Fiducial inference in the classical errors-in-variables model

Abstract

For the slope parameter of the classical errors-in-variables model, existing interval estimations with finite length will have confidence level equal to zero because of the Gleser–Hwang effect. Especially when the reliability ratio is low and the sample size is small, the Gleser–Hwang effect is so serious that it leads to the very liberal coverages and the unacceptable lengths of the existing confidence intervals. In this paper, we obtain two new fiducial intervals for the slope. One is based on a fiducial generalized pivotal quantity and we prove that this interval has the correct asymptotic coverage. The other fiducial interval is based on the method of the generalized fiducial distribution. We also construct these two fiducial intervals for the other parameters of interest of the classical errors-in-variables model and introduce these intervals to a hybrid model. Then, we compare these two fiducial intervals with the existing intervals in terms of empirical coverage and average length. Simulation results show that the two proposed fiducial intervals have better frequency performance. Finally, we provide a real data example to illustrate our approaches.

Appendix 1: The derivation of FGPQ for \(\beta _1\)

Since the structural equation is $(n-1)\mathbf{T }=\mathbf{KWK }^T$, it turns out that

$$\begin{aligned} \left\{ \begin{aligned} (n-1)S_{xx}&=\frac{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}{\sigma _{yy}}E_{11}+2\sqrt{\frac{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}{\sigma _{yy}}}\frac{\sigma _{xy}}{\sqrt{\sigma _{yy}}}E_{12}+\frac{\sigma _{xy}^2}{\sigma _{yy}}E_{22}\\ (n-1)S_{xy}&=\sqrt{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}E_{12}+\sigma _{xy}E_{22}\\ (n-1)S_{yy}&=\sigma _{yy}E_{22} \end{aligned} \right. . \end{aligned}$$

Let $\varvec{\eta }\triangleq (\sigma _{xx},\sigma _{xy},\sigma _{yy})^T$, $\mathbf{S }\triangleq (S_{xx},S_{xy},S_{yy})^T$ and $\mathbf{E }\triangleq (E_{11},E_{12},E_{22})^T$. On the one hand, we can obtain $\mathbf{E }$ from the above structural equation, i.e.,

$$\begin{aligned} \left\{ \begin{aligned} E_{11}&=\frac{\sigma _{yy}(n-1)S_{xx}}{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}-2\frac{\sigma _{xy}(n-1)S_{xy}}{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}+\frac{\sigma _{xy}^2(n-1)S_{yy}}{\sigma _{yy}(\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2)}\\ E_{12}&=\frac{(n-1)S_{xy}}{\sqrt{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}}-\frac{\sigma _{xy}(n-1)S_{yy}}{\sigma _{yy}\sqrt{\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2}}\\ E_{22}&=\frac{(n-1)S_{yy}}{\sigma _{yy}} \end{aligned} \right. . \end{aligned}$$

(15)

On the other hand, we can derive $\varvec{\eta }$ from (15), that is

$$\begin{aligned} \left\{ \begin{aligned} \sigma _{xx}&=(n-1)\left[ \frac{E_{22}(S_{xx}S_{yy}-S_{xy}^2)}{S_{yy}(E_{11}E_{22}-E_{12}^2)}+\frac{1}{S_{yy}E_{22}}(S_{xy}-\frac{\sqrt{S_{xx}S_{yy}-S_{xy}^2}}{\sqrt{E_{11}E_{22}-E_{12}^2}}E_{12})^2\right] \\ \sigma _{xy}&=\frac{n-1}{E_{22}}\left( S_{xy}-\frac{\sqrt{S_{xx}S_{yy}-S_{xy}^2}}{\sqrt{E_{11}E_{22}-E_{12}^2}}E_{12}\right) \\ \sigma _{yy}&=\frac{(n-1)S_{yy}}{E_{22}} \end{aligned} \right. . \end{aligned}$$

(16)

Due to $\lambda =1$ and (2), we have

$$\begin{aligned} \left( \begin{matrix} \sigma _{xx}&{}\quad \sigma _{xy}\\ \sigma _{yx}&{}\quad \sigma _{yy}\\ \end{matrix} \right) = \left( \begin{matrix} \sigma ^{2}+\sigma _{\epsilon }^{2}&{}\quad \beta _1\sigma ^{2}\\ \beta _1\sigma ^{2}&{}\quad \beta _1^{2}\sigma ^{2}+\sigma _{\epsilon }^{2}\\ \end{matrix} \right) . \end{aligned}$$

Thus,

$$\begin{aligned} \beta _1= \left\{ \begin{aligned}&\frac{\sigma _{yy}-\sigma _{xx}+\sqrt{(\sigma _{yy}- \sigma _{xx})^2+4 \sigma _{xy}^2}}{2\sigma _{xy}}~~&\sigma _{xy}\ne 0\\&0&\sigma _{xy}=0 \end{aligned} \right. . \end{aligned}$$

(17)

Combining (15), (16) and (17) with the structural method, we get the FGPQ of $\beta _1$. Let $(E_{11}^*,E_{12}^*,E_{22}^*)$ be the independent copy of $(E_{11},E_{12},E_{22})$. Replacing $(E_{11},E_{12},E_{22})$ in (16) by $(E_{11}^*,E_{12}^*,E_{22}^*)$, we can get $\sigma _{xx}^*,\sigma _{xy}^*$ and $\sigma _{yy}^*$. Note that $P(\mathscr {\sigma }_{xy}^*=0)=0$, the case of $\beta _1=0$ in (17) need not to be considered.

1.1 Appendix 2: Proof of the lemma and theorem

Let $\overset{P}{\longrightarrow }$ denote that a sequence of random variables converges in probability, $\overset{a.e}{\longrightarrow }$ denote that a sequence of random variables converges almost everywhere, $\overset{W}{\longrightarrow }$ denote that a sequence of distribution functions converges weakly and $\overset{L}{\longrightarrow }$ denote that a sequence of random variables converges in law.

Lemma 1

Let $\{A_n,n=1,\ldots \}$ be a sequence of real random vector and $A_n\overset{L}{\rightarrow }A\sim N(\mathbf{0 },\varvec{\varOmega })$. $Z_n$ is a scalar function of $A_n$ with $Z_n=f(A_n)$, where f is a continuous function. $A_n^*$ is an independent copy of $A_n$, $Z_n^*=f({A_n^*}')$ and $\epsilon _n(A_n^*,A_n)\overset{P}{\longrightarrow }0$. Then

$$\begin{aligned} P(Z_n^*\le Z_n+\epsilon _n(A_n^*,A_n)|A_n) \overset{L}{\longrightarrow } U(0, 1). \end{aligned}$$

where U(0, 1) denotes a uniform distribution on interval (0, 1).

Proof

Because $A_n\overset{L}{\rightarrow }A\sim N(\mathbf{0 },\varvec{\varOmega })$, so there exist $F_n$ and F, such that $A_n\sim F_n$ and $A\sim F$ with $F_n \overset{W}{\longrightarrow } F$, where F is the distribution of $N(\mathbf{0 },\varvec{\varOmega })$. Let $A_n^*$ and $A^*$ be the independent copy of $A_n$ and A , respectively. Then $(A_n^*,A_n)\overset{L}{\longrightarrow }(A^*,A)$, where $(A_n^*,A_n)\sim F_n\times F_n$ and $(A^*,A)\sim F\times F$. Together with Skorokhod representation theorem, there exist $A',~{A^*}',~A_n'$ and ${A_n^*}'$, such that $({A_n^*}',A_n')\sim F_n\times F_n$ and $({A^*}',A')\sim F\times F$ with $({A_n^*}',A_n') \overset{a.e}{\longrightarrow }({A^*}',A')$. Let $\mathscr {F}_n\triangleq \sigma (A_1', A_2', \ldots , A_n')$, i.e, the minimum $\sigma$-field such that $A'_1, A'_2, \ldots , A'_n$ are measurable. And let $Z=f(A),Z^*=f(A^*),Z'=f(A'),{Z^*}'=f({A_n^*}'),Z_n'=f(A_n'),{Z_n^*}'=f({A_n^*}')$, then

$$\begin{aligned}&P\left( Z_n^*\le Z_n+\epsilon _n(A_n^*,A_n)|A_n\right) \\&\quad \overset{d}{=}P\left( {Z_n^*}'\le {Z_n}'+\epsilon _n({A_n^*}',A_n')|A_n'\right) \\&\quad =P\left( {Z_n^*}'\le {Z_n'}+\epsilon _n({A_n^*}',A_n')|\mathscr {F}_n\right) \\&\quad =P\left( {Z^*}'+{Z_n^*}'-{Z^*}'\le Z'+{Z_n}'-Z'+\epsilon _n({A_n^*}',A_n')|\mathscr {F}_n\right) \\&\quad =P\left( {Z^*}'\le Z'+\epsilon _n'({A_n^*}',A_n')|\mathscr {F}_n\right) \\&\quad =E\left( I({Z^*}'\le Z'+\epsilon _n'({A_n^*}',A_n'))|\mathscr {F}_n\right) \\&\quad \overset{P}{\longrightarrow } E\left( I({Z^*}'\le Z')|\mathscr {F}_\infty \right) \\&\quad =P\left( {Z^*}'\le Z'|\mathscr {F}_\infty \right) \sim U(0, 1), \end{aligned}$$

where $\overset{d}{=}$ denotes that both sides of the equality have the same distribution and $\epsilon _n'({A_n^*}',A_n')=\epsilon _n({A_n^*}',A_n')+Z_n'-Z'-({Z_n^*}'-{Z^*}')\overset{P}{\longrightarrow } 0$. The last but one formula is given by a version of convergence in probability of Theorem 5.5.9 in Durrett (2010) with $h_n=I({Z^*}'\le Z'+\epsilon _n'({A_n^*}',A_n'))$, $h=I({Z^*}'\le Z')$ and $g=1$. It remains to prove that $h_n\overset{P}{\longrightarrow } h$.

Let $W={Z^*}'-Z'$, then $h_n=I(W\le \epsilon _n')$ and $h=I(W\le 0)$. Because W is a continuous random variable, for any fixed $b>0$, there exists a $c>0$, such that $P\{|W|\le c\}<b$. For every $\delta >0$,

$$\begin{aligned} \begin{aligned}&P\{|h_n-h|<\delta \}\\&\quad =P\left\{ |I\left( W\le \epsilon _n'\right) -I(W\le 0)|<\delta \right\} \\&\quad \le P\left\{ |I\left( W\le \epsilon _n'\right) -I(W\le 0)|<\delta \cap |\epsilon _n'|<c\cap |W|>c\right\} \\&\qquad + P\left\{ |I\left( W\le \epsilon _n'\right) -I(W\le 0)|<\delta \cap |\epsilon _n'|\ge c\right\} \\&\qquad + P\{|I\left( W\le \epsilon _n'\right) -I(W\le 0)|<\delta \cap |W|\le c\}\\&\quad \le P\{|I\left( W\le \epsilon _n'\right) -I(W\le 0)|<\delta \cap |\epsilon _n'|<c\cap |W|>c\}+ P\{|\epsilon _n'|\ge c\}+b\\&\quad =P\{|I(W\le \epsilon _n')-I(W\le 0)|<\delta \cap |\epsilon _n'|<c\cap W>c\}\\&\qquad +P\{|I(W\le \epsilon _n')-I(W\le 0)|<\delta \cap |\epsilon _n'|<c\cap W<-c\}+ P\{|\epsilon _n'|\ge c\}+b\\&\quad =P\{|\epsilon _n'|\ge c\}+b\xrightarrow []{n\rightarrow \infty }b. \end{aligned} \end{aligned}$$

Let $b\rightarrow 0$, and the proof is complete.

Lemma 2

(Fuller 1987, Theorem 1.C.1) Let $\{\mathbf{Z}_n\}$ be a sequence of independent identically distributed p-dimensional random vectors with mean $\varvec{\mu }_{\mathbf{Z }}$, covariance matrix $\varvec{\varSigma }_{\mathbf{ZZ }}$, and finite fourth moments. Let $\bar{\mathbf{Z }}=n^{-1}\sum _{t=1}^n \mathbf{Z }_t$ be the sample mean, $m_{\mathbf{ZZ }}=(n-1)^{-1}\sum _{t=1}^n (\mathbf{Z }_t-\bar{\mathbf{Z }})(\mathbf{Z }_t-\bar{\mathbf{Z }})^T$ the sample covariance matrix,

$$\begin{aligned} \begin{aligned} vech~\mathbf{m }_{\mathbf{ZZ }}&=(m_{ZZ11},m_{ZZ21},\ldots ,m_{ZZp1},m_{ZZ22},\ldots ,m_{ZZpp})^T,\\ vech~\varvec{\varSigma }_{\mathbf{ZZ }}&=(\sigma _{ZZ11},\sigma _{ZZ21},\ldots ,\sigma _{ZZp1},\sigma _{ZZ22},\ldots ,\sigma _{ZZpp})^T,\\ \mathbf{a }_t&=\left( \left( \mathbf{Z }_t-\varvec{\mu }_{\mathbf{Z }}\right) ^T,\left[ vech\left\{ \left( \mathbf{Z }_t-\varvec{\mu }_{\mathbf{Z }}\right) \left( \mathbf{Z }_t-\varvec{\mu }_{\mathbf{Z }}\right) ^T-\varvec{\varSigma }_{\mathbf{ZZ }}\right\} \right] ^T\right) ^T, \end{aligned} \end{aligned}$$

and $\varvec{\varOmega }=E\{\mathbf{a}_t \mathbf{a}_t^T\}$, where vech denotes half-vectorization. Then

$$\begin{aligned} n^{1/2}\left[ (\bar{\mathbf{Z }}-\varvec{\mu }_{\mathbf{Z }})^T,(vech~\mathbf{m }_{\mathbf{ZZ }}-vech~\varvec{\varSigma }_{\mathbf{ZZ }})^T\right] ^T\overset{L}{\longrightarrow }N(\mathbf{0 },\varvec{\varOmega }). \end{aligned}$$

Proof of Theorem

Proof

Because $P_{\beta _1}(\hat{\beta }_{1_{\alpha /2}}\le \beta _1 \le \hat{\beta }_{1_{1-\alpha /2}})=P_{\beta _1}(\beta _1 \le \hat{\beta }_{1_{1-\alpha /2}})-P_{\beta _1}(\beta _1<\hat{\beta }_{1_{\alpha /2}})$, if we prove that as $n\rightarrow \infty$,

$$\begin{aligned} P_{\beta _1}(\beta _1 \le \hat{\beta }_{1_\gamma })\rightarrow \gamma , \end{aligned}$$

(18)

the assertion follows. Let $F(r,\mathbf{s })$ be the distribution function of $\mathscr {R}_{\beta _1}$ conditional on $\mathbf{S }=\mathbf{s }$. Taking $F(r,\mathbf{s })$ on both sides of the inequality in (18), that is,

$$\begin{aligned} P_{\beta _1}\left( F(\beta _1,\mathbf{s }) \le F(\hat{\beta }_{1_\gamma },\mathbf{s })\right)&=P_{\beta _1}\left( F(\beta _1,\mathbf{s })\le \gamma \right) \\&=P_{\beta _1}\left( P(\mathscr {R}_{\beta _1}\le \beta _1|\mathbf{S }=\mathbf{s })\le \gamma \right) . \end{aligned}$$

What is left is to show that $P_{\beta _1}(P(\mathscr {R}_{\beta _1}\le \beta _1|\mathbf{S })\le \gamma )\rightarrow \gamma$, i.e.,

$$\begin{aligned} P(\mathscr {R}_{\beta _1}\le \beta _1|\mathbf{S })\overset{L}{\longrightarrow } U(0,1). \end{aligned}$$

In particular, by (7), we have

$$\begin{aligned}&P(\mathscr {R}_{\beta _1}\le \beta _1|\mathbf{S })\nonumber \\&\quad =P\left( \left[ \mathscr {R}_{yy}-\mathscr {R}_{xx}+\sqrt{(\mathscr {R}_{yy}- \mathscr {R}_{xx})^2+4 \mathscr {R}_{xy}^2}\right] \Big /(2\mathscr {R}_{xy})\le \beta _1|\mathbf{S })\right) \nonumber \\&\quad =P\left( 2\mathscr {R}_{xy}{\Big /}\left[ \sqrt{(\mathscr {R}_{yy}- \mathscr {R}_{xx})^2+4 \mathscr {R}_{xy}^2}-(\mathscr {R}_{yy}-\mathscr {R}_{xx})\right] \le \beta _1|\mathbf{S })\right) \nonumber \\&\quad =P\left( 2\mathscr {R}_{xy}-\beta _1\left[ \sqrt{(\mathscr {R}_{yy}- \mathscr {R}_{xx})^2+4 \mathscr {R}_{xy}^2}-(\mathscr {R}_{yy}-\mathscr {R}_{xx})\right] \le 0|\mathbf{S }\right) . \end{aligned}$$

(19)

Let $f\triangleq 2\mathscr {R}_{xy}-\beta _1\left[ \sqrt{(\mathscr {R}_{yy}- \mathscr {R}_{xx})^2+4 \mathscr {R}_{xy}^2}-(\mathscr {R}_{yy}-\mathscr {R}_{xx})\right]$. By first order Taylor expansion of f at $\mathbf{S }=\mathbf{S }^*=\varvec{\eta }=(\sigma _{xx},\sigma _{xy},\sigma _{yy})$ with

$$\begin{aligned} \begin{aligned} f\big |_{(\varvec{\eta },\varvec{\eta })}&=0,~\frac{\partial f}{\partial S_{xx}^*}\Bigg |_{(\varvec{\eta },\varvec{\eta })}=-\frac{\partial f}{\partial S_{xx}}\Bigg |_{(\varvec{\eta },\varvec{\eta })}=C,\\ \frac{\partial f}{\partial S_{xy}^*}\Bigg |_{(\varvec{\eta },\varvec{\eta })}&=-\frac{\partial f}{\partial S_{xy}}\Bigg |_{(\varvec{\eta },\varvec{\eta })}=\frac{C(\sigma _{yy}-\sigma _{xx})}{\sigma _{xy}},~\frac{\partial f}{\partial S_{yy}^*}\Bigg |_{(\varvec{\eta },\varvec{\eta })}=-\frac{\partial f}{\partial S_{yy}}\Bigg |_{(\varvec{\eta },\varvec{\eta })}=-C, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} C=\frac{2(\sigma _{xx}\sigma _{yy}-\sigma _{xy}^2)\sigma _{yy}\sigma _{xy}}{\sqrt{(\sigma _{yy}- \sigma _{xx})^2+4 \sigma _{xy}^2}}. \end{aligned}$$

Therefore, the probability in (19) is equivalent to

$$\begin{aligned} \begin{aligned} P\Bigg \{&\left. C(S_{xx}^*-\sigma _{xx})+\frac{C(\sigma _{yy}-\sigma _{xx})}{\sigma _{xy}}(S_{xy}^*-\sigma _{xy})-C(S_{yy}^*-\sigma _{yy})\right. \\ -&C(S_{xx}-\sigma _{xx})-\frac{C(\sigma _{yy}-\sigma _{xx})}{\sigma _{xy}}(S_{xy}-\sigma _{xy})+C(S_{yy}-\sigma _{yy}) +\epsilon _n\le 0 \Big |\mathbf{S }\Bigg \}, \end{aligned} \end{aligned}$$

(20)

where

$$\begin{aligned} \begin{aligned} \epsilon _n&=o_p(|S_{xx}-\sigma _{xx}|+|S_{xy}-\sigma _{xy}|+|S_{yy}-\sigma _{yy}|\\&\quad +|S_{xx}^*-\sigma _{xx}|+|S_{xy}^*-\sigma _{xy}|+|S_{yy}^*-\sigma _{yy}|)\\&=o_p(n^{-1/2}). \end{aligned} \end{aligned}$$

Thus, (20) is equivalent to

$$\begin{aligned}&P\Bigg \{\sqrt{n}\left[ \left( S_{xx}^*-\sigma _{xx}\right) +\frac{\sigma _{yy}-\sigma _{xx}}{\sigma _{xy}}\left( S_{xy}^*-\sigma _{xy}\right) -(S_{yy}^*-\sigma _{yy})\right] \nonumber \\&\quad \le \sqrt{n}\left[ \left( S_{xx}-\sigma _{xx}\right) +\frac{\sigma _{yy}-\sigma _{xx}}{\sigma _{xy}}\left( S_{xy}-\sigma _{xy}\right) -(S_{yy}-\sigma _{yy})\right] +\frac{\epsilon _n}{C} \Big |\mathbf{S }\Bigg \}.\qquad \qquad \end{aligned}$$

(21)

By Lemma 2, we conclude that

$$\begin{aligned} A_n\triangleq \sqrt{n}\left( (S_{xx},S_{xy},S_{yy})^T-(\sigma _{xx},\sigma _{xy},\sigma _{yy})^T\right) \overset{L}{\longrightarrow }N_3(\mathbf{0 },\varvec{\varOmega }_{ss}), \end{aligned}$$

where $\varvec{\varOmega }_{ss}$ is a $3\times 3$ matrix formed by taking a block of the entries of the last three rows and columns from $\varvec{\varOmega }$ in Lemma 2 with $\mathbf{Z }_t=(x_t,~y_t)^T$.

Finally, let $f(u,v,w)=u+\frac{\sigma _{yy}-\sigma _{xx}}{\sigma _{xy}}v-w$, $\epsilon _n'(A_n^*,A_n)=\frac{\sqrt{n}\epsilon _n}{C}\overset{P}{\longrightarrow }0$. By Lemma 1, we see at once that (21) converges to U(0, 1) in law.