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Robust estimators for one-shot device testing data under gamma lifetime model with an application to a tumor toxicological data

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Abstract

Due to its flexibility, gamma distribution is commonly used for lifetime data analysis in reliability and survival studies, and especially in one-shot device testing data. In the study of such data, inducing more failures by accelerated life tests is a common practice, to obtain more lifetime information within a relatively short period of time. In this paper, we develop weighted minimum density power divergence estimators, as a natural extension of the classical maximum likelihood estimator, in the analysis of one-shot device testing data, under accelerated life tests based on gamma lifetime distribution. Wald-type test statistics, based on these estimators, are also developed. Through a Monte Carlo simulation study, the suggested estimators and tests are shown to be robust alternatives to the maximum likelihood estimators and the classical Wald tests based on them. Finally, these procedures are applied to a mice tumor toxicological data for illustrative purpose.

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Acknowledgements

We would like to thank the editor, the associate editor, and the referees for their helpful comments and suggestions. This research is partially supported by Grant MTM2015-67057-P and Grant FPU16/03104 from Ministerio de Economia y Competitividad and Ministerio de Educacion, Cultura y Deporte (Spain).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    N. Balakrishnan

  2. Department of Statistics and O.R. I, Complutense University of Madrid, Madrid, Spain

    E. Castilla & L. Pardo

  3. Department of Financial and Actuarial Economics and Statistics, Complutense University of Madrid, Madrid, Spain

    N. Martín

Authors
  1. N. Balakrishnan
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  2. E. Castilla
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  3. N. Martín
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  4. L. Pardo
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Correspondence to E. Castilla.

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Proofs of the main results

Proofs of the main results

1.1 Proof of Theorem 1

From (8), we have

$$\begin{aligned} d_{KL}(\widehat{\varvec{p}},\varvec{p}(\varvec{\theta }))&=\sum _{i=1}^{I}\sum _{j=1}^{2}\frac{n_{ij}}{K}\log \left( \dfrac{\frac{n_{ij} }{K_{i}}}{F_{j}(IT_{i};\,\varvec{x}_i,\varvec{\theta })}\right) \\&=\sum _{i=1}^{I}\sum _{j=1}^{2}\frac{n_{ij}}{K}\log \left( \frac{n_{i}}{K_{i} }\right) -\sum _{i=1}^{I}\sum _{j=1}^{2}\frac{n_{ij}}{K}\log \left( F_{j}(IT_{i};\,\varvec{x}_i,\varvec{\theta })\right) \\&=c-\frac{1}{K}\sum _{i=1}^{I}\left\{ n_{i}\log \left( F(IT_{i};\,\varvec{x}_i,\varvec{\theta })\right) +(K_{i}-n_{i})\log \left( R(IT_{i};\,\varvec{x}_i,\varvec{\theta })\right) \right\} \\&=c-\frac{1}{K}\log \left( \prod _{i=1}^{I}F(IT_{i};\,\varvec{x}_i,\varvec{\theta })^{n_{i}}R^{K_{i}-n_{i}}(IT_{i};\,\varvec{x}_i,\varvec{\theta })\right) \\&=c-\frac{1}{K}\log \mathcal {L}(\varvec{\theta } ) , \end{aligned}$$

where \(c=\sum _{i=1}^{I}\sum _{j=1}^{2}\frac{n_{ij}}{K}\log \left( \frac{n_{i} }{K_{i}}\right) \) does not depend on the vector of parameters \(\varvec{a} \).

1.2 Proof of Theorem 6

Since

$$\begin{aligned} \frac{\partial }{\partial \varvec{\theta }}\sum _{i=1}^{I}\frac{K_{i}}{K}d_{\beta }^{*}(\widehat{\varvec{p}}_{i},\varvec{\pi }_{i}(\varvec{\theta }))&=\sum _{i=1}^{I}\frac{K_{i}}{K}\frac{\partial }{\partial \varvec{\theta } }d_{\beta }^{*}(\widehat{\varvec{p}}_{i},\varvec{\pi }_{i} (\varvec{\theta }))\\ \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial }{\partial \varvec{\theta }}d_{\beta }^{*} (\widehat{\varvec{p}}_{i},\varvec{\pi }_{i}(\varvec{\theta }))\\&\quad =\left( \frac{\partial }{\partial \varvec{\theta }}\pi _{i1}^{\beta +1} (\varvec{\theta })+\frac{\partial }{\partial \varvec{\theta }}\pi _{i2} ^{\beta +1}(\varvec{\theta })\right) -\frac{\beta +1}{\beta }\left( \widehat{p}_{i1}\frac{\partial }{\partial \varvec{\theta }}\pi _{i1}^{\beta }(\varvec{\theta })+\widehat{p}_{i2}\frac{\partial }{\partial \varvec{\theta } }\pi _{i2}^{\beta }(\varvec{\theta })\right) \\&\quad =\left( \beta +1\right) \left( \pi _{i1}^{\beta }(\varvec{\theta } )-\pi _{i2}^{\beta }(\varvec{\theta })-\widehat{p}_{i1}\pi _{i1}^{\beta -1}(\varvec{\theta })+\widehat{p}_{i2}\pi _{i2}^{\beta -1}(\varvec{\theta })\right) \frac{\partial }{\partial \varvec{\theta }}\pi _{i1}(\varvec{\theta })\\&\quad =\left( \beta +1\right) \left( \left( \pi _{i1}(\varvec{\theta })-\widehat{p}_{i1}\right) \pi _{i1}^{\beta -1}(\varvec{\theta })-\left( \pi _{i2}(\varvec{\theta })-\widehat{p}_{i2}\right) \pi _{i2}^{\beta -1}(\varvec{\theta })\right) \frac{\partial }{\partial \varvec{\theta }} \pi _{i1}(\varvec{\theta })\\&\quad =\left( \beta +1\right) \left( \left( \pi _{i1}(\varvec{\theta })-\widehat{p}_{i1}\right) \pi _{i1}^{\beta -1}(\varvec{\theta })+\left( \pi _{i1}(\varvec{\theta })-\widehat{p}_{i1}\right) \pi _{i2}^{\beta -1}(\varvec{\theta })\right) \frac{\partial }{\partial \varvec{\theta }} \pi _{i1}(\varvec{\theta })\\&\quad =\left( \beta +1\right) \left( \pi _{i1}(\varvec{\theta } )-\widehat{p}_{i1}\right) \left( \pi _{i1}^{\beta -1}(\varvec{\theta } )+\pi _{i2}^{\beta -1}(\varvec{\theta })\right) \frac{\partial }{\partial \varvec{\theta }}\pi _{i1}(\varvec{\theta })\\&\quad =\left( \beta +1\right) \left( \pi _{i1}(\varvec{\theta } )-\widehat{p}_{i1}\right) \left( \pi _{i1}^{\beta -1}(\varvec{\theta } )+\pi _{i2}^{\beta -1}(\varvec{\theta })\right) f\left( IT_{i};\,\varvec{x}_i,\varvec{\theta }\right) \frac{\partial f\left( IT_{i};\,\varvec{x}_i,\varvec{\theta }\right) }{\partial \varvec{\theta }}, \end{aligned}$$

we obtain

$$\begin{aligned} \frac{\partial }{\partial \varvec{\theta }}\sum _{i=1}^{I}\frac{K_{i}}{K}d_{\beta }^{*}(\widehat{\varvec{p}}_{i},\varvec{\pi }_{i}(\varvec{\theta }))& = \frac{\beta +1}{K}\sum _{i=1}^{I}\left( K_{i}\pi _{i1}(\varvec{\theta })-n_{i}\right) \left( \pi _{i1}^{\beta -1}(\varvec{\theta })+\pi _{i2}^{\beta -1}(\varvec{\theta })\right) \\&\quad f\left( IT_{i};\,\varvec{x}_i,\varvec{\theta }\right) \frac{\partial f\left( IT_{i};\,\varvec{x}_i,\varvec{\theta }\right) }{\partial \varvec{\theta }}. \end{aligned}$$

In a similar way, we can get the derivative with respect to \(\varvec{b}\), and then, the required results follow.

1.3 Proof of Theorem 7

Let us denote

$$\begin{aligned} \varvec{u}_{ij}(\varvec{\theta })&=\left( \frac{\partial \log \pi _{ij}(\varvec{\theta })}{\partial \varvec{a}},\frac{\partial \log \pi _{ij}(\varvec{\theta })}{\partial \varvec{b}}\right) ^{T}= \left( \frac{1}{\pi _{ij}(\varvec{\theta })}\frac{\partial \pi _{ij}(\varvec{\theta })}{\partial \varvec{a}},\frac{1}{\pi _{ij}(\varvec{\theta })}\frac{\partial \pi _{ij}(\varvec{\theta } )}{\partial \varvec{b}}\right) ^{T}\\&=\left( \frac{(-1)^{j+1}}{\pi _{ij}(\varvec{\theta })}l_{i}\varvec{x} _{i},\frac{(-1)^{j+1}}{\pi _{ij}(\varvec{\theta })}s_{i}\varvec{x} _{i}\right) ^{T}, \end{aligned}$$

with

$$\begin{aligned} l_{i}& = \,\alpha _{i} \left\{ - \Psi \left( \alpha _{i}\right) \pi _{i1}(\varvec{\theta }) +\log \left( \frac{IT_i}{\lambda _{i}}\right) \pi _{i1}(\varvec{\theta })\right. \nonumber \\&\quad \left. -\,\frac{\left( \frac{IT_i}{\lambda _{i}}\right) ^{\alpha _i}}{\alpha _i^2 \Gamma (\alpha _i)} {_2}F_2\left( \alpha _i,\alpha _i;1+\alpha _i,1+\alpha _i;-\frac{IT_i}{\lambda _{i}}\right) \right\} \end{aligned}$$
(27)

and

$$\begin{aligned} s_{i}=-f\left( IT_{i};\,\varvec{x}_i,\varvec{\theta }\right) IT_{i};\, \end{aligned}$$
(28)

see Balakrishnan and Ling (2014) for more details.

Upon using Theorem 3.1 of Ghosh and Basu (2013), we have

$$\begin{aligned} \sqrt{K}\left( \widehat{\varvec{\theta }}_{\beta }-\varvec{\theta } _{0}\right) \overset{\mathcal {L}}{\underset{K\mathcal {\rightarrow } \infty }{\longrightarrow }}\mathcal {N}\left( \varvec{0}_{2J},\varvec{J} _{\beta }^{-1}(\varvec{\theta }_{0})\varvec{K}_{\beta } (\varvec{\theta }_{0})\varvec{J}_{\beta }^{-1}(\varvec{\theta } _{0})\right) , \end{aligned}$$

where

$$\begin{aligned} \varvec{J}_{\beta }(\varvec{\theta })&=\sum _{i=1}^{I} \sum _{j=1}^{2}\frac{K_i}{K}\varvec{u}_{ij}(\varvec{\theta })\varvec{u}_{ij} ^{T}(\varvec{\theta })\pi _{ij}^{\beta +1}(\varvec{\theta }),\\ \varvec{K}_{\beta }(\varvec{\theta })&=\left( \sum _{i=1}^{I}\sum _{j=1}^{2}\frac{K_i}{K}\varvec{u}_{ij}(\varvec{\theta })\varvec{u} _{ij}^{T}(\varvec{\theta })\pi _{ij}^{2\beta +1}(\varvec{\theta } )-\sum _{i=1}^{I}\frac{K_i}{K}\varvec{\xi }_{i,\beta }(\varvec{\theta })\varvec{\xi }_{i,\beta }^{T}(\varvec{\theta })\right) , \end{aligned}$$

with

$$\begin{aligned} \varvec{\xi }_{i,\beta }(\varvec{\theta })&=\sum _{j=1}^{2}\varvec{u} _{ij}(\varvec{\theta })\pi _{ij}^{\beta +1}(\varvec{\theta })\\&=\left( l_{i}\varvec{x}_{i},s_{i}\varvec{x}_{i}\right) ^{T} \sum _{j=1}^{2}(-1)^{j+1}\pi _{ij}^{\beta }(\varvec{\theta }). \end{aligned}$$

Now, for \(\varvec{u}_{ij}(\varvec{\theta })\varvec{u} _{ij}^{T}(\varvec{\theta })\), we have

$$\begin{aligned} \varvec{u}_{ij}(\varvec{\theta })\varvec{u}_{ij}^{T}(\varvec{\theta })=\frac{1}{\pi _{ij}^{2}(\varvec{\theta })}\left( \begin{array} [c]{cc} l_{i}^{2}\varvec{x}_{i}^{T}\varvec{x}_{i} &\quad l_{i}s_{i}\varvec{x} _{i}^{T}\varvec{x}_{i}\\ l_{i}s_{i}\varvec{x}_{i}^{T}\varvec{x}_{i} &\quad s_{i}^{2}\varvec{x} _{i}^{T}\varvec{x}_{i} \end{array} \right) =\frac{1}{\pi _{ij}^{2}(\varvec{\theta })}\varvec{M}_{i}, \end{aligned}$$

with

$$\begin{aligned} \varvec{M}_{i}=\left( \begin{array} [c]{cc} l_{i}^{2}\varvec{x}_{i}^{T}\varvec{x}_{i} &\quad l_{i}s_{i}\varvec{x} _{i}^{T}\varvec{x}_{i}\\ l_{i}s_{i}\varvec{x}_{i}^{T}\varvec{x}_{i} &\quad s_{i}^{2}\varvec{x} _{i}^{T}\varvec{x}_{i} \end{array} \right) . \end{aligned}$$
(29)

It then follows that

$$\begin{aligned} \varvec{J}_{\beta }(\varvec{\theta })&=\sum _{i=1} ^{I}\frac{K_i}{K}\varvec{M}_{i}\sum _{j=1}^{2}\pi _{ij}^{\beta -1}(\varvec{\theta })\\&=\sum _{i=1}^{I}\frac{K_i}{K}\varvec{M}_{i}\left( \pi _{i1}^{\beta -1}(\varvec{\theta })+\pi _{i2}^{\beta -1}(\varvec{\theta })\right) . \end{aligned}$$

In a similar manner,

$$\begin{aligned} \varvec{\xi }_{i,\beta }(\varvec{\theta })\varvec{\xi }_{i,\beta } ^{T}(\varvec{\theta })=\varvec{M}_{i}\left( \sum _{j=1}^{2} (-1)^{j+1}\pi _{ij}^{\beta }(\varvec{\theta })\right) ^{2} \end{aligned}$$

and

$$\begin{aligned} \varvec{K}_{\beta }(\varvec{\theta })=\sum _{i=1} ^{I}\frac{K_i}{K}\varvec{M}_{i}\left( \sum _{j=1}^{2}\pi _{ij}^{2\beta -1} (\varvec{\theta })-\left( \sum _{j=1}^{2}(-1)^{j+1}\pi _{ij}^{\beta }(\varvec{\theta })\right) ^{2}\right) . \end{aligned}$$

Since

$$\begin{aligned} \sum _{j=1}^{2}\pi _{ij}^{2\beta -1}(\varvec{\theta })-\left( \sum _{j=1} ^{2}(-1)^{j+1}\pi _{ij}^{\beta }(\varvec{\theta })\right) ^{2}=\pi _{i1}(\varvec{\theta })\pi _{i2}(\varvec{\theta })\left( \pi _{i1} ^{\beta -1}(\varvec{\theta })+\pi _{i2}^{\beta -1}(\varvec{\theta })\right) ^{2}, \end{aligned}$$

we have

$$\begin{aligned} \varvec{K}_{\beta }(\varvec{\theta })=\sum _{i=1} ^{I}\frac{K_i}{K}\varvec{M}_{i}\pi _{i1}(\varvec{\theta })\pi _{i2}(\varvec{\theta })\left( \pi _{i1}^{\beta -1}(\varvec{\theta })+\pi _{i2}^{\beta -1}(\varvec{\theta })\right) ^{2}. \end{aligned}$$

1.4 Proof of Theorem 9

Let \(\varvec{\theta }_{0}\in \Theta \) be the true value of parameter \( \varvec{\theta }\). It is clear that

$$\begin{aligned} \varvec{m}\left( \widehat{\varvec{\theta }}_{\beta }\right)&= {}\, \varvec{ m}\left( \varvec{\theta }_{0}\right) +\varvec{M}^{T}\left( \widehat{ \varvec{\theta }}_{\beta }\right) \left( \widehat{\varvec{\theta }}_{\beta }- \varvec{\theta }_{0}\right) +o_{p}\left( \left\| \widehat{\varvec{\theta }} _{\beta }-\varvec{\theta }_{0}\right\| \right) \\& = \,\varvec{M}^{T}\left( \widehat{\varvec{\theta }}_{\beta }\right) \left( \widehat{\varvec{\theta }}_{\beta }-\varvec{\theta }_{0}\right) +o_{p}\left( K^{-1/2}\right) . \end{aligned}$$

But, under \(H_{0},\)

$$\begin{aligned} \sqrt{K}\left( \widehat{\varvec{\theta }}_{\beta }-\varvec{\theta }_{0}\right) \underset{K\rightarrow \infty }{\overset{\mathcal {L}}{\longrightarrow }} \mathcal {N}\left( \varvec{0}_{J+1},\varvec{\Sigma }_{\beta }\left( \widehat{\varvec{\theta }}_{\beta }\right) \right) . \end{aligned}$$

Therefore, under \(H_{0},\)

$$\begin{aligned} \sqrt{K}\varvec{m}\left( \widehat{\varvec{\theta }}_{\beta }\right) \underset{K\rightarrow \infty }{\overset{\mathcal {L}}{\longrightarrow }} \mathcal {N}\left( \varvec{0}_{r},\varvec{M}^{T}\left( \varvec{\theta } _{0}\right) \varvec{\Sigma }_{\beta }\left( \widehat{\varvec{\theta }} _{\beta }\right) \varvec{M}\left( \varvec{\theta }_{0}\right) \right) \end{aligned}$$

and taking into account that \(rank(\varvec{M}\left( \varvec{\theta } _{0}\right) )=r\), we get

$$\begin{aligned} K\varvec{m}\left( \widehat{\varvec{\theta }}_{\beta }\right) ^{T}\left( \varvec{M}^{T}\left( \varvec{\theta }_{0}\right) \varvec{\Sigma } _{\beta }\left( \varvec{\theta }_{0}\right) \varvec{M}\left( \varvec{\theta } _{0}\right) \right) ^{-1}\varvec{m}\left( \widehat{\varvec{\theta }} _{\beta }\right) \underset{K\rightarrow \infty }{\overset{\mathcal {L}}{ \longrightarrow }}\chi _{r}^{2}. \end{aligned}$$

But, \(\left( \varvec{M}\left( \widehat{\varvec{\theta }}_{\beta }\right) ^{T}\varvec{\Sigma }_{\beta }\left( \widehat{\varvec{\theta }}_{\beta }\right) \varvec{M}\left( \widehat{\varvec{\theta }}_{\beta }\right) \right) ^{-1}\) is a consistent estimator of \(\left( \varvec{M}\left( \varvec{\theta }_{0}\right) ^{T}\varvec{\Sigma }_{\beta }\left( \varvec{\theta }_{0}\right) \varvec{M}\left( \varvec{\theta }_{0}\right) \right) ^{-1}\). Therefore,

$$\begin{aligned} W_{K}\left( \widehat{\varvec{\theta }}_{\beta }\right) \underset{K\rightarrow \infty }{\overset{\mathcal {L}}{\longrightarrow }}\chi _{r}^{2}. \end{aligned}$$

1.5 Proof of Theorem 10

Under the assumption that

$$\begin{aligned} \widehat{\varvec{\theta }}_{\beta }\underset{K\rightarrow \infty }{\overset{P}{ \longrightarrow }}\varvec{\theta }^{*}, \end{aligned}$$

the asymptotic distribution of \(\ell _{\beta }\left( \widehat{\varvec{\theta }} _{1},\widehat{\varvec{\theta }}_{2}\right) \) coincides with the asymptotic distribution of \(\ell _{\beta }\left( \widehat{\varvec{\theta }}_{1}, \varvec{\theta }^{*}\right) .\) A first-order Taylor expansion of \(\ell _{\beta }\left( \widehat{\varvec{\theta }}_{\beta },\varvec{\theta }\right) \) at \(\widehat{\varvec{\theta }}_{\beta }\), around \(\varvec{\theta }^{*}\), gives

$$\begin{aligned} \left( \ell _{\beta }\left( \widehat{\varvec{\theta }}_{\beta },\varvec{\theta } ^{*}\right) -\ell _{\beta }\left( \varvec{\theta }^{*},\varvec{\theta } ^{*}\right) \right) =\left. \frac{\partial \ell _{\beta }\left( \varvec{\theta },\varvec{\theta }_{*}\right) }{\partial \varvec{\theta }^{T}} \right| _{\varvec{\theta }=\varvec{\theta }_{*}}\left( \widehat{ \varvec{\theta }}_{\beta }-\varvec{\theta }^{*}\right) +o_{p}(K^{-1/2}). \end{aligned}$$

Now, the result follows since

$$\begin{aligned} \sqrt{K}\left( \widehat{\varvec{\theta }}_{\beta }-\varvec{\theta }^{*}\right) \underset{K\rightarrow \infty }{\overset{\mathcal {L}}{ \longrightarrow }}\mathcal {N}\left( \varvec{0}_{J+1},\varvec{\Sigma } _{\beta }\left( \varvec{\theta }^{*}\right) \right) . \end{aligned}$$

1.6 Proof of Theorem 12

A Taylor series expansion of \(\varvec{m}(\widehat{\varvec{\theta }}_{\beta })\) around \(\varvec{\theta }_{n}\) yields

$$\begin{aligned} \varvec{m}(\widehat{\varvec{\theta }}_{\beta })=\varvec{m}(\varvec{ \theta }_{n})+\varvec{M}^{T}(\varvec{\theta }_{n})(\widehat{\varvec{\theta }} _{\beta }-\varvec{\theta }_{n})+o\left( \left\| \widehat{\varvec{\theta }} _{\beta }-\varvec{\theta }_{n}\right\| \right) . \end{aligned}$$

From (23), we have

$$\begin{aligned} \varvec{m}(\widehat{\varvec{\theta }}_{\beta })=K^{-1/2}\varvec{M}^{T}( \varvec{\theta }_{0})\varvec{d}+\varvec{M}^{T}(\varvec{\theta }_{n})( \widehat{\varvec{\theta }}_{\beta }-\varvec{\theta }_{n})+o\left( \left\| \widehat{\varvec{\theta }}_{\beta }-\varvec{\theta }_{n}\right\| \right) +o\left( \left\| \varvec{\theta }_{n}-\varvec{\theta }_{0}\right\| \right) . \end{aligned}$$

As

$$\begin{aligned} \sqrt{K}(\widehat{\varvec{\theta }}_{\beta }-\varvec{\theta }_{n})\underset{ n\rightarrow \infty }{\overset{\mathcal {L}}{\longrightarrow }}\mathcal {N}( \varvec{0}_{J+1},\varvec{\Sigma }_{\beta }(\varvec{\theta }_{0})) \end{aligned}$$

and \(\sqrt{K}\left( o\left( \left\| \widehat{\varvec{\theta }}_{\beta }- \varvec{\theta }_{n}\right\| \right) +o\left( \left\| \varvec{\theta }_{n}- \varvec{\theta }_{0}\right\| \right) \right) =o_{p}\left( 1\right) \), we have

$$\begin{aligned} \sqrt{n}\varvec{m}(\widehat{\varvec{\theta }}_{\beta })\underset{ n\rightarrow \infty }{\overset{\mathcal {L}}{\longrightarrow }}\mathcal {N}( \varvec{M}^{T}(\varvec{\theta }_{0})\,\varvec{d},\varvec{M}^{T}( \varvec{\theta }_{0})\varvec{\Sigma }_{\beta }(\varvec{\theta }_{0}) \varvec{M}(\varvec{\theta }_{0})). \end{aligned}$$

We can observe from the relationship \(\varvec{d}^{*}=\varvec{M}( \varvec{\theta }_{0})^{T}\varvec{d}\), if \(\varvec{m}(\varvec{\theta } _{n})=n^{-1/2}\varvec{d}^{*}\), that

$$\begin{aligned} \sqrt{n}\varvec{m}(\widehat{\varvec{\theta }}_{\beta })\underset{ n\rightarrow \infty }{\overset{\mathcal {L}}{\longrightarrow }}\mathcal {N}( \varvec{d}^{*},\varvec{M}^{T}(\varvec{\theta }_{0})\varvec{ \Sigma }_{\beta }(\varvec{\theta }_{0})\varvec{M}(\varvec{\theta }_{0})). \end{aligned}$$

So, the quadratic form here is

$$\begin{aligned} W_{K}\left( \widehat{\varvec{\theta }}_{\beta }\right) =\varvec{Z}^{T} \varvec{Z} \end{aligned}$$

with

$$\begin{aligned} \varvec{Z}=\sqrt{n}\varvec{m}(\widehat{\varvec{\theta }}_{\beta })\left( \varvec{M}^{T}(\varvec{\theta }_{0})\varvec{\Sigma }_{\beta }( \varvec{\theta }_{0})\varvec{M}(\varvec{\theta }_{0})\right) ^{-1/2} \end{aligned}$$

and

$$\begin{aligned} \varvec{Z}\underset{n\rightarrow \infty }{\overset{\mathcal {L}}{ \longrightarrow }}\mathcal {N}\left( \left( \varvec{M}^{T}(\varvec{\theta } _{0})\varvec{\Sigma }_{\beta }(\varvec{\theta }_{0})\varvec{M}( \varvec{\theta }_{0})\right) ^{-1/2}\varvec{M}(\varvec{\theta }_{0})^{T} \varvec{d},\varvec{I}_{r}\right) , \end{aligned}$$

where \(\varvec{I}_{r}\) is the identity matrix or order r. Hence, the required the result follows immediately, and the non-centrality parameter is

$$\begin{aligned} \varvec{d}^{T}\varvec{M}(\varvec{\theta }_{0})\left( \varvec{M} ^{T}(\varvec{\theta }_{0})\varvec{\Sigma }_{\beta }(\varvec{\theta }_{0}) \varvec{M}(\varvec{\theta }_{0})\right) ^{-1}\varvec{M}(\varvec{\theta } _{0})^{T}\varvec{d}=\varvec{d}^{*T}\left( \varvec{M}^{T}( \varvec{\theta }_{0})\varvec{\Sigma }_{\beta }(\varvec{\theta }_{0}) \varvec{M}(\varvec{\theta }_{0})\right) ^{-1}\varvec{d}^{*}. \end{aligned}$$

1.7 Proof of Theorem 13

The influence function of \(W_K(\widehat{\varvec{\theta }}_{\beta })\), with respect to the \(j_0\)th observation of the \(i_0\)th group of observations, is defined as

$$\begin{aligned}\mathcal {IF}\left( i_{0},j_{0},x,T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }}\right) =\left. \frac{\partial W_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) }{\partial \varepsilon }\right| _{\varepsilon =0^{+}}\end{aligned}$$

where

$$\begin{aligned}&\frac{\partial W_{K}\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) }{\partial \varepsilon }\\&\quad =2\varvec{m}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \left( \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \varvec{\Sigma }\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \varvec{M}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \right) ^{-1}\\&\qquad \times \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \frac{\partial \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) }{\partial \varepsilon }\\&\qquad +\varvec{m}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \frac{\partial }{\partial \varepsilon }\left( \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \varvec{\Sigma }\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \varvec{M}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \right) ^{-1} \\&\qquad \times \varvec{m}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \end{aligned}$$

and

$$\begin{aligned}&\left. \frac{\partial W_{K}\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) }{\partial \varepsilon }\right| _{\varepsilon =0^{+}}\\&\quad =2\varvec{m}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \left( \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \varvec{\Sigma }\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \varvec{M}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \right) ^{-1}\\&\qquad \times \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \mathcal {IF}\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \\&\qquad +\varvec{m}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \left. \frac{\partial }{\partial \varepsilon }\left( \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \varvec{\Sigma }\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \varvec{M}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \right) ^{-1}\right| _{\varepsilon =0^{+}} \\&\qquad \times \varvec{m}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x}\right) \right) \end{aligned}$$

For \(\varvec{\theta }=\varvec{\theta }_0\), \(\varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta }_0,x}\right) =\varvec{\theta }_0\) and \(\varvec{m}^T(\varvec{\theta }_0)=0\). Therefore,

$$\begin{aligned} \mathcal {IF}\left( i_{0},j_{0},x,T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right) =0, \end{aligned}$$

in a similar way, we have

$$\begin{aligned} \mathcal {IF}\left( i_{0},j_{0},\underline{x},T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right) =0 \end{aligned}$$

In order to get the second-order influence function, with respect to the jth observation of the ith group of observation, we must get

$$\begin{aligned} \mathcal {IF}_2\left( i_{0},j_{0},x,T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right) =\left. \frac{\partial ^2}{\partial \varepsilon } W_K(\varvec{F}_{\underline{K}},\varvec{\theta }_0,x,\varepsilon )\right| _{\varepsilon =0^{+}} \end{aligned}$$

We can express

$$\begin{aligned} \frac{\partial ^2 W_K(\varvec{F}_{\underline{K}},\varvec{\theta },x,\varepsilon )}{\partial \varepsilon }&=\varvec{l}(K,\varvec{\theta },x,\varepsilon ) +2\frac{\partial \varvec{U}_\beta (\varvec{F}_{\underline{K}},\varvec{\theta },x,\varepsilon )}{\partial \varepsilon }\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \\&\quad \times \left( \varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \varvec{\Sigma }\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \right. \\&\qquad \left. \varvec{M}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \right) ^{-1}\varvec{M}^T\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \\&\quad \times \frac{\partial \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) }{\partial \varepsilon } \end{aligned}$$

With \(\varvec{l}(K,\varvec{\theta },x,\varepsilon )\), we denote all the terms which contain the expression \(\varvec{m}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta },x,\varepsilon }\right) \right) \), because for \(\varvec{\theta }=\varvec{\theta }_0\) and \(\varepsilon =0\), we have \(\varvec{m}\left( \varvec{U}_{\beta }\left( \varvec{F}_{\underline{K},\varvec{\theta }_0,x}\right) \right) =0\). Therefore, we have

$$\begin{aligned} \mathcal {IF}_2\left( i_{0},j_{0},x,T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right)&=\left. \frac{\partial ^2}{\partial \varepsilon } W_K(\varvec{F}_{\underline{K}},\varvec{\theta }_0,x,\varepsilon )\right| _{\varepsilon =0^{+}}\\&=2\mathcal {IF}^T\left( i_{0},j_{0},x,T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right) \varvec{M}(\varvec{\theta }_0)\left( \varvec{M}^T(\varvec{\theta }_0)\varvec{\Sigma }(\varvec{\theta }_0)\varvec{M}(\varvec{\theta }_0) \right) ^{-1}\\&\quad \times \varvec{M}^T(\varvec{\theta }_0)\mathcal {IF}\left( i_{0},j_{0},x,T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right) . \end{aligned}$$

In a similar way, we obtain the expression of \(\mathcal {IF}_2\left( \underline{x},T_{\beta },\varvec{F}_{\underline{K},\varvec{\theta }_0}\right) \).

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Balakrishnan, N., Castilla, E., Martín, N. et al. Robust estimators for one-shot device testing data under gamma lifetime model with an application to a tumor toxicological data. Metrika 82, 991–1019 (2019). https://doi.org/10.1007/s00184-019-00718-5

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