On combining unbiased and possibly biased correlated estimators

Abstract

We study estimators that combine an unbiased estimator with a possibly biased correlated estimator of a mean vector. The combined estimators are shrinkage-type estimators that shrink the unbiased estimator towards the biased estimator. Conditions under which the combined estimator dominates the original unbiased estimator are given. Models studied include normal models with a known covariance structure, scale mixtures of normals, and more generally elliptically symmetric models with a known covariance structure. Elliptically symmetric models with a covariance structure known up to a multiple are also considered.

Appendices

Appendix A

1.1 A.1. Proof of Lemma 2 in Sect. 2

Proof

Let $Z\sim N_p(\mu , I_p)$, then $Y=Z'Z$ has a $\chi ^2_p(\mu '\mu )$ distribution with density:

$$\begin{aligned} f(y|p,\mu '\mu )=\sum _{k=0}^{\infty } P_k\left( \frac{\mu '\mu }{2}\right) f_{p+2k}(y)\, [13] \end{aligned}$$

where $P_k$ is the Poisson density, $\frac{e^{-\frac{\mu '\mu }{2}}(\frac{\mu '\mu }{2})^k }{k!}$, and $f_{p+2k}$ is the density of a $\chi ^2_{p+2k}$, $\frac{y^{\frac{p+2k}{2}-1}e^{-\frac{y}{2}}}{\Gamma (\frac{p+2k}{2})2^{\frac{p+2k}{2}}}$, so that Y is a Poisson mixture of central $\chi ^2$ densities. Supposing that $p\ge 3$, by Tonelli’s theorem

$$\begin{aligned} E\left[ \frac{1}{Y}\right] =\sum _{k=0}^{\infty }P_k\left( \frac{\mu '\mu }{2}\right) E\left[ \frac{1}{\chi ^2_{p+2k}}\right] = \end{aligned}$$

$$\begin{aligned} \sum _{k=0}^{\infty }P_k\left( \frac{\mu '\mu }{2}\right) \frac{1}{p+2k-2}=E_{\frac{\mu '\mu }{2}}\left[ \frac{1}{p+2k-2}\right] , \end{aligned}$$

(51)

where the expectation in (51) is taken with respect to a Poisson distribution with parameter $\frac{\mu '\mu }{2}$. In order for the expectation of the random variable $\frac{1}{Y}$ to exist, $p\ge 3$ is necessary since for $k=0$, $E\left[ \frac{1}{\chi ^2_p}\right] < \infty$ implies $p\ge 3$. Furthermore for $p\ge 3$ and $l>0$,

$$\begin{aligned} E\left[ \frac{1}{\chi ^2_{p+l}}\right]< & {} E\left[ \frac{1}{\chi ^2_{p}}\right] =\frac{1}{p-2}\\ {\text {so that}} \\ E\left[ \frac{1}{Y}\right]< & {} \sum _{k=0}^{\infty }P_k\left( \frac{\mu '\mu }{2}\right) \left[ \frac{1}{p-2}\right] = \frac{1}{p-2} \end{aligned}$$

which is finite. By the convexity of the function $g(k)=\frac{1}{2k+p-2}$, Jensen’s inequality implies

$$\begin{aligned} \frac{1}{p-2+\mu '\mu }=\frac{1}{p-2+2Ek} \le E\left[ \frac{1}{p+2k-2}\right] \le E\left[ \frac{1}{p-2}\right] =\frac{1}{p-2}. \end{aligned}$$

(52)

Casella and Hwang (1982) gives the more refined upper bound

$$\begin{aligned} E_{\mu ' \mu }\left[ \frac{1}{Z'Z}\right] \le \frac{1}{p-2}\left( \frac{p+2}{p+2+\mu '\mu }\right) \end{aligned}$$

(53)

for $p \ge 3$. When $p>3$, further refinement on the upper bound for the expectation in expression (53) is given in Green and Strawdermann (1991)

$$\begin{aligned} E_{\mu '\mu }\left[ \frac{1}{Z'Z}\right] \le \frac{1}{p-4+\mu '\mu }. \end{aligned}$$

(54)

Combining the bounds from expression (54) and (53), the following expression was established in Green and Strawdermann (1991) when $p\ge 4$

$$\begin{aligned} \frac{1}{p-2+\mu '\mu }\le E_{\mu '\mu }\left[ \frac{1}{Z'Z}\right] \le \text {min}\left\{ \frac{p+2}{(p-2)(p+2+\mu ' \mu )},\frac{1}{p-4+\mu '\mu }\right\} . \end{aligned}$$

(55)

Suppose now that $X\sim N_p(\mu ,\Sigma )$ and let $Y=\Sigma ^{-\frac{1}{2}}X\sim N_p(\Sigma ^{-\frac{1}{2}}\mu ,I_p)$. For any orthogonal matrix U with rank p, $Z=U'Y\sim N_p(U'\Sigma ^{-\frac{1}{2}}\mu ,I_p).$ Using the spectral value decomposition of $\Sigma$ to get

$$\begin{aligned} P'\Sigma P=\Lambda ={\text {Diag}}(\lambda _1,\lambda _2,\ldots ,\lambda _p), \end{aligned}$$

where P is the orthogonal matrix of eigenvectors of $\Sigma$ and $\Lambda$ is the diagonal matrix of eigenvalue of $\Sigma$, and denoting $\lambda _{(1)}$ and $\lambda _{(p)}$ as the largest and smallest eigenvalues of $\Sigma$, the following inequalities exist:

$$\begin{aligned} \lambda _{(1)}E[Z'Z]\le E[X'X]= E[X'\Sigma ^{-\frac{1}{2}}PP'\Sigma PP'\Sigma ^{-\frac{1}{2}}X] = E[Z'\Lambda Z]\le \lambda _{(p)}E[Z'Z] \end{aligned}$$

with $Z=P'\Sigma ^{-\frac{1}{2}}X$. Noting that the function $\phi (x)=\frac{1}{x}$ is convex for $x>0$,

$$\begin{aligned}{} & {} \text {max}\left\{ \frac{1}{\lambda _{p}}E\left[ \frac{1}{Z'Z}\right] ,\frac{1}{tr(\Lambda )+\mu '\mu }\right\} \\ {}{} & {} \quad \le E\left[ \frac{1}{Z'\Lambda Z}\right] = E\left[ \frac{1}{X'X}\right] \le \frac{1}{\lambda _{(1)}}E\left[ \frac{1}{Z'Z}\right] \le \infty \end{aligned}$$

provided $p \ge 3$, where $Z \sim N_p(P'\Sigma ^{-\frac{1}{2}}\mu ,I_p)$.

From expression (55) then, when $p \ge 4$,

$$\begin{aligned}{} & {} \text {max} \left\{ \frac{1}{\text {tr}(\Lambda ) + \mu '\mu },\frac{1}{\lambda _{(p)}}\left[ \frac{1}{p-2+\mu '\Sigma ^{-1}\mu }\right] \right\} \le E\left[ \frac{1}{X'X}\right] , \end{aligned}$$

(56)

$$\begin{aligned}{} & {} \quad \le \text {min}\left\{ \frac{1}{\lambda _{(1)}}\left[ \frac{1}{p-4+\mu '\Sigma ^{-1}\mu }\right] ,\frac{1}{\lambda _{(1)}}\left[ \frac{p+2}{(p-2)(p+2+\mu '\Sigma ^{-1}\mu }\right] \right\} , \end{aligned}$$

(57)

and for $p\ge 3$,:

$$\begin{aligned}{} & {} \text {max} \left\{ \frac{1}{tr(\Lambda ) + \mu '\mu },\frac{1}{\lambda _{(p)}}\left[ \frac{1}{p-2+\mu '\Sigma ^{-1}\mu }\right] \right\} \le E\left[ \frac{1}{X'X}\right] \\{} & {} \quad \le \frac{1}{\lambda _{(1)}}\left[ \frac{p+2}{(p-2)(p+2+\mu '\Sigma ^{-1}\mu )}\right] . \end{aligned}$$

$\square$

1.2 A.2. Proof of Lemma 5 in Sect. 2

Proof

Let $q_{ij}$ denote the $(i,j)^{th}$ entry in Q.

$$\begin{aligned} \frac{\partial }{\partial x_i} \frac{r(\Vert x-y \Vert ^2_{Q})(x_i-y_i)}{\Vert x-y \Vert ^2_{Q}}= \end{aligned}$$

(by an application of Lemma 4 on $\Vert x-y\Vert ^2_{Q}$)

$$\begin{aligned}{} & {} \frac{\Vert x-y \Vert ^2_{Q}[2r'(\Vert x-y \Vert ^2_{Q})\sum _{j=1}^pq_{ij}(x_j-y_j)(x_i-y_i)+r(\Vert x-y \Vert ^2_{Q})]}{(\Vert x-y \Vert ^2_{Q})2} \\{} & {} \quad - \frac{2r(\Vert x-y \Vert ^2_{Q})\sum _{j=1}^pq_{ij}(x_j-y_j)(x_i-y_i)}{(\Vert x-y \Vert ^2_{Q})2}, \end{aligned}$$

so that

$$\begin{aligned}{} & {} \frac{r^2( \Vert x-y \Vert ^2_Q)}{\Vert x-y \Vert ^2_Q}-2\text {div}_x\left( \frac{r(\Vert x-y\Vert ^2_Q)}{\Vert x-y\Vert ^2_Q} (x-y)\right) \nonumber \\{} & {} \quad = \frac{r^2(\Vert x-y\Vert ^2_{Q})}{\Vert x-y \Vert ^2_{Q})}-2\left[ \frac{(p-2)r(\Vert x-y \Vert ^2_{Q})}{\Vert x-y \Vert ^2_{Q}}+2r'(\Vert x-y \Vert ^2_{Q}) \right] \nonumber \\{} & {} \qquad {\text{(by } \text{ assumption } \text{ ii) }}\nonumber \\{} & {} \quad \le \frac{r^2(\Vert x-y \Vert ^2_{Q})-2(p-2)r(\Vert x-y \Vert ^2_{Q})}{\Vert x-y \Vert ^2_{Q}} \nonumber \\{} & {} \quad = \frac{-r(\Vert x-y \Vert ^2_{Q})[2(p-2)-r(\Vert x-y \Vert ^2_{Q})]}{\Vert x-y \Vert ^2_{Q}} \le 0. \end{aligned}$$

(58)

$\square$

1.3 A.3. Proof of Corollary 2 in Sect. 2

Proof

Let $Z=B(Y-AX)$. Using expression (14), the risk of the estimator $\delta$, $R(\delta ,\theta ,\eta )$ can be expressed as:

$$\begin{aligned}{} & {} R(X,\theta ) +E\left[ E\left[ \frac{a^2}{\Vert X-\theta _0\Vert _{Q^*}}-2\text {div}_x\left( \frac{a}{\Vert X-\theta _0 \Vert ^2_{Q^*}}(X-\theta _0)\right) \mid Z=\theta _0\right] \right] \nonumber \\{} & {} \quad = \text {tr}(Q\Sigma _{11})+E\left[ E\left[ \frac{a^2}{\Vert X-\theta _0 \Vert ^2_{Q^*}}-2\frac{(p-2)a}{\Vert X-\theta _0 \Vert ^2_{Q^*}}\mid Z=\theta _0\right] \right] \nonumber \\{} & {} \quad = \text {tr}(Q\Sigma _{11}) -a(2(p-2)-a)E\left[ \frac{1}{\Vert X-Z \Vert ^2_{Q^*}}\right] . \end{aligned}$$

(59)

Since $Q>0$ and symmetric, $Q^*= \Sigma _{11}^{-1} Q^{-1}\Sigma _{11}^{-1}$ has a symmetric square root denoted by $Q^{*}{}^{\frac{1}{2}}$. Let

$$\begin{aligned} \Sigma ^*= \Sigma _{11}+B(\Sigma _{22}-\Sigma _{21}\Sigma _{11}^{-1}\Sigma _{12})B'. \end{aligned}$$

Making the change of variables,

$$\begin{aligned} W=X-Z=X-B(Y-AX) \sim N_p(\mu _{\theta \eta }, \Sigma ^*), \end{aligned}$$

and

$$\begin{aligned} V=Q^*{}^{\frac{1}{2}}W \sim N_p(Q^*{}^{\frac{1}{2}}\mu _{\theta \eta }, Q^*{}^{\frac{1}{2}}\Sigma ^*Q^*{}^{\frac{1}{2}}), \end{aligned}$$

(60)

in (59) implies

$$\begin{aligned} R(\delta ,\theta ,\eta )= tr(Q\Sigma _{11})-a(2(p-2)-a)E\left[ \frac{1}{V'V}\right] . \end{aligned}$$

(61)

Since V has a multivariate normal distribution, whose parameters are given in (60), an application of Lemma 2 to the $E\left[ \frac{1}{V'V}\right]$ in (61) implies the result since any eigenvalue of $Q^*{}^{\frac{1}{2}}[\Sigma _{11}+B(\Sigma _{22}-\Sigma _{21}\Sigma _{11}^{-1}\Sigma _{12})B']Q^*{}^{\frac{1}{2}}$ is also an eigenvalue of $[\Sigma _{11}+B(\Sigma _{22}-\Sigma _{21}\Sigma _{11}^{-1}\Sigma _{12})B']Q^*$. $\square$

1.4 A.4. Proof of Lemma 6 in Sect. 3.1

Proof

Let $Y=\frac{1}{\sigma }\Sigma ^{-\frac{1}{2}}X$, and $Z = P'Y$ where $P'\Sigma ^{\frac{1}{2}}Q\Sigma ^{\frac{1}{2}}P =\Lambda$ where $\Lambda$ is the diagonal matrix of eigenvalues and P is the associated column matrix of eigenvectors of $\Sigma ^{\frac{1}{2}}Q\Sigma ^{\frac{1}{2}}$. Then, $Y \sim N_p(\frac{1}{\sigma }\Sigma ^{-\frac{1}{2}}\mu , I)$ and $Z\sim N_p(\frac{1}{\sigma }P'\Sigma ^{-\frac{1}{2}}\mu ,I)$ so that:

$$\begin{aligned}{} & {} E\left[ \frac{X'QX}{\sigma ^2}\right] = E\left[ Y'\Sigma ^{\frac{1}{2}}Q\Sigma ^{\frac{1}{2}}Y\right] =E[(P'Y)'P'\Sigma ^{\frac{1}{2}}Q\Sigma ^{\frac{1}{2}}P(P'Y)]\\{} & {} \quad = E[Z'\Lambda Z] = E\left[ \sum _{i=1}^p \lambda _i Z_i^2\right] . \end{aligned}$$

Now, $\lbrace Z_i \rbrace _{i=1}^p$ is an an independent collection of random variables, where

$$\begin{aligned} Z_i \sim N\left( \frac{\mu ^*_i}{\sigma }, 1\right) \end{aligned}$$

and where $\mu ^*=P'\Sigma ^{-\frac{1}{2}}\mu ,$ and thus $\lbrace Z_i^2 \rbrace _{i=1}^p$ is a collection of independent random variables with

$$\begin{aligned} Z_i^2 \sim \chi ^2_1\left( \frac{\mu _i^{*2}}{\sigma ^2}=\nu _i\right) . \end{aligned}$$

Since a non-central chi-squared random variable has a monotone likelihood ratio (Lehmann & Romano, 2005), $\chi ^2_1(\nu _i)$ is stochastically increasing in the parameter $\nu _i$, and thus is stochastically decreasing in $\sigma ^2$. Let

$$\begin{aligned} U(z_1^2,\ldots ,z_p^2) =\sum _{i=1}^p \lambda _iz_i^2. \end{aligned}$$

Since U is increasing in each of its coordinates, $\lbrace Z_i^2 \rbrace _{i=1}^p$ is independent collection of random variables, and each $Z_i^2$ is stochastically decreasing in $\sigma ^2$, $U(Z_1^2,Z_2^2,\ldots ,Z_p^2)=\sum _{i=1}^p\lambda _iZ_i^2$ is stochastically decreasing in $\sigma ^2$, establishing the result. $\square$

1.5 A.5. Proof of Lemma 7 in Sect. 3.2

Proof

$$\begin{aligned}{} & {} E[(X-\theta )'\Sigma _{11}^{-1}g(X,Y)]\nonumber \\{} & {} \quad =\int \left[ \sum _{i=1}^p\int \left[ \sum _{j=1}^p \sigma ^*_{ij}(x_j-\theta _j)g_i(x,y)\right] \right. \nonumber \\ {}{} & {} \quad \left. f(\Vert x-\theta \Vert ^2_{\Sigma _{11}^{-1}}+\Vert y-(\theta +\eta ) \Vert ^2_{\Sigma _{22}^{-1}}) {\text {d}}x\right] {\text {d}}y\nonumber \\{} & {} \quad =\int \left[ \sum _{i=1}^p \int g_i(x,y) \left\{ -\frac{\partial }{\partial x_i}(\frac{1}{2}\int _{t}^{\infty }f(u){\text {d}}u)\right\} {\text {d}}x\right] {\text {d}}y\nonumber \\{} & {} \quad =\int \left[ \sum _{i=1}^p \int \frac{\partial }{\partial x_i}g_i(x,y)F(t){\text {d}}x\right] {\text {d}}y, \end{aligned}$$

(62)

where $t=\Vert x-\theta \Vert ^2_{\Sigma _{11}^{-1}}+\Vert y-(\theta +\eta ) \Vert ^2_{\Sigma _{22}^{-1}}$, and $\sigma _{ij}^{*}$ is the i, $j^{th}$ element of $\Sigma _{11}^{-1}$. Upon dividing and multiplying (62) by f(t),

$$\begin{aligned}{} & {} \int \int \text {div}_x(g(x,y))\frac{F(t)}{f(t)}f(t){\text {d}}x{\text {d}}y\nonumber \\{} & {} \quad = E\left[ \text {div}_x(g(X,Y))\frac{F\left( \Vert X-\theta \Vert ^2_{\Sigma _{11}^{-1}}+\Vert Y-(\theta +\eta ) \Vert ^2_{\Sigma _{22}^{-1}}\right) }{f\left( \Vert X-\theta \Vert ^2_{\Sigma _{11}^{-1}}+\Vert Y-(\theta +\eta ) \Vert ^2_{\Sigma _{22}^{-1}}\right) }\right] \end{aligned}$$

(63)

establishing the result. $\square$

1.6 A.6. Proof of Lemma 8 in Sect. 4

Proof

Let $t=\frac{1}{\sigma ^2}(\Vert x-\theta \Vert ^2_{\Sigma _{11}^{-1}} + \Vert y-\eta ^* \Vert ^2_{\Sigma _{22}^{-1}}+\Vert u \Vert ^2)$. Denoting

$$\begin{aligned} F(t) = \frac{1}{2}\int _t^{\infty }f(u){\text {d}}u, \end{aligned}$$

the partial derivative of F(t) with respect to the $i^{th}$ coordinate of u is

$$\begin{aligned} \frac{\partial }{\partial u_i}F(t)= \frac{\partial }{\partial u_i} \frac{1}{2}\int _t^{\infty }f(u){\text {d}}u = -f(t)\frac{u_i}{\sigma ^2}. \end{aligned}$$

(64)

Therefore,

$$\begin{aligned}{} & {} E\left[ \frac{h(X,Y,S)}{\sigma ^2}\right] \nonumber \\ {}{} & {} \quad =\int _{\Re ^{2p+k}} \frac{h\left( x,y,\Vert u \Vert ^2\right) }{\sigma ^2}f\left( \frac{1}{\sigma ^2}(\Vert x-\theta \Vert ^2_{\Sigma _{11}^{-1}} + \Vert y-\eta ^* \Vert ^2_{\Sigma _{22}^{-1}}+\Vert u \Vert ^2)\right) {\text {d}}u{\text {d}}x{\text {d}}y\nonumber \\{} & {} \quad = \int _{\Re ^{2p+k}} \frac{u'}{\sigma ^2} \frac{uh(x,y,\Vert u \Vert ^2)}{\Vert u \Vert ^2}f\left( \frac{1}{\sigma ^2}(\Vert x-\theta \Vert ^2_{\Sigma _{11}^{-1}} + \Vert y-\eta ^* \Vert ^2_{\Sigma _{22}^{-1}}+\Vert u \Vert ^2)\right) \nonumber \\ {}{} & {} \quad {\text {d}}u{\text {d}}x{\text {d}}y. \end{aligned}$$

(65)

Let $g_{x,y}(u) = \frac{uh(x,y,\Vert u \Vert ^2)}{\Vert u \Vert ^2}$ with $i^{th}$ coordinate

$$\begin{aligned} g_{x,y}(u)_i = \frac{u_ih(x,y,\Vert u \Vert ^2)}{\Vert u \Vert ^2} \end{aligned}$$

in (65). By the weak differentiability of g, and the expression for the partial derivative of F(t) in (64), expression (65) satisfies

$$\begin{aligned}{} & {} \sum _{i=1}^k \int _{\Re ^{2p+k}} \frac{u_i}{\sigma ^2}g_{x,y}(u)_i f\left( \frac{1}{\sigma ^2}(\Vert x-\theta \Vert ^2_{\Sigma _{11}^{-1}} + \Vert y-\eta ^* \Vert ^2_{\Sigma _{22}^{-1}}+\Vert u \Vert ^2)\right) {\text {d}}u{\text {d}}x{\text {d}}y, \nonumber \\\end{aligned}$$

(66)

$$\begin{aligned}{} & {} \quad = \sum _{i=1}^k\int _{\Re ^{2p+k}}g_{x,y}(u)_i \left( -\frac{\partial }{\partial u_i} F(t)\right) {\text {d}}u{\text {d}}x{\text {d}}y, \end{aligned}$$

(67)

$$\begin{aligned}{} & {} \quad = \sum _{i=1}^k \int _{\Re ^{2p+k}} \left( \frac{\partial }{\partial u_i}g_{x,y}(u)_i\right) F(t){\text {d}}u{\text {d}}x{\text {d}}y \end{aligned}$$

(68)

$$\begin{aligned}{} & {} \quad = \int _{\Re ^{2p+k}} \text {div}_u(g_{x,y}(u))\frac{F(t)}{f(t)}f(t){\text {d}}u{\text {d}}x{\text {d}}y \end{aligned}$$

$$\begin{aligned}{} & {} \quad = E\left[ \text {div}_u(g_{x,y}(u))\frac{F(t)}{f(t)}\right] , \end{aligned}$$

(69)

where the equality from (67) to (68) is justified by the weak differentiability of $g_{x,y}(u)$ for all (x, y).

Since

$$\begin{aligned} \text {div}_u(g_{x,y}(u))=(k-2)\frac{h(x,y,s)}{S}+2\frac{\partial }{\partial S}h(x,y,s) \end{aligned}$$

expression (69) is equivalent to

$$\begin{aligned} E\left[ \left( (k-2)\frac{h(X,Y,S)}{S}+2\frac{\partial }{\partial S}h(X,Y,S)\right) \frac{F(t)}{f(t)}\right] \end{aligned}$$

establishing the result. $\square$

Appendix B

1.1 B.1. Development of UMVUE Estimator

When $\eta =0$ the density of the pair of random variables $\begin{pmatrix} X\\ Y \end{pmatrix}$ is

$$\begin{aligned} f(x,y)= \vert 2\pi \Sigma \vert ^{-\frac{1}{2}}\exp \left( -\frac{1}{2}\left( \begin{pmatrix} x-\theta \\ y-\theta \end{pmatrix}'\Sigma ^{-1} \begin{pmatrix} x-\theta \\ y-\theta \end{pmatrix} \right) \right) . \end{aligned}$$

(70)

Let

$$\begin{aligned} X_{2.1}= Y-\Sigma _{21}\Sigma _{11}^{-1}X \end{aligned}$$

so that X is independent of $X_{2.1}$, and

$$\begin{aligned} Y^* =(I-\Sigma _{21}\Sigma _{11}^{-1})^{-1}X_{2.1} \end{aligned}$$

(71)

so that,

$$\begin{aligned} E[Y^*]=(I-\Sigma _{21}\Sigma _{11}^{-1})^{-1}(I-\Sigma _{21}\Sigma _{11}^{-1})\theta =\theta \end{aligned}$$

(72)

$Y^*$ is an unbiased estimator of $\theta$, with variance

$$\begin{aligned} \text {var}(Y^*)=\Sigma _{Y^*}=(I-\Sigma _{21}\Sigma _{11}^{-1})^{-1}[\Sigma _{22}-\Sigma _{21}\Sigma _{11}^{-1}\Sigma _{12}](I-\Sigma _{21}\Sigma _{11}^{-1})^{-t}. \end{aligned}$$

(73)

The joint density of $\begin{pmatrix} X\\ Y^*\end{pmatrix}$ is

$$\begin{aligned} f(x,y^*)=k\exp \left( -\frac{1}{2}\left( \begin{pmatrix}x-\theta \\ y^*-\theta \end{pmatrix}' \begin{pmatrix} \Sigma _{11}^{-1} &{} 0 \\ 0 &{} \Sigma _{Y^*}^{-1} \end{pmatrix} \begin{pmatrix} x-\theta \\ y^*-\theta \end{pmatrix}\right) \right) , \end{aligned}$$

(74)

where k is the normalizing constant in (74). By expanding the quadratic form in the exponential of density (74), we can find a complete sufficient statistic for the parameter $\theta$ that will be unbiased and thus will be the UMVUE estimator. Expanding the quadratic form in (74) yields

$$\begin{aligned} x'\Sigma _{11}^{-1}x+\theta '\Sigma _{11}^{-1} \theta -2\theta '\Sigma _{11}^{-1}x +y^{*}{}'\Sigma _{Y^*}^{-1}y^*+ \theta '\Sigma _{Y^*}^{-1} \theta -2\theta '\Sigma _{Y^*}^{-1}y^*, \end{aligned}$$

(75)

which implies the density in (74) is an exponential family of the form

$$\begin{aligned} h(x,y^*)g(\theta )e^{\theta 'T(X,Y^*)} \end{aligned}$$

(76)

with complete sufficient statistic

$$\begin{aligned} T(X,Y^*)= \Sigma _{11}^{-1}X+\Sigma _{Y^*}^{-1}Y^*. \end{aligned}$$

(77)

Thus, the UMVUE estimator of $\theta$, $\delta _c$, is

$$\begin{aligned} \delta _c(X,Y)=(\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1})^{-1}(\Sigma _{11}^{-1}X+\Sigma _{Y^*}^{-1}Y^*), \end{aligned}$$

(78)

with variance

$$\begin{aligned} \Sigma _{\delta _c}=\text {Cov}(\delta _c,\delta _c)=(\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1})^{-1}. \end{aligned}$$

(79)

When the loss is of the form $L_Q(d,\theta )=(d-\theta )'Q(d-\theta )$, the risk $\delta _c$ when $Y^*$ is unbiased for $\theta$ is

$$\begin{aligned}{} & {} E[(\delta _c-\theta )'Q(\delta _c-\theta )]=tr(Q\Sigma _{\delta _c})=tr(Q(\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1})^{-1}) \\{} & {} \quad = \text {tr}(Q[\Sigma _{11}^{-1}+(I-\Sigma _{21}\Sigma _{11}^{-1})'[\Sigma _{22}-\Sigma _{21}\Sigma _{11}^{-1}\Sigma _{12}]^{-1}(I-\Sigma _{21}\Sigma _{11}^{-1})]^{-1} ). \end{aligned}$$

When the researcher is mistaken and $Y^*$ is a biased for $\theta$ ($\eta \ne 0$), with bias

$$\begin{aligned}{} & {} E[\delta _c-\theta ]= E[(\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1})^{-1}(\Sigma _{11}^{-1}\theta +\Sigma _{Y^*}^{-1}(\theta +\eta ))-\theta ] \\{} & {} \quad = (\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1})^{-1}\Sigma _{Y^*}^{-1}\eta , \end{aligned}$$

the risk of using $\delta _c$ as an estimator of $\theta$ will be

$$\begin{aligned} E[\Vert \delta _c(X,Y)-\theta \Vert ^2_{Q}]= \text {tr}(Q\Sigma _{\delta _c})+\eta '\Sigma _{Y^*}^{-1}(\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1} )^{-1}Q(\Sigma _{11}^{-1}+\Sigma _{Y^*}^{-1} )^{-1}\Sigma _{Y^*}^{-1}\eta , \end{aligned}$$

which is unbounded in $\eta$ unlike the estimators develop in Sect. 2 which have bounded risk.

1.2 B.2. Comparison of Risks Between $\delta _c$ (50) and $\delta _1$(45)

To compare the risk $\delta _1(X,Y)$ and $\delta _c(X,Y)$, we first compare the risk of using $\delta _c$ when $\eta =0$, and show that $\delta _c$ will have uniformly smaller risk than $\delta _1$ when $\eta =0$. We then give sufficient conditions for when the risk of $\delta _1$ will dominate the risk of $\delta _c$ by comparing the upper bound for the risk of $\delta _1(X,Y)$ developed in Corollary 2 to the exact risk of using $\delta _c$. When $\eta '\eta =0$,

$$\begin{aligned} p\sigma ^2-\frac{(p-2)\sigma ^2B}{A+B} \le R(\delta _1(X,Y), \theta , \eta ) \le p\sigma ^2-\frac{(p-2)\sigma ^2B}{A+B}, \end{aligned}$$

where

$$\begin{aligned} A:= \sigma ^2\tau ^2(1-\varrho ^2) \end{aligned}$$

and

$$\begin{aligned} B:= (\sigma ^2-\varrho \sigma \tau )^2, \end{aligned}$$

by Corollary 2. In comparison, the exact risk of $\delta _c$

$$\begin{aligned} R(\delta _c(X,Y),\theta ,\eta ) = \frac{p\sigma ^2A}{A+B}=\frac{p\sigma ^2(\sigma ^2\tau ^2(1-\varrho ^2))}{\sigma ^2\tau ^2(1-\varrho ^2)+(\sigma ^2-\varrho \sigma \tau )^2} \end{aligned}$$

so that

$$\begin{aligned}{} & {} R(\delta _1(X,Y),\theta , 0)-R(\delta _c(X,Y),\theta , 0)=p\sigma ^2-\frac{(p-2)\sigma ^2B}{A+B}- \frac{p\sigma ^2A}{A+B} \\{} & {} \quad = p\sigma ^2\frac{B}{A+B}-(p-2)\sigma ^2\frac{B}{A+B}= \frac{2\sigma ^2B}{A+B} \ge 0 \end{aligned}$$

implying that $R(\delta _c(X,Y), \theta , 0 ) < R(\delta _1(X,Y), \theta , 0)$.

When the upper bound for the risk of $\delta _1$ given by Corollary 2 is less than the risk of $\delta _c$

$$\begin{aligned}{} & {} p\sigma ^2-\frac{(p-2)\sigma ^2B}{(p-2)[B+A]+\sigma ^2\eta '\eta }< \frac{p\sigma ^2A}{A+B}+ \frac{\eta '\eta B^2}{[A+B]^2}\nonumber \\{} & {} \quad \Leftrightarrow \frac{p\sigma ^2B}{A+B} -\frac{(p-2)\sigma ^2B}{(p-2)[B+A]+\sigma ^2\eta '\eta } < \frac{\eta '\eta B^2}{[A+B]^2} \nonumber \\{} & {} \quad \Leftrightarrow p\sigma ^2B(A+B)[(p-2)(B+A)+\sigma ^2\eta '\eta ]- (p-2)\sigma ^2B[A+B]^2\nonumber \\ \end{aligned}$$

(80)

$$\begin{aligned}{} & {} \quad < \eta '\eta B^2[(p-2)(B+A)+\sigma ^2\eta '\eta ]. \end{aligned}$$

(81)

Let $x =\eta '\eta$ where $x \in [0,\infty )$ and

$$\begin{aligned} Q(x)= ax^2+bx+c. \end{aligned}$$

For

$$\begin{aligned} a= & {} \sigma ^2B^2,\\ c= & {} -(p-2)\sigma ^2B(A+B)^2[p-1], \end{aligned}$$

and

$$\begin{aligned} b= B(B+A)[(p-2)B-p\sigma ^4], \end{aligned}$$

expressions (80)–(81) will be satisfied once $Q(x)>0$. Since $a>0$ and $c<0$, Q(x) will have 2 real roots denoted by $r_1$ and $r_2$ respectively, where $r_1 <0$ and $r_2>0$. Since $a>0$, $x >r_2$ implies $Q(x)>0$ and so a sufficient condition for $R(\delta _1(X,Y), \theta ,\eta ) < R(\delta _c(X,Y), \theta ,\eta )$ is

$$\begin{aligned}{} & {} \eta ' \eta > \frac{-(\sigma ^2-\varrho \sigma \tau )^2[(\sigma ^2-\varrho \sigma \tau )^2+\sigma ^2\tau ^2(1-\varrho ^2)]((p-2)(\sigma ^2-\varrho \sigma \tau )^2-p\sigma ^4)}{2\sigma ^2(\sigma ^2-\varrho \sigma \tau )^4}\\{} & {} \quad + \frac{\sqrt{(\sigma ^2-\varrho \sigma \tau )^4( (\sigma ^2-\varrho \sigma \tau )^2+\sigma ^2\tau ^2(1-\varrho ^2))^2 [((p-2)(\sigma ^2-\varrho \sigma \tau )^2-p\sigma ^4 )^2+4\sigma ^4(p-2)(p-1)(\sigma ^2-\varrho \sigma \tau )^2]}}{2\sigma ^2(\sigma ^2-\varrho \sigma \tau )^4}. \end{aligned}$$