On the Simultaneous Estimation of Weibull Reliability Functions

Abstract

Under the homogeneity assumption of k reliability functions from  a two-parameter Weibull distribution, we have developed the asymptotic theory for the simultaneous estimation of reliability functions. Improved estimation strategies based on the Graybill Deal, preliminary test and the James-Stein type shrinkage principles are discussed. Using the sequence of local alternatives and squared error loss function, we have derived the asymptotic distributional quadratic bias and risk of the said estimators. A comparison in terms of risk has also been carried out among the listed estimators relative to the benchmark maximum likelihood estimator. To be more specific, we have pointed out the regions where our suggested estimators perform better than other estimators. A comprehensive Monte-Carlo simulation study is presented to validate the behavior of various estimation methods in terms of simulated relative efficiency along with a real-data application.

Appendices

Appendix

Following lemma due to Judge and Bock [10] is very useful for the smooth derivation of ADB, ADQB, and ADQR expressions of our estimators.

Lemma 3

Let the k-dimensional random vector $\varvec{x}$ follows a multivariate normal distribution with mean $\varvec{\mu }_{x}$ and covariance $\varvec{I}_{k}$. Then, for any measurable function $\phi (\cdot )$, we have

$$\begin{aligned} E\left[ \varvec{x}\phi (\varvec{x}'\varvec{x})\right]&=\varvec{\mu }_{x}E\left[ \phi \left( \chi ^{2}_{k+2}(\varDelta )\right) \right] \end{aligned}$$

(36)

$$\begin{aligned} E\left[ \varvec{x}\varvec{x}'\phi (\varvec{x}'\varvec{x})\right]&=\varvec{I}_{k}E\left[ \phi \left( \chi ^{2}_{k+2}(\varDelta )\right) \right] +\varvec{\mu }_{x}\varvec{\mu }'_{x}E\left[ \phi \left( \chi ^{2}_{k+4}(\varDelta )\right) \right] , \end{aligned}$$

(37)

where $\varDelta$ is the non-centrality parameter.

Proof of Theorem 1

Let $W_i=\sqrt{n_i}(\hat{\gamma }_i-\gamma _i)$ then from Lemma 1 and under the sequence of local alternatives $\mathscr {H}_n$, we know that $\sqrt{n_i}\left( \hat{\gamma }_i-\gamma _i\right) \xrightarrow {D}\mathscr {N}(0,v_0)$, which gives $\sqrt{n}\sqrt{\lambda _{i,n}}\left( \hat{\gamma _i}-\gamma _i\right) \xrightarrow {D}\mathscr {N}\left( 0,v_{0}\right)$, or $\sqrt{n}\left( \hat{\gamma _i}-\gamma _i\right) \xrightarrow {D}\mathscr {N}\left( 0,v_{0}p^{-1}_i\right)$. Thus, assuming independence among $W_i$’s and using matrix form we write

$$\begin{aligned} \varvec{W}_n=\sqrt{n}\left( \hat{\varvec{\gamma }}^{UE}-\varvec{\gamma }\right) \xrightarrow {D}\varvec{W}\sim \mathscr {N}_k(\varvec{0},\varvec{V}), \end{aligned}$$

(38)

where $\varvec{V}=v_{0}\varvec{\varOmega }^{-1}$, $v_0=(\gamma _0\ln \gamma _0)^{2}\left[ 1.109-0.514\ln \left( -\ln \gamma _0\right) +0.608\left\{ \ln \left( -\ln \gamma _0\right) \right\} ^{2}\right]$, $\varvec{\varOmega }=\text {diag}\left( p_{1},p_{2},\cdots ,p_{k}\right)$, and fixing $\lim (\lambda _{i,n})=p_i$ to be constant. Under $\mathscr {H}_n$ we have $\varvec{X}_n=\sqrt{n}(\hat{\varvec{\gamma }}^{UE}-\varvec{\gamma }_0)=\varvec{W}_n+\varvec{\xi }$, which is a linear function of $\varvec{W}_n$ and thus

$$\begin{aligned} \varvec{X}_n=\sqrt{n}(\hat{\varvec{\gamma }}^{UE}-\varvec{\gamma }_0)\xrightarrow {D}\varvec{X}\sim \mathscr {N}_k(\varvec{\xi },\varvec{V}). \end{aligned}$$

(39)

Similarly, we write $\varvec{Z}_n=\sqrt{n}(\hat{\varvec{\gamma }}^{RE}-\varvec{\gamma }_0)=\varvec{1}_k\varvec{1}'_k\varvec{\varOmega }_n\varvec{X}_n$, which is again a linear function of $\varvec{X}_n$. Assuming $\varvec{p}'\varvec{\xi }=0$, $\varvec{\varOmega }_n\xrightarrow {P}\varvec{\varOmega }=\text {diag}(p_1,p_2,\cdots ,p_k)$ and invoking Slutsky’s theorem, we say that

$$\begin{aligned} \varvec{Z}_n=\sqrt{n}(\hat{\varvec{\gamma }}^{RE}-\varvec{\gamma }_0)\xrightarrow {D}\varvec{Z}\sim \mathscr {N}_k(\varvec{0},v_0\varvec{1}_k\varvec{1}'_k). \end{aligned}$$

(40)

Also, we have $\varvec{Y}_n=\sqrt{n}(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})=\varvec{X}_n-\varvec{Z}_n=(\varvec{I}_k-\varvec{1}_k\varvec{1}'_k\varvec{\varOmega }_n)\varvec{X}_n=\varvec{H}_n\varvec{X}_n$, which is a linear function of $\varvec{X}_n$. Since, $\varvec{H}_n\xrightarrow {P}\varvec{H}_0=\varvec{I}_k-\varvec{1}_k\varvec{1}'_k\varvec{\varOmega }$ and by Slutsky’s theorem

$$\begin{aligned} \varvec{Y}_n=\sqrt{n}(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})\xrightarrow {D}\varvec{Y}\sim \mathscr {N}_k(\varvec{\beta },\varvec{B}_0), \end{aligned}$$

(41)

where $\varvec{\beta }=\varvec{H}_0\varvec{\xi }$ and $\varvec{B}_0=\varvec{H}_0\varvec{V}\varvec{H}_0'=\varvec{V}\varvec{H}_0'$.

For the joint distributions, we firstly note that $\begin{pmatrix} \varvec{X}_n \\ \varvec{Y}_n \end{pmatrix}= \begin{pmatrix} \varvec{I}_k \\ \varvec{H}_n \end{pmatrix}\varvec{X}_n=\varvec{C}_n\varvec{X}_n$, which is a linear combination of $\varvec{X}_n$ and by (39) and (41)

$$\begin{aligned} \begin{pmatrix} \varvec{X}_n \\ \varvec{Y}_n \end{pmatrix}\xrightarrow {D} \begin{pmatrix} \varvec{X} \\ \varvec{Y} \end{pmatrix}\sim \mathscr {N}_{2k} \begin{Bmatrix} \begin{pmatrix} \varvec{\xi } \\ \varvec{\beta } \end{pmatrix}, \begin{pmatrix} \varvec{V} &{} \varvec{B}_0\\ \varvec{B}_0' &{} \varvec{B}_0 \end{pmatrix} \end{Bmatrix}. \end{aligned}$$

(42)

Final, we have $\begin{pmatrix} \varvec{Z}_n \\ \varvec{Y}_n \end{pmatrix}= \begin{pmatrix} \varvec{1}_k\varvec{1}'_k\varvec{\varOmega }_n \\ \varvec{H}_n \end{pmatrix}\varvec{X}_n=\varvec{F}_n\varvec{X}_n$, which is again a linear function of $\varvec{X}_n$ and by (40) and (41)

$$\begin{aligned} \begin{pmatrix} \varvec{Z}_n \\ \varvec{Y}_n \end{pmatrix}\xrightarrow {D} \begin{pmatrix} \varvec{Z} \\ \varvec{Y} \end{pmatrix}\sim \mathscr {N}_{2k} \begin{Bmatrix} \begin{pmatrix} \varvec{0} \\ \varvec{\beta } \end{pmatrix}, \begin{pmatrix} v_0\varvec{1}_k\varvec{1}'_k &{} \varvec{0}\\ \varvec{0} &{} \varvec{B}_0 \end{pmatrix} \end{Bmatrix}. \end{aligned}$$

(43)

Proof of Theorem 2

Firstly, we compte ADB of the estimators under consideration as follows:

$$\begin{aligned} \varvec{b}(\hat{\varvec{\gamma }}^{LS})&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{LS}-\varvec{\gamma }_{n})\right] =\lim _{n\rightarrow \infty }E\left[ \sqrt{n}\left\{ \hat{\varvec{\gamma }}^{UE}-c(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})-\varvec{\gamma }_0-\varvec{\xi }/\sqrt{n}\right\} \right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \varvec{X}_n-\varvec{\xi }-c\varvec{Y}_n\right] =E(\varvec{X})-\varvec{\xi }-cE(\varvec{Y})\nonumber \\&=-c\varvec{\beta }. \end{aligned}$$

(44)

By replacing $c=0$, and $c=1$ in (44), one can obtain ADB of $\hat{\varvec{\gamma }}^{UE}$ and $\hat{\varvec{\gamma }}^{RE}$, respectively. Secondly, by utilizing (36) we write

$$\begin{aligned} \varvec{b}(\hat{\varvec{\gamma }}^{SP})&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{SP}-\varvec{\gamma }_{n})\right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}\left\{ \hat{\varvec{\gamma }}^{UE}-c(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})I(\mathscr {D}_n<d_{n,\alpha })-\varvec{\gamma }_0-\varvec{\xi }/\sqrt{n}\right\} \right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \varvec{X}_n-\varvec{\xi }-c\varvec{Y}_nI(\mathscr {D}_n<d_{n,\alpha })\right] \nonumber \\&=E(\varvec{X})-\varvec{\xi }-cE[\varvec{Y}I(\chi ^{2}_{k-1}(\varDelta )<\chi ^{2}_{k-1,\alpha })]=-c E[\varvec{Y}I(\chi ^{2}_{k-1}(\varDelta )<\chi ^{2}_{k-1,\alpha })]\nonumber \\&=-c\varvec{\beta }\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta ). \end{aligned}$$

(45)

For $c=1$, (45) gives ADB of $\hat{\varvec{\gamma }}^{PT}$. In the similar way, we may write

$$\begin{aligned} \varvec{b}(\hat{\varvec{\gamma }}^{JS})&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n})\right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}\left\{ \hat{\varvec{\gamma }}^{UE}-(k-3)\mathscr {D}_{n}^{-1}(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})-\varvec{\gamma }_0-\varvec{\xi }/\sqrt{n}\right\} \right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \varvec{X}_n-\varvec{\xi }-(k-3)\varvec{Y}_n\mathscr {D}_{n}^{-1}\right] =E(\varvec{X})-\varvec{\xi }-(k-3)E[\varvec{Y}\chi ^{-2}_{k-1}(\varDelta )]\nonumber \\&=-(k-3)\varvec{\beta }E[\chi ^{-2}_{k+1}(\varDelta )]. \end{aligned}$$

(46)

Finally, we have

$$\begin{aligned} \varvec{b}(\hat{\varvec{\gamma }}^{S+})&=\lim _{n\rightarrow \infty }\mathbb {E}\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{S+}-\varvec{\gamma }_{n})\right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}\left\{ \hat{\varvec{\gamma }}^{JS}-\left( 1-(k-3)\mathscr {D}_{n}^{-1}\right) I(\mathscr {D}_n<(k-3))(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})-\varvec{\gamma }_{n}\right\} \right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}\left\{ \hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n}-\left( 1-(k-3)\mathscr {D}_{n}^{-1}\right) I(\mathscr {D}_n<(k-3))(\hat{\varvec{\gamma }}^{UE}-\hat{\varvec{\gamma }}^{RE})\right\} \right] \nonumber \\&=\varvec{b}(\hat{\varvec{\gamma }}^{JS})-\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_nI(\mathscr {D}_n<(k-3))\right] +(k-3)\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_n\mathscr {D}_{n}^{-1}I(\mathscr {D}_n<(k-3))\right] \nonumber \\&=\varvec{b}(\hat{\varvec{\gamma }}^{JS})-E\left[ \varvec{Y}I(\chi ^{2}_{k-1}(\varDelta )<(k-3))\right] +(k-3)E\left[ \varvec{Y}\chi ^{-2}_{k-1}(\varDelta )I(\chi ^{2}_{k-1}(\varDelta )<(k-3))\right] \nonumber \\&=\varvec{b}(\hat{\varvec{\gamma }}^{JS})-\varvec{\beta }\varPsi _{k+1}(k-3;\varDelta )+(k-3)\varvec{\beta }E\left[ \chi ^{-2}_{k+1}(\varDelta )I(\chi ^{2}_{k+1}(\varDelta )<(k-3))\right] \nonumber \\&=-\varvec{\beta }\left[ \varPsi _{k+1}(k-3;\varDelta )+(k-3)E[\chi ^{-2}_{k+1}(\varDelta )I(\chi ^{2}_{k+1}(\varDelta )>(k-3))]\right] . \end{aligned}$$

(47)

Using these ADB expressions together with (14) completes our proof.

Proof of Theorem 3

To compute ADQR as mentioned in (15), we first need AMSEM of the estimator. Therefore, we first have

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{LS})&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{LS}-\varvec{\gamma }_{n})\sqrt{n}(\hat{\varvec{\gamma }}^{LS}-\varvec{\gamma }_{n})'\right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \left\{ (\varvec{X}_n-\varvec{\xi })-c\varvec{Y}_n\right\} \left\{ (\varvec{X}_n-\varvec{\xi })-c\varvec{Y}_n\right\} '\right] \nonumber \\&=V(\varvec{X})-2\underbrace{c E[(\varvec{X}-\varvec{\xi })\varvec{Y}']}_{T_1}+\, c^2\underbrace{E(\varvec{Y}\varvec{Y}')}_{T_0}, \end{aligned}$$

(48)

where

$$\begin{aligned} T_0=E(\varvec{Y}\varvec{Y}')=V(\varvec{Y})+E(\varvec{Y})E(\varvec{Y}')=\varvec{B}_0+\varvec{\beta }\varvec{\beta }', \end{aligned}$$

(49)

and by conditional expectation argument of a multivariate normal distribution, we write

$$\begin{aligned} T_1&=E[E\{(\varvec{X}-\varvec{\xi })|\varvec{Y}\}\varvec{Y}']=E[\{\varvec{0}+\varvec{B}_0\varvec{B}_0^{-1}(\varvec{Y}-\varvec{\beta })\}\varvec{Y}']\nonumber \\&=E(\varvec{Y}\varvec{Y}'-\varvec{\beta }\varvec{Y}')=E(\varvec{Y}\varvec{Y}')-\varvec{\beta }E(\varvec{Y}')=\varvec{B}_0. \end{aligned}$$

(50)

Replacing (49) and (50) in (48), we have the AMSEM of $\hat{\varvec{\gamma }}^{LS}$ i.e.,

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{LS})=\varvec{V}-c(2-c)\varvec{B}_0+c^2\varvec{\beta }\varvec{\beta }'. \end{aligned}$$

(51)

Thus, by (15) and letting $\varvec{\beta }'\varvec{Q}\varvec{\beta }=\varDelta _{\beta }$ we write

$$\begin{aligned} R(\hat{\varvec{\gamma }}^{LS};\varvec{Q})=\text {tr}(\varvec{Q}\varvec{V})-c(2-c)\text {tr}(\varvec{Q}\varvec{B}_0)+c^2\varDelta _{\beta }. \end{aligned}$$

(52)

Replacement of $c=0$, and $c=1$ in (52) yields ADQR of $\hat{\varvec{\gamma }}^{UE}$ and $\hat{\varvec{\gamma }}^{RE}$. Secondly,

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{SP})=&\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{SP}-\varvec{\gamma }_{n})\sqrt{n}(\hat{\varvec{\gamma }}^{SP}-\varvec{\gamma }_{n})'\right] \nonumber \\ =&\lim _{n\rightarrow \infty }E\left[ \left\{ (\varvec{X}_n-\varvec{\xi })-c\varvec{Y}_n I(\mathscr {D}_n<d_{n,\alpha })\right\} \left\{ (\varvec{X}_n-\varvec{\xi })-c\varvec{Y}_nI(\mathscr {D}_n<d_{n,\alpha })\right\} '\right] \nonumber \\ =&\lim _{n\rightarrow \infty }V(\varvec{X}_n)-2c\lim _{n\rightarrow \infty }E\left[ (\varvec{X}_n-\varvec{\xi })\varvec{Y}'_nI(\mathscr {D}_n<d_{n,\alpha })\right] +c^2\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_n\varvec{Y}'_nI(\mathscr {D}_n<d_{n,\alpha })\right] \nonumber \\ =&V(\varvec{X})-2c\underbrace{E\left[ (\varvec{X}-\varvec{\xi })\varvec{Y}'I(\chi ^{2}_{k-1}(\varDelta )<\chi ^{2}_{k-1,\alpha })\right] }_{T_3}+\, c^2\underbrace{E\left[ \varvec{Y}\varvec{Y}'I(\chi ^2_{k-1}(\varDelta )<\chi ^2_{k-1,\alpha })\right] }_{T_2}, \end{aligned}$$

(53)

such that by (37) we write $T_2$ as

$$\begin{aligned} T_2=\varvec{B}_0\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta )+\varvec{\beta }\varvec{\beta }'\varPsi _{k+3}(\chi ^{2}_{k-1,\alpha };\varDelta ), \end{aligned}$$

(54)

and invoking conditional expectation argument along with (36) and (37), we write

$$\begin{aligned} T_3&=E\left[ E\bigl \{(\varvec{X}-\varvec{\xi })\big |\varvec{Y}\bigr \}\varvec{Y}'I(\chi ^{2}_{k-1}(\varDelta )<\chi ^{2}_{k-1,\alpha })\right] \nonumber \\&=E\left[ (\varvec{Y}-\varvec{\beta })\varvec{Y}'I(\chi ^{2}_{k-1}(\varDelta )<\chi ^{2}_{k-1,\alpha })\right] \nonumber \\&=\underbrace{E\left[ \varvec{Y}\varvec{Y}'I(\chi ^2_{k-1}(\varDelta )<\chi ^2_{k-1,\alpha })\right] }_{T_2}-\varvec{\beta }E\left[ \varvec{Y}'I(\chi ^2_{k-1}(\varDelta )<\chi ^2_{k-1,\alpha })\right] \nonumber \\&=\varvec{B}_0\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta )+\varvec{\beta }\varvec{\beta }'\varPsi _{k+3}(\chi ^{2}_{k-1,\alpha };\varDelta )-\varvec{\beta }\varvec{\beta }'\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta ). \end{aligned}$$

(55)

Replacing (54), (55) in (53) and collecting alike terms gives

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{SP})&=\varvec{V}-c(2-c)\varvec{B}_0\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta )\nonumber \\&+c\varvec{\beta }\varvec{\beta }'\left[ 2\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta )-(2-c)\varPsi _{k+3}(\chi ^{2}_{k-1,\alpha };\varDelta )\right] . \end{aligned}$$

(56)

Using $c=1$ in (56) we have the AMSEM of $\hat{\varvec{\gamma }}^{PT}$ i.e.,

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{PT})=\varvec{V}-\varvec{B}_0\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta )+\, \varvec{\beta }\varvec{\beta }'\left[ 2\varPsi _{k+1}(\chi ^{2}_{k-1,\alpha };\varDelta )-\varPsi _{k+3}(\chi ^{2}_{k-1,\alpha };\varDelta )\right] . \end{aligned}$$

Thirdly, we have

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{JS})&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n})\sqrt{n}(\hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n})'\right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \left\{ (\varvec{X}_n-\varvec{\xi })-(k-3)\varvec{Y}_n\mathscr {D}_{n}^{-1}\right\} \left\{ (\varvec{X}_n-\varvec{\xi })-(k-3)\varvec{Y}_n\mathscr {D}_{n}^{-1}\right\} '\right] \nonumber \\&=\lim _{n\rightarrow \infty }V(\varvec{X}_n)-2(k-3)\lim _{n\rightarrow \infty }E\left[ (\varvec{X}_n-\varvec{\xi })\varvec{Y}'_n\mathscr {D}_n^{-1}\right] +\,(k-3)^2\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_n\varvec{Y}'_n\mathscr {D}_n^{-2}\right] \nonumber \\&=V(\varvec{X})-2(k-3)\underbrace{E\left[ (\varvec{X}-\varvec{\xi })\varvec{Y}'\chi ^{-2}_{k-1}(\varDelta )\right] }_{T_5}+(k-3)^2\underbrace{E\left[ \varvec{Y}\varvec{Y}'\chi ^{-4}_{k-1}(\varDelta )\right] }_{T_4}.\nonumber \\ \end{aligned}$$

(57)

Using (37), $T_4$ becomes

$$\begin{aligned} T_4=\varvec{B}_0'E[\chi ^{-4}_{k+1}(\varDelta )]+\varvec{\beta }\varvec{\beta }'E[\chi ^{-4}_{k+3}(\varDelta )]. \end{aligned}$$

(58)

By virtue of conditional expectation argument and and result (2.2.13e) mentioned in Saleh [15], we have

$$\begin{aligned} T_5=\varvec{B}_0E[\chi ^{-2}_{k+1}(\varDelta )]-2\varvec{\beta }\varvec{\beta }'E[\chi ^{-4}_{k+3}(\varDelta )]. \end{aligned}$$

(59)

Substituting (58), (59) in (57) and collecting alike terms gives

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{JS})&=\varvec{V}-(k-3)\varvec{B}_0\Bigl [2E[\chi ^{-2}_{k+1}(\varDelta )]-(k-3)E[\chi ^{-4}_{k+1}(\varDelta )]\Bigr ]\nonumber \\&+\>(k-3)(k+1)\varvec{\beta }\varvec{\beta }'E[\chi ^{-4}_{k+3}(\varDelta )]. \end{aligned}$$

At the end, we consider

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{S+})&=\lim _{n\rightarrow \infty }E\left[ \sqrt{n}(\hat{\varvec{\gamma }}^{S+}-\varvec{\gamma }_{n})\sqrt{n}(\hat{\varvec{\gamma }}^{S+}-\varvec{\gamma }_{n})'\right] \nonumber \\&=\lim _{n\rightarrow \infty }E\left[ \begin{aligned} \left\{ \sqrt{n}\left( \hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n}\right) -\{1-(k-3)\mathscr {D}^{-1}_n\}\varvec{Y}_{n}I\left( \mathscr {D}_n<(k-3)\right) \right\} \\ \left\{ \sqrt{n}\left( \hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n}\right) -\{1-(k-3)\mathscr {D}^{-1}_n\}\varvec{Y}_{n}I\left( \mathscr {D}_n<(k-3)\right) \right\} ' \end{aligned} \right] \nonumber \\&=\varvec{\varSigma }(\hat{\varvec{\gamma }}^{JS})+\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_{n}\varvec{Y}'_{n}\{1-(k-3)\mathscr {D}^{-1}_n\}^{2}I\left( \mathscr {D}_n<(k-3)\right) \right] \nonumber \\&\>\>\>\>\>-2\lim _{n\rightarrow \infty }E\left[ \sqrt{n}\left( \hat{\varvec{\gamma }}^{JS}-\varvec{\gamma }_{n}\right) \varvec{Y}'_{n}\{1-(k-3)\mathscr {D}^{-1}_n\}I\left( \mathscr {D}_n<(k-3)\right) \right] \nonumber \\&=\varvec{\varSigma }(\hat{\varvec{\gamma }}^{JS})-\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_{n}\varvec{Y}'_{n}\{1-(k-3)\mathscr {D}^{-1}_n\}^{2}I\left( \mathscr {D}_n<(k-3)\right) \right] \nonumber \\&\>\>\>\>\>+2\lim _{n\rightarrow \infty }E\left[ \varvec{Y}_n\varvec{Y}'_n\{1-(k-3)\mathscr {D}^{-1}_n\}I\left( \mathscr {D}_n<(k-3)\right) \right] \nonumber \\&\>\>\>\>\>-2\lim _{n\rightarrow \infty }E\left[ \left( \varvec{X}_n-\varvec{\xi }\right) \varvec{Y}'_{n}\{1-(k-3)\mathscr {D}^{-1}_n\}I\left( \mathscr {D}_n<(k-3)\right) \right] \nonumber \\&=\varvec{\varSigma }(\hat{\varvec{\gamma }}^{JS})-\underbrace{E\left[ \varvec{Y}\varvec{Y}'\{1-(k-3)\chi ^{-2}_{k-1}(\varDelta )\}^{2}I\left( \chi ^{2}_{k-1}(\varDelta )<(k-3)\right) \right] }_{T_{8}}\nonumber \\&\>\>\>\>\>+2\underbrace{E\left[ \varvec{Y}\varvec{Y}'\{1-(k-3)\chi ^{-2}_{k-1}(\varDelta )\}I\left( \chi ^{2}_{k-1}(\varDelta )<(k-3)\right) \right] }_{T_{7}}\nonumber \\&\>\>\>\>\>-2\underbrace{E\left[ \left( \varvec{X}-\varvec{\xi }\right) \varvec{Y}'\{1-(k-3)\chi ^{-2}_{k-1}(\varDelta )\}I\left( \chi ^{2}_{k-1}(\varDelta )<(k-3)\right) \right] }_{T_{6}}. \end{aligned}$$

(60)

Applying (37) to $T_7$ and $T_8$, conditional expectation argument to $T_6$ and collecting alike terms, reduces (60) to

$$\begin{aligned} \varvec{\varSigma }(\hat{\varvec{\gamma }}^{S+})&=\varvec{\varSigma }(\hat{\varvec{\gamma }}^{JS})-\varvec{B}_0E\left[ \{1-(k-3)\chi ^{-2}_{k+1}(\varDelta )\}^{2}I(\chi ^{-2}_{k+1}(\varDelta )<(k-3))\right] \nonumber \\&+\>\varvec{\beta }\varvec{\beta }'\left[ \begin{aligned} 2E[\{1-(k-3)\chi ^{-2}_{k+1}(\varDelta )\}I(\chi ^{-2}_{k+1}(\varDelta )<(k-3))]\\ -E[\{1-(k-3)\chi ^{-2}_{k+3}(\varDelta )\}^{2}I(\chi ^{-2}_{k+3}(\varDelta )<(k-3))] \end{aligned}\right] . \end{aligned}$$

A similar treatment mentioned earlier for obtaining $R(\hat{\varvec{\gamma }}^{LS};\varvec{Q})$ applied to these AMSEM completes the proof.