Measurement Error Models for Replicated Data Under Asymmetric Heavy-Tailed Distributions

Abstract

Replicated data with measurement errors are frequently presented in economical, environmental, chemical, medical and other fields. In this paper, we discuss a replicated measurement error model under the class of scale mixtures of skew-normal distributions, which extends symmetric heavy and light tailed distributions to asymmetric cases. We also consider equation error in the model for displaying the matching degree between the true covariate and response. Explicit iterative expressions of maximum likelihood estimates are provided via the expectation–maximization type algorithm. Empirical Bayes estimates are conducted for predicting the true covariate and response. We study the effectiveness as well as the robustness of the maximum likelihood estimations through two simulation studies. The method is applied to analyze a continuing survey data of food intakes by individuals on diet habits.

Appendix

1.1 The PDF of Some SMSN Distributions and the Conditional Moments

The pdf of some important SMSN distributions and the properties about conditional moments are as follows:

(1) The multivariate skew-t distribution $ST _m(\varvec{\mu }, {\varvec{\Sigma }}, \varvec{\lambda };\nu )$:

$\kappa (u)=1/u$, $U\sim Gamma (\nu /2,\nu /2)$ with $\nu > 0$. The pdf of $\varvec{Y}$ is given by

$$\begin{aligned} f(\varvec{y})=2t_m(\varvec{y}|\varvec{\mu },{\varvec{\Sigma }};\nu )T\left( \sqrt{\frac{m+\nu }{d+\nu }}A;\nu +m\right) , \end{aligned}$$

where $d=(\varvec{y}-\varvec{\mu })^{\top }{\varvec{\Sigma }}^{-1}(\varvec{y}-\varvec{\mu })$, $t_m(\cdot |\varvec{\mu }, {\varvec{\Sigma }};\nu )$ and $T(\cdot ;\nu )$ denote the pdf of m-dimensional Student-t distribution and the cdf of standard univariate t distribution, respectively. The skew-normal distribution is the limiting case when $\nu \rightarrow +\infty$.

The conditional moments take the forms

$$\begin{aligned} u_r=&\,\frac{f_0(\varvec{y})}{f(\varvec{y})}\frac{2^{r+1}\Gamma ((\nu +m+2r)/2)(\nu +d)^{-r}}{\Gamma ((\nu +m)/2)}T\left( \sqrt{\frac{m+\nu +2r}{d+\nu }}A;\nu +m+2r\right) ,\\ \eta _r=&\,\frac{f_0(\varvec{y})}{f(\varvec{y})}\frac{2^{(r+1)/2}\Gamma ((\nu +m+r)/2)}{\pi ^{1/2}\Gamma ((\nu +m)/2)}\frac{(\nu +d)^{(\nu +m)/2}}{(\nu +d+A^2)^{(\nu +m+r)/2}}. \end{aligned}$$

where $f_0(\varvec{y})=\int \nolimits _0^{\infty }\phi _m(\varvec{y}|\varvec{\mu },\kappa (u){\varvec{\Sigma }})d H (u)$, i.e. the pdf of the class of SMN distribution when $\varvec{\lambda }=\varvec{0}$.

(2) The multivariate skew-slash distribution $SS _m(\varvec{\mu }, {\varvec{\Sigma }}, \varvec{\lambda };\nu )$:

$\kappa (u)=1/u$, $U\sim Beta (\nu , 1)$ with $0<u<1$ and $\nu > 0$. The pdf of $\varvec{Y}$ is given by

$$\begin{aligned} f(\varvec{y})=2\nu \int \nolimits _0^{1}u^{\nu -1}\phi _m(\varvec{y}|\varvec{\mu },u^{-1}{\varvec{\Sigma }})\Phi (u^{1/2}A)d u. \end{aligned}$$

When $\nu \rightarrow +\infty$, the skew-slash distribution reduces to the skew-normal one.

The conditional moments take the forms

$$\begin{aligned} u_r&=\frac{f_0(\varvec{y})}{f(\varvec{y})}\frac{2\Gamma ((2\nu +m+2r)/2)}{\Gamma ((2\nu +m)/2)}\Big (\frac{2}{d}\Big )^{r}\frac{P_1((2\nu +m+2r)/2,d/2)}{P_1((2\nu +m)/2,d/2)}\\&\quad \times E [\Phi (S^{1/2}A)],\\ \eta _r&=\frac{f_0(\varvec{y})}{f(\varvec{y})}\frac{2^{{(r+1)}/2}\Gamma ((2\nu +m+r)/2)}{\pi ^{1/2}\Gamma ((2\nu +m)/2)}\frac{d^{(2\nu +m)/2}}{(d+A^2)^{(2\nu +m+r)/2}}\\&\quad \times \frac{P_1((2\nu +m+r)/2,(d+A^2)/2)}{P_1((2\nu +m)/2,d/2)}, \end{aligned}$$

where $S\sim Gamma ((2\nu +m+2r)/2,d/2)I _{(0,1)}$ and $P_x(a,b)$ denotes the cdf of the $Gamma (a,b)$ distribution evaluated at x.

(3) The multivariate skew-contaminated normal distribution $SCN _m(\varvec{\mu }, {\varvec{\Sigma }}, \varvec{\lambda };\nu ,\gamma )$:

When $\kappa (u)=1/u$ and U follows a discrete random probability function $h(u;\nu ,\gamma )=\nu I _{(u=\gamma )}+(1-\nu ) I _{(u=1)}$ with given parameter vector $\varvec{\nu }=(\nu ,\gamma )^{\top }$ and $0<\nu<1, 0<\gamma \leqslant 1$, we get the multivariate skew-contaminated normal distribution with the pdf as

$$\begin{aligned} f(\varvec{y})=2\left\{ \nu \phi _m\left( \varvec{y}|\varvec{\mu },\gamma ^{-1}{\varvec{\Sigma }}\right) \Phi \left( \gamma ^{1/2}A\right) +(1-\nu )\phi _m\left( \varvec{y}|\varvec{\mu },{\varvec{\Sigma }}\right) \Phi (A)\right\} . \end{aligned}$$

The SN distribution is a special case as $\gamma =1$.

The conditional moments take the forms

$$\begin{aligned}&u_r=\frac{2}{f(\varvec{y})}\left\{ \nu \gamma ^{r}\phi _m\left( \varvec{y}|\varvec{\mu },\gamma ^{-1}{\varvec{\Sigma }}\right) \Phi \left( \gamma ^{1/2}A\right) +(1-\nu )\phi _m\left( \varvec{y}|\varvec{\mu },{\varvec{\Sigma }}\right) \Phi (A)\right\} ,\\&\eta _r=\frac{2}{f(\varvec{y})}\left\{ \nu \gamma ^{r/2}\phi _m\left( \varvec{y}|\varvec{\mu },\gamma ^{-1}{\varvec{\Sigma }}\right) \phi \left( \gamma ^{1/2}A\right) +(1-\nu )\phi _m\left( \varvec{y}|\varvec{\mu },{\varvec{\Sigma }}\right) \phi (A)\right\} . \end{aligned}$$

1.2 The First Derivatives of $d_t$, $A_t$ and $\log |{\varvec{\Sigma }}|$ with Respect to $\varvec{\theta }$

By direct calculations, we have the first derivatives of $d_t$, $A_t$ and $\log |{\varvec{\Sigma }}|$ as follows:

for $d_t$:

$$\begin{aligned} \frac{\partial d_t}{\partial \theta _i}= -2{(\varvec{Z}_t-\varvec{\mu })}^{\top }{\varvec{\Sigma }}^{-1}\frac{\partial \varvec{\mu }}{\partial \theta _i} -{(\varvec{Z}_t-\varvec{\mu })}^\top {\varvec{\Sigma }}^{-1}\frac{\partial {\varvec{\Sigma }}}{\partial \theta _i}{\varvec{\Sigma }}^{-1}(\varvec{Z}_t-\varvec{\mu }), \end{aligned}$$

for $A_t$:

$$\begin{aligned} \frac{\partial A_t}{\partial \theta _i}=\bigg (\frac{\partial \psi }{\partial \theta _i}\varvec{b}^{\top } +\psi \frac{\partial \varvec{b}^{\top }}{\partial \theta _i}-\psi \varvec{b}^{\top }{\varvec{\Sigma }}^{-1} \frac{\partial {\varvec{\Sigma }}}{\partial \theta _i}\bigg ){\varvec{\Sigma }}^{-1}(\varvec{Z}_t-\varvec{\mu }) -\psi \varvec{b}^{\top }{\varvec{\Sigma }}^{-1}\frac{\partial \varvec{\mu }}{\partial \theta _i}, \end{aligned}$$

where $\psi =\frac{\lambda _x\phi _x}{\sqrt{\phi _x+\lambda _x^2\Lambda _x}}$.

for $\log |{\varvec{\Sigma }}|$:

$$\begin{aligned} \frac{\partial \log |{\varvec{\Sigma }}|}{\partial \beta }=&2q\beta \phi _{\delta }\phi _x/\tau ,\\ \frac{\partial \log |{\varvec{\Sigma }}|}{\partial \phi _x}=&\left[ p(\phi _{\varepsilon }+q\phi _e)+q\beta ^2\phi _{\delta }\right] /\tau ,\\ \frac{\partial \log |{\varvec{\Sigma }}|}{\partial \phi _{\delta }}=&(p-1)/\phi _{\delta }+(\phi _{\varepsilon }+q\phi _e+q\beta ^2\phi _x)/\tau ,\\ \frac{\partial \log |{\varvec{\Sigma }}|}{\partial \phi _{\varepsilon }}=&(q-1)/\phi _{\varepsilon }+(\phi _{\delta }+p\phi _x)/\tau ,\\ \frac{\partial \log |{\varvec{\Sigma }}|}{\partial \phi _e}=&q(\phi _{\delta }+p\phi _x)/\tau , \end{aligned}$$

where $|{\varvec{\Sigma }}|=\phi _{\delta }^{p-1}\phi _{\varepsilon }^{q-1}\tau$, $\tau =(\phi _{\delta }+p\phi _x)(\phi _{\varepsilon }+q\phi _e)+q\beta ^2\phi _{\delta }\phi _x$.

In addition, we also need to calculate the following derivation:

for $\varvec{\mu }$:

$$\begin{aligned} \frac{\partial \varvec{\mu }}{\partial \mu _x}=\varvec{b}, \frac{\partial \varvec{\mu }}{\partial \alpha }=\varvec{c}, \frac{\partial \varvec{\mu }}{\partial \beta }=\mu _x\varvec{c}. \end{aligned}$$

for ${\varvec{\Sigma }}$:

$$\begin{aligned} \frac{\partial {\varvec{\Sigma }}}{\partial \beta }= & {} \phi _x\left( \varvec{c}\varvec{b}^{\top }+\varvec{b}\varvec{c}^{\top }\right) , \frac{\partial {\varvec{\Sigma }}}{\partial \phi _x}=\varvec{b}\varvec{b}^{\top }, \frac{\partial {\varvec{\Sigma }}}{\partial \phi _{\delta }}=\varvec{D}\left\{ \left( \varvec{1}_p^{\top },\varvec{0}_q^{\top }\right) \right\} , \frac{\partial {\varvec{\Sigma }}}{\partial \phi _{\varepsilon }}\\= & {} \varvec{D}(\varvec{c}), \frac{\partial {\varvec{\Sigma }}}{\partial \phi _e}=\varvec{c}\varvec{c}^{\top }. \end{aligned}$$

for $\varvec{b}$:

$$\begin{aligned} \frac{\partial \varvec{b}}{\partial \beta }=\varvec{c}. \end{aligned}$$

for $\psi$:

$$\begin{aligned}&\frac{\partial \psi }{\partial \beta }=q\beta c^{-2}\psi ^3/(\phi _{\varepsilon }+q\phi _e), \quad \frac{\partial \psi }{\partial \lambda _x}=\phi _x^2(\phi _x+\lambda _x^2\Lambda _x)^{-3/2},\\&\frac{\partial \psi }{\partial \phi _x}=\frac{1}{2}\psi \phi _x^{-1}+\frac{1}{2}\psi ^3\phi _x^{-1}c^{-2}\varvec{b}^{\top }{\varvec{\Sigma }}_1^{-1}\varvec{b}, \quad \frac{\partial \psi }{\partial \phi _{\delta }}=-\frac{1}{2}p\psi ^3c^{-2}\phi _{\delta }^{-2},\\&\frac{\partial \psi }{\partial \phi _{\varepsilon }}=-\frac{1}{2}\psi ^3c^{-2}q\beta ^2{(\phi _{\varepsilon }+q\phi _e)}^{-2},\quad \frac{\partial \psi }{\partial \phi _e}=-\frac{1}{2}q^2\psi ^3c^{-2}\beta ^2{(\phi _{\varepsilon }+q\phi _e)}^{-2}. \end{aligned}$$