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H-relative error estimation for multiplicative regression model with random effect

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Abstract

Relative error approaches are more of concern compared to absolute error ones such as the least square and least absolute deviation, when it needs scale invariant of output variable, for example with analyzing stock and survival data. A relative error estimation procedure based on the h-likelihood is developed to avoid heavy and intractable integration for a multiplicative regression model with random effect. Statistical properties of the parameters and random effect in the model are studied. Numerical studies including simulation and real examples show the proposed estimation procedure performs well.

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Acknowledgements

The authors are grateful to the Editor, the Associate Editor, and the anonymous referees for comments and suggestions that lead to improvements in the paper. This research is partially supported by funds of the State Key Program of National Natural Science of China (No. 11231010) and National Natural Science of China (No. 11471302).

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Authors and Affiliations

  1. Department of Statistics and Finance, Management School, University of Science and Technology of China, Hefei, China

    Zhanfeng Wang, Zhuojian Chen & Zimu Chen

Authors
  1. Zhanfeng Wang
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  2. Zhuojian Chen
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  3. Zimu Chen
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Corresponding author

Correspondence to Zhanfeng Wang.

Appendix: Proofs of the main results

Appendix: Proofs of the main results

To prove Theorems, we need the following conditions,

  • \(A_{1}\): For each \(i\in \{1,\ldots ,K\}\), \(n_{i}/\sum _{j=1}^K n_j \rightarrow \lambda _i>0\), as \(n_{j} \rightarrow \infty \), \(j=1,\ldots ,K\).

  • \(A_{2}\): \(\Vert X_{ij}\Vert <\infty \) for \(j=1,\ldots ,n_i, i=1,\ldots ,K\).

  • \(A_{3}\): As \(n_i \rightarrow \infty , K/n_i \rightarrow O(1)\).

Proof of Theorem 1

Similar to Paik et al. (2015), we have a limiting normal distribution of \(\hat{\nu }_{i}\). Here we focus on derivation of asymptotical variance of \(\hat{\nu }_{i}\). Let (\(\hat{\varvec{\theta }}\), \(\hat{\nu }\)) be one solution of

where \(W\{\varvec{\theta },\nu ;\varvec{{Y}}\}=(h_{1}^{(1)}\{\varvec{\theta },\nu _{1};Y_{1}\},\ldots , h_{K}^{(1)}\{\varvec{\theta },\nu _{K};Y_{K}\})^\top \) and \(\nu =(\nu _1,\ldots , \nu _K)^\top \). By Taylor expansion, we get

$$\begin{aligned} h_{i}^{(1)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}-h_{i}^{(1)}\{\varvec{\theta },\nu _{0i};Y_{i}\} \approx B_{21i}(\varvec{\hat{\theta }}-\varvec{\theta }), \\ \frac{\partial }{\partial \varvec{\theta }}m(\varvec{\hat{\theta }};\varvec{{Y}})-\frac{\partial }{\partial \varvec{\theta }}m(\varvec{\theta };\varvec{{Y}}) \approx -A_{11}(\varvec{\hat{\theta }}-\varvec{\theta }). \end{aligned}$$

where \(A_{11}=E \{ -\frac{\partial ^2}{\partial \varvec{\theta } \partial \varvec{\theta }^{\top }}m(\varvec{\theta };\varvec{{Y}}_i) \}\), \(B_{21i} = E \{ \frac{\partial }{\partial \varvec{\theta }^{\top }}h_{i}^{(1)} \{ \varvec{\theta }, \nu _{i};\varvec{{Y}}_{i} \} | \nu _{i} = \nu _{0i} \}\). Thus, we can get

$$\begin{aligned} h_{i}^{(1)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}-h_{i}^{(1)}\{\varvec{\theta },\nu _{0i};Y_{i}\}=B_{21i}A_{11}^{-1}\frac{\partial }{\partial \varvec{\theta }}m(\varvec{\theta };\varvec{{Y}}). \end{aligned}$$

Under A3, we can write

$$\begin{aligned} h_{i}^{(1)}\{\varvec{\hat{\theta }},\hat{\nu _i};Y_{i}\}&=0 \approx h_{i}^{(1)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}+(\hat{\nu _i}-\nu _{0i})h_{i}^{(2)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}\\&\quad +\frac{1}{2}(\hat{\nu _i}-\nu _{0i})^2h_{i}^{(3)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}, \end{aligned}$$

and

$$\begin{aligned} \sqrt{n_i}(\hat{\nu _i}-\nu _{0i})=\{-h_{i}^{(2)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}+O_{p}(n_i^{-\frac{1}{2}})\}^{-1}\sqrt{n_i}h_{i}^{(1)}\{\varvec{\hat{\theta }},\nu _{0i};Y_{i}\}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \sqrt{n_i}(\hat{\nu _i}-\nu _{0i})=I(\varvec{\theta },\nu _{0i})^{-1}\sqrt{n_i}[h_{i}^{(1)}\{\varvec{\theta },\nu _{0i};Y_{i}\}+B_{21i}A_{11}^{-1}\frac{\partial }{\partial \varvec{\theta }}m(\varvec{\theta };\varvec{{Y}})]+o_p(1). \end{aligned}$$

It follows that

$$\begin{aligned}&Var\{\sqrt{n_i}(\hat{\nu _i}-\nu _{0i})\}\\&\quad =I(\varvec{\theta },\nu _{0i})^{-1}+n_iI(\varvec{\theta },\nu _{0i})^{-1}B_{21i}A_{11}^{-1}Var\left[ \frac{\partial }{\partial \varvec{\theta }}m(\varvec{\theta };\varvec{{Y}})| \nu _i=\nu _{0i}\right] A_{11}^{-1}B_{21i}^{\top }I(\varvec{\theta },\nu _{0i})^{-1}\\&\quad \quad -\,2n_{i}I(\varvec{\theta },\nu _{0i})^{-1}B_{21i}^{\top }A_{11}^{-1}Cov\left[ \frac{\partial }{\partial \varvec{\theta }}m(\varvec{\theta };\varvec{{Y}}), h_{i}^{(1)}\{\varvec{\theta },\nu _i; \varvec{{Y}}_i | \nu _i=\nu _{0i}\}\right] . \end{aligned}$$

\(\square \)

Laplace approximation

From Tierney and Kadane (1986), we show that

$$\begin{aligned}&\exp \{m_i(\varvec{\theta };\varvec{{Y}}_i)\}=\int \exp [n_ih_{i}\{\varvec{\theta },\nu _{i};\varvec{{Y}}_{i}\}]d\nu _{i}\\&\qquad =\exp [n_ih_{i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]\sqrt{2\pi } \tau _{i} n_{i}^{-\frac{1}{2}}[1-C_{n_{i}}\{\varvec{\theta },\hat{\nu }_{i}\}]+O(n_{i}^{-2}), \end{aligned}$$

where

$$\begin{aligned}&\tau _{i}^2 = -[h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^{-1},\\&C_{n_{i}}\{\varvec{\theta },\hat{\nu }_{i}\}=J_{1i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}/{(8n_{i})}-{5} J_{2i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}/{(24n_{i})},\\&J_{1i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}=-\,{h_{i}^{(4)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}}/{[h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^2},\\&J_{2i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}=-\,{[h_{i}^{(3)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^2}/{[h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^3}. \end{aligned}$$

Under model (3), it easily shows that

$$\begin{aligned}&h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}\\&\quad =n_i^{-1}\sum _{j=1}^{n_{i}}-Y_{ij}\exp (-X_{ij}^T\beta -\nu _{i})-\exp (X_{ij}^T\beta +\nu _{i})Y_{ij}^{-1} -\frac{1}{n_{i}\sigma ^2}\\&\quad =h_{i}^{(4)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}-\frac{1}{n_{i}\sigma ^2},|h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}|>|h_{i}^{(3)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}|,\\&J_{1i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}=-\,{h_{i}^{(4)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}}/{[h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^2}\\&\quad \quad =-\,h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}^{-1}<1,\\&J_{2i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}=-\,{[h_{i}^{(3)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^2}/{[h_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}]^3}\\&\quad \quad<-\,h_{i}^{(3)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}^{-1}<1, \end{aligned}$$

which show that \(J_{1i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}\) and \(J_{2i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}\) have the order \(O_{p}(1)\). Thence, we obtain

$$\begin{aligned} m(\varvec{\theta };\varvec{{Y}})&= \sum _{i=1}^K m_i(\varvec{\theta };\varvec{{Y}}_i)\nonumber \\&=\sum _{i=1}^{K}[n_ih_{i}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}-\frac{1}{2}\sum _{i=1}^{K}\log [-n_ih_{i}^{(2)}\{\varvec{\theta },\hat{\nu }_{i};\varvec{{Y}}_{i}\}/2\pi ] \nonumber \\&\quad +\sum _{i=1}^{K}\log [1-C_{n_{i}}\{\varvec{\theta },\hat{\nu }_{i}\}]+O_p(n^{-1})\nonumber \\&=p_{\nu }(H)+O_{p}(n^{-1}), \end{aligned}$$
(A.1)

where \(p_{\nu }(H)\) is also called the adjusted profile likelihood.

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Wang, Z., Chen, Z. & Chen, Z. H-relative error estimation for multiplicative regression model with random effect. Comput Stat 33, 623–638 (2018). https://doi.org/10.1007/s00180-018-0798-7

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