mathrm{G}_{\theta^{\prime}}^{-1\mathrm{i}}(\mathrm{u}\vert \mathrm{y}_{\mathrm{i}}),\ \mathrm{y}_{\mathrm{i}}]\right\Vert \leq \mathrm{C}_{\mathrm{i}}(\mathrm{u})l_{\mathrm{i}}(\Vert \theta-\theta^{\prime}\Vert)\end{array}$$

∀θ,θ′ ∈ Ω, u ∈ (0,1), and

$$E_{\mathrm{G}_{\theta}^{\mathrm{i}}(\mathrm{x} \vert \mathrm{y}_{\mathrm{i}})}[\left\Vert\mathcal{H}_{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\Vert \mathrm{w}_{\theta}^{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}}) \log ^{+}\{\left\Vert\mathcal{H}_{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\right\Vert \mathrm{w}_{\theta}^{\mathrm{i}}(\mathrm{X}, \mathrm{y}_{\mathrm{i}})\}]<\infty$$

C4. ${\rm{f}}_\theta ^{\rm{i}}({{\rm{y}}_{\rm{i}}}),\;\;\;\;\;\;\;{\mathbb{S}_{\rm{i}}}({{\rm{y}}_{\rm{i}}}\vert\theta )$ and ℍ_i(y_i ∣ θ) are twice continuously differentiable in $\theta ,\;\;\;\;\;{\rm{f}}_\theta ^{\rm{i}}({{\rm{y}}_{\rm{i}}})$ is positive, and ℍ_T({y_i} ∣ θ) is non-singular on Ω.

Conditions (C1) − (C3) are needed for the uniform SLLN corresponding to the summands w_ijk, S_ijk w_ijk, ${\widetilde{\rm{S}}_{{\rm{ijk}}}}{{\rm{w}}_{{\rm{ijk}}}}$ and ${{\cal H}_{{\rm{ijk}}}}{{\rm{w}}_{{\rm{jk}}}}$ respectively. If ${\rm{G}}_\theta ^{\rm{i}}( \cdot \vert{\rm{y}})$ does not depend on θ, these conditions take a simpler form.

Condition (C2b) is only required if ${\widetilde{\rm{S}}_{\rm{i}}}({\rm{x}},{\rm{y}}\vert\theta ) \ne {{\rm{S}}_{\rm{i}}}({\rm{x}},{\rm{y}}\vert\theta )$.

Proof of Theorem 1. For simplicity of presentation, we assume that θ is a real parameter and N = 1. In that case, can omit the subscript i throughout and denote ${\mathbb{S}_{\rm{T}}} = {\mathbb{S}_{\rm{1}}}$ by $\mathbb{S}$ etc., throughout the proof. K will stand for a generic constant.

Note that we can write ${{\rm{w}}_{{\rm{ijk}}}} = {\rm{w}}_{{\theta _{\rm{j}}}}^{\rm{i}}[{\rm{G}}_{{\theta _{\rm{j}}}}^{{\rm{ - 1i}}}({{\rm{U}}_{{\rm{jk}}}},{\rm{y}}),\;{\rm{y}}]$, where U_ijk are i.i.d. U[0,1]. It is easy to see that by (C1), the conditions BD, P-SLLN and S-LIP in the uniform SLLN of Andrews (1992) are satisfied; here, uniformity refers to uniform in θ ∈ Ω. Hence, by Theorem 3 of the same paper,

$${1 \over {\rm{M}}}\sum\limits_{{\rm{k}} = 1}^{\rm{M}} {{{\rm{w}}_{{\rm{ijk}}}}} \;\;\mathop \to \limits^{{\rm{a}}.{\rm{s}}{\rm{.}}} \;\;\phi _{{\theta _{\rm{j}}}}^{\rm{i}}({\rm{y}})\;\;\;,$$

uniformly in j ≥ 1, as M → ∞, provided the θ_j′s lie in Ω; $\phi _\theta ^{\rm{i}}( \cdot )$ is defined in (12). In the same way, by (C2) and (C3),

$$\begin{array}{l}{\frac{1}{\mathrm{M}} \sum\limits_{\mathrm{k}=1}^{\mathrm{M}} \mathrm{S}_{\text{ik}}\mathrm{w}_{\text{jk}}\ \ \overset{\mathrm{a}.\mathrm{s}.}{\rightarrow}\ \ \phi_{\theta_{\mathrm{i}}}(\mathrm{y}) \mathbb{S}(\mathrm{y} \vert \theta_{\mathrm{i}})}, \\ {\frac{1}{\mathrm{M}} \sum\limits_{\mathrm{k}=1}^{\mathrm{M}} \tilde{\mathrm{S}}_{\text{ik}} \mathrm{w}_{\text{jk}}\ \ \overset{\mathrm{a}.\mathrm{s}.}{\rightarrow} \ \ \phi_{\theta_{\mathrm{i}}}(\mathrm{y}) \tilde{\mathbb{S}}\left(\mathrm{y} \vert \theta_{\mathrm{i}}\right)},\end{array}$$

and

$$\frac{1}{\mathrm{M}} \sum_{\mathrm{k}=1}^{\mathrm{M}} \mathcal{H}_{\text{jk}}\ \mathrm{w}_{\text{jk}}\ \ \ \overset{\mathrm{a}.\mathrm{s}.}{\rightarrow}\ \ \ \phi_{\theta_{\mathrm{j}}}(\mathrm{y})\left[\mathbb{H}\left(\mathrm{y} \vert \theta_{\mathrm{j}}\right)-\mathbb{S}\left(\mathrm{y} \vert \theta_{\mathrm{j}}\right) \tilde{\mathbb{S}}^{\mathrm{T}}(\mathrm{y} \vert \theta_{\mathrm{j}})\right],$$

uniformly in j ≥ 1, as M → ∞, provided the θ_j′s lie in Ω. Therefore

$$\mathrm{r}_{\mathrm{i}} \equiv\left\vert\frac{\widehat{\mathbb{S}}_{\mathrm{j}}}{\widehat{\mathbb{H}}_{\mathrm{j}}}-\frac{\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}} \phi_{\theta_{\mathrm{j}^{\prime}}}(\mathrm{y}) \mathbb{S}\left(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}}\right)}{\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}} \phi_{\theta_{\mathrm{j}^{\prime}}}(\mathrm{y}) \mathbb{H}\left(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}}\right)}\right\vert \ \ \ \overset{\mathrm{a}.\mathrm{s}.}{\rightarrow} 0,$$

((A1.1))

uniformly in j ≥ 1, as M → ∞, provided the θ_j′s lie in Ω.

Next, note that by Taylor expanding both the numerator and the denominator of the second ratio in r_j we get

$$\left\vert\frac{\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \phi_{\theta_{\mathrm{j}^{\prime}}}(\mathrm{y}) \mathbb{S}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}{\mathrm{j}^{\prime}\sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \phi_{\theta_{\mathrm{j}^{\prime}}}(\mathrm{y})\mathbb{H}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}-\frac{\mathbb{S}(\mathrm{y} \vert \overline{\theta})}{\mathbb{H}(\mathrm{y} \vert \overline{\theta})}\right\vert \leq \text{Kj}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}}(\theta_{\mathrm{j}^{\prime}}-\overline{\theta})^{2}.$$

((A1.2))

Similarly,

$$\left\vert \mathrm{j}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \frac{\mathbb{S}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}{\mathbb{H}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}-\frac{\mathbb{S}(\mathrm{y} \vert \overline{\theta})}{\mathbb{H}(\mathrm{y} \vert \overline{\theta})}\right\vert \leq \mathrm{K}\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}}(\theta_{\mathrm{j}^{\prime}}-\overline{\theta})^{2}.$$

((A1.3))

Combining (A1.1), (A1.2) and (A1.3) we get

$$\begin{array}{lll}\left\vert\frac{\widehat{\mathbb{S}}_{\mathrm{j}}}{\widehat{\mathbb{H}}_{\mathrm{j}}}-\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \frac{\mathbb{S}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}{\mathbb{H}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}\right\vert & \leq & \mathrm{r}_{\mathrm{j}}+\mathrm{K}\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}}(\theta_\mathrm{{j}^{\prime}}-\overline{\theta})^{2} \\ & \leq & \mathrm{r}_{\mathrm{j}}+\mathrm{K}\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}}(\theta_{\mathrm{j}^{\prime}}-\widehat{\theta})^{2}, \end{array}$$

((A1.4))

provided the θ_j′s lie in Ω.

By a Taylor expansion of $({\rm{y}}\vert\widehat\theta )$ around θ_j we get

$$0=\mathbb{S}(\mathrm{y} \vert \widehat{\theta})=\mathbb{S}(\mathrm{y} \vert \theta_{\mathrm{j}})+(\widehat{\theta}-\theta_{\mathrm{j}}) \mathbb{H}(\mathrm{y} \vert \theta_{\mathrm{j}})+\frac{1}{2}(\widehat{\theta}-\theta_{\mathrm{j}})^{2} \mathbb{S}^{\prime \prime}(\mathrm{y} \vert \theta_{\mathrm{i}}^{\ast}),$$

((A1.5))

where $\theta _{\rm{j}}^ {\ast}$ lies between θ_j and $\widehat\theta$. Dividing (A1.5) by ℍ(y ∣ θ_j) and averaging, we obtain

$$\widehat{\theta}=\overline{\theta}_{\mathrm{j}}-\mathrm{j}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \frac{\mathbb{S}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}{ \mathbb{H}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}-(2 \mathrm{j})^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}}(\widehat{\theta}-\theta_{\mathrm{j}^{\prime}}) \frac{\mathbb{S}^{\prime \prime}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}}^{\ast})}{\mathbb{H}(\mathrm{y} \vert \theta_{\mathrm{j}^{\prime}})}.$$

((A1.6))

Using ${\theta _{{\text{j + 1}}}} = {\bar \theta _\text{j}}/{ \mathbb{\hat{H}}_\text{j}}$, (A1.4) and (A1.6) we obtain

$$\vert\widehat\theta - {\theta _{{\rm{j}} + 1}}\vert\; \le \;{{\rm{r}}_{\rm{j}}} + {\rm{K}}\;{{\rm{j}}^{ - 1}}\sum\limits_{{\rm{j}}\prime = 1}^{\rm{j}} {} {({\theta _{{\rm{j}}\prime }} - \widehat\theta )^2},$$

provided the θ_j′s lie in Ω.

Let P > max(2K, η⁻¹). Then re-write the above inequality as

$${{\rm{a}}_{{\rm{j}} + 1}}\; \le \;{\epsilon_{\rm{j}}} + {{\rm{K}} \over {\rm{P}}}\;{{\rm{j}}^{ - 1}}\sum\limits_{{\rm{j}}\prime = 1}^{\rm{j}} {} {\rm{a}}_{{\rm{j}}\prime }^2\;,$$

((A1.7))

where ${{\rm{a}}_{\rm{j}}} = {\rm{P}}\vert\widehat\theta - {\theta _{\rm{j}}}\vert$ and ∊_j = r_jP, j ≥ 1. By (A1.1), on a set of probability one, select M large enough so that ${\epsilon_{\rm{j}}} < {1 \over 2},{\kern 1pt} \forall {\rm{j}} \ge {\rm{1}}$, provided the θ_j′s lie in Ω. Suppose that the starting value ${\theta _1} \in \Omega = [\widehat\theta - \eta ,\widehat\theta + \eta ]$. Then a₁ < 1 and using (A1.7) we find that ${{\rm{a}}_2} \le ({1 \over 2} + {{\rm{K}} \over {\rm{P}}}) < 1$ which implies that θ₂ ∈ Ω. Inductively using (A1.7) we find that a_j ∈ Ω and $0 \le {{\rm{a}}_{\rm{j}}} \le ({1 \over 2} + {{\rm{K}} \over {\rm{P}}})$, ∀j ≥ 1. Therefore, applying “limsup” to both sides of (A1.7) and using the fact that for a_j ≥ 0,

$$\mathop {\lim \sup }\limits_{{\rm{j}}\; \to \;\infty } \;{{\rm{j}}^{ - 1}}\sum\limits_{{\rm{j}}\prime = 1}^{\rm{j}} {} {\rm{a}}_{{\rm{j}}\prime }^2 \le {(\mathop {\lim \sup }\limits_{{\rm{j}}\; \to \;\infty } \;{{\rm{a}}_{\rm{j}}})^2}\;,$$

and that ${{\rm{r}}_{\rm{j}}}\mathop \to \limits^{{\rm{a}}.{\rm{s}}.} 0$ as j → ∞ (for any given replication size M), we conclude that on a set of probability one,

$$\lim \sup\limits_{{\rm{j}} \to \infty } \,{{\rm{a}}_{\rm{j}}} \le {1 \over 2}{(\lim \sup\limits_{{\rm{j}} \to \infty } \,{{\rm{a}}_{\rm{j}}})^2},$$

which immediately implies $\lim \sup\limits_{{\rm{j}} \to \infty } \,{{\rm{a}}_{\rm{j}}} = 0$, since $\lim \sup\limits_{{\rm{j}} \to \infty } \,{{\rm{a}}_{\rm{j}}} \in [0,1)$. Hence, convergence of θ_j to $\widehat\theta$ is established. For the case N > 1 note that while equations (A1.1) — (A1.3) become considerably more complicated, equation (A1.4) remains unchanged. □

Regularity assumptions for Theorem 2: Assume the conditions for Theorem 1 plus

N1. The matrix $\sum\limits_{\rm{i}} {{{\rm{V}}^{\rm{i}}}}$ is positive definite.

Proof of Theorem 2. Write

$$\sqrt {\rm{j}} ({\theta _{{\rm{j}} + 1}} - \widehat\theta ) = \sqrt {\rm{j}} \,({\overline \theta _{\rm{j}}} - \widehat\theta - \mathbb{\widehat{H}}_{\rm{j}}^{ - 1} {\mathbb{\widehat{S}}_{\rm{j}}})$$

$$\begin{array}{l}{=\widehat{\mathbb{H}}_{\mathrm{j}}^{-1}\{\mathbb{H}_{\mathrm{T}}(\{ \mathrm{y}_{\mathrm{i}}\} \vert \widehat{\theta})-\widehat{\mathbb{H}}_{\mathrm{j}}\} \sqrt{\mathrm{j}}(\overline{\theta}_{\mathrm{j}}-\widehat{\theta})} \\ {+\widehat{\mathbb{H}}_{\mathrm{j}}^{-1}[\sqrt{\mathrm{j}}\{\mathbb{H}_{\mathrm{T}}(\{\mathrm{y}_{\mathrm{i}}\} \vert \widehat{\theta})(\overline{\theta}_{\mathrm{j}}-\widehat{\theta})-\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{i}} \mathbb{S}_{\mathrm{T}}( \{\mathrm{y}_{\mathrm{i}}\} \vert \theta_{\mathrm{j}^{\prime}})\}]} \\ {-\widehat{\mathbb{H}}_{\mathrm{j}}^{-1} \sqrt{\mathrm{j}} \{\widehat{\mathbb{S}}_{\mathrm{j}}-\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \mathbb{S}_{\mathrm{T}}(\{\mathrm{y}_{\mathrm{i}}\} \vert \theta_{\mathrm{j}^{\prime}})\}}.\end{array}$$

((A2.1))

Since ${\mathbb{\widehat{H}}_{\rm{j}}}\mathop \to \limits^P {\mathbb{H}_{\rm{T}}}(\{ {{\rm{y}}_{\rm{i}}}\} \vert{\widehat\theta })$ and $\sqrt {\rm{j}} \,({\overline \theta _{\rm{j}}} - \widehat\theta )$ is O_p(1), the first term of (A2.1) converges to zero in probability.

Taylor expanding ${\mathbb{S}_{\rm{T}}}(\{ {{\rm{y}}_{\rm{i}}}\} \vert{\theta _{\rm{j}}})$ around $\widehat\theta$ and averaging, we obtain

$${{\rm{j}}^{ - 1}}\sum\limits_{{\rm{j}}\prime = 1}^{\rm{j}} {} {\mathbb{S}_{\rm{T}}}(\{ {{\rm{y}}_{\rm{i}}}\} {\rm{\vert}}{\theta _{{\rm{j}}\prime }}) = {\mathbb{H}_{\rm{T}}}(\{ {{\rm{y}}_{\rm{i}}}\} {\rm{\vert}}\hat \theta )\;({\bar \theta _{\rm{j}}} - \hat \theta ) + {{\rm{O}}_{\rm{p}}}({{\rm{j}}^{ - 1}}\sum\limits_{{\rm{j}}\prime = 1}^{\rm{j}} {} {\rm{\vert}}{\theta _{{\rm{j}}\prime }} - \hat \theta {{\rm{\vert}}^2})\;;$$

therefore, the second term on the right hand side of (A2.1) is o_p(1), since ${{\rm{j}}^{1/2}}\sum\limits_{{\rm{j}}\prime = 1}^j {{\rm{|}}{\theta _{{\rm{j}}\prime }} - \widehat\theta {\rm{|}} = {{\rm{O}}_{\rm{p}}}}$

From (5), (6) and (8), we have

$$\widehat{\mathbb{S}}_{\mathrm{j}} \equiv \sum_{\mathrm{i}=1}^{\mathrm{N}} \frac{\frac{1}{\mathrm{j}\cdot\mathrm{M}} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \sum\limits_{\mathrm{k}=1}^{\mathrm{M}} \mathrm{S}_{\mathrm{ij}^{\prime}\mathrm{k}} \mathrm{w}_{\mathrm{ij}^{\prime}\mathrm{k}}}{\frac{1}{\mathrm{j}\cdot\mathrm{M}} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \sum\limits_{\mathrm{k}=1}^{\mathrm{M}} \mathrm{w}_{\mathrm{ij}^{\prime}\mathrm{k}}} \equiv \sum_{\mathrm{i}=1}^{\mathrm{N}} \frac{\mathrm{N}_{\mathrm{i}}}{\mathrm{D}_{\mathrm{i}}}.$$

It is easy to verify the conditional Lindeberg condition for linear combinations of (N_i, D_i). By the martingale central limit theorem (Corollary 3.1 of Hyde and Hall, 1980), the asymptotic distribution of $\sqrt {\rm{j}} \,\sum\limits_{{\rm{i}} = 1}^{\rm{N}} {({{\rm{N}}_{\rm{i}}},{{\rm{D}}_{\rm{i}}})}$ is the multivariate normal distribution with mean vector

$$\sqrt{\mathrm{j}}\left(\sum_{\mathrm{i}=1}^{\mathrm{N}} \mathrm{j}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}}\mathbb{S}_{\mathrm{i}}(\mathrm{y}_{\mathrm{i}} \vert \theta_{\mathrm{j}^{\prime}}) \phi_{\theta_{\mathrm{j}^{\prime}}}^{\mathrm{i}}(\mathrm{y}_{\mathrm{i}}), \sum\limits_{\mathrm{i}=1}^{\mathrm{N}}\mathrm{j}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \phi_{\theta_{\mathrm{j}^{\prime}}}^{\mathrm{i}}(\mathrm{y}_{\mathrm{i}})\right)$$

and variance-covariance matrix ${{\rm{M}}^{ - 1}}\sum\limits_{{\rm{i}} = 1}^{\rm{N}} {{{\rm{V}}^{\rm{i}}}}$, where the Vⁱ is defined in (15)–(18). By the delta method, we have

$$\sqrt {\rm{j}} \,({\mathbb{\widehat{S}}_{\rm{j}}} - {\mu _{\rm{j}}})\;\mathop \to \limits^d \;{\rm{N(0,\mathbb{V}}}{\rm{)}}\,\,\,\,{\rm{,}}$$

((A2.2))

where $\mu_{\mathrm{j}}=\sum_{\mathrm{i}=1}^{\mathrm{N}} \frac{\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \mathbb{S}_{\mathrm{i}}(\mathrm{y}_{\mathrm{i}} \vert \theta_{\mathrm{j}^{\prime}}) \phi_{\theta_{\mathrm{j}^{\prime}}}^{\mathrm{i}}(\mathrm{y}_{\mathrm{i}})}{\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \phi_{\theta_{\mathrm{j}^{\prime}}}^{\mathrm{i}}(\mathrm{y}_{\mathrm{i}})}$ and where $\mathbb{V}$ is given in equation (19). Using a Taylor series argument as in the proof of Theorem 1, we get for each i

$$\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \frac{\mathbb{S}_{\mathrm{i}}(\mathrm{y}_{\mathrm{i}} \vert \theta_{\mathrm{j}^{\prime}}) \phi_{\theta_{\mathrm{j}^{\prime}}}^{\mathrm{i}}(\mathrm{y}_{\mathrm{i}})}{\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{i}} \phi_{\theta_{\mathrm{j}^{\prime}}}^{\mathrm{i}}(\mathrm{y}_{\mathrm{i}})}=\mathrm{j}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \mathbb{S}_{\mathrm{i}}(\mathrm{y}_{\mathrm{i}} \vert \theta_{\mathrm{j}^{\prime}})+\mathrm{O}_{\mathrm{p}}(\mathrm{j}^{-1} \sum_{\mathrm{j}^{\prime}=1}^{\mathrm{i}}(\theta_{\mathrm{j}^{\prime}}-\widehat{\theta})^{2});$$

hence,

$$\sqrt{\mathrm{j}}\ \left\{\mu_{\mathrm{j}}-\mathrm{j}^{-1} \sum\limits_{\mathrm{j}^{\prime}=1}^{\mathrm{j}} \mathbb{S}(\{ \mathrm{y}_{\mathrm{i}}\} \vert \theta_{\mathrm{j}^{\prime}})\right\}=o_{\mathrm{p}}(1).$$

((A2.3))

Combining (A2.1), (A2.2) and (A2.3) we find where $\sqrt {\rm{j}} \;({\theta _{{\rm{j}} + 1}} - \widehat\theta )\;\;\mathop \to \limits^d \;\;{\rm{N}}(0,\Sigma )$,

$$\Sigma = {\mathbb{H}^{ - 1}}(\{ {{\rm{y}}_{\rm{i}}}\} \vert\widehat\theta ) \cdot \mathbb{V}\cdot {\mathbb{H}^{ - {\rm{T}}}}(\{ {{\rm{y}}_{\rm{i}}}\} \vert\widehat\theta ).$$

□