Abstract
Asymptotically correct simultaneous confidence bands (SCBs) are proposed in both multiplicative and additive form to compare variance functions of two samples in the nonparametric regression model based on deterministic designs. The multiplicative SCB is based on two-step estimation of ratio of the variance functions, which is as efficient, up to order \(n^{-1/2}\), as an infeasible estimator if the two mean functions are known a priori. The additive SCB, which is the log transform of the multiplicative SCB, is location and scale invariant in the sense that the width of SCB is free of the unknown mean and variance functions of both samples. Simulation experiments provide strong evidence that corroborates the asymptotic theory. The proposed SCBs are used to analyze several strata pressure data sets from the Bullianta Coal Mine in Erdos City, Inner Mongolia, China.
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Acknowledgements
This research is part of the first author’s dissertation under the supervision of the second author, and has been supported exclusively by National Natural Science Foundation of China award 11771240. The authors are grateful to Professor Yaodong Jiang for providing the strata pressure data, and two Reviewers for thoughtful comments.
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Appendix
Appendix
The following is a reformulation of Theorems 11.1.5 and 12.3.5 in Leadbetter et al. (1983).
Lemma 1
If a Gaussian process \(\varsigma \left( s\right) ,0\le s\le T\) is stationary with mean zero and variance one, and covariance function statisfying
for some constant \(C>0,0<\alpha \le 2.\) Then as \(T\rightarrow \infty \),
where \(a_{T}=\left( 2\log T\right) ^{1/2}\) and
with \(H_{1}=1,H_{2}=\pi ^{-1/2}\).
Lemmas 2–4 are from Cai et al. (2019).
Lemma 2
Under Assumption (A6), for \(s=1,2\), as \(n\rightarrow \infty \),
Lemma 3
Under Assumptions (A2), (A6) and (A7), for \(s=1,2\), as \( n\rightarrow \infty \),
Lemma 4
Under Assumptions (A2)–(A4), (A6), (A7), for \(s=1,2\), as \(n\rightarrow \infty ,\)
Denote
Lemma 5
Under Assumptions (A2)–(A4), (A6), (A7), as \( n\rightarrow \infty ,\)
Consequently,
where \(a_{h}\) and \(v_{n}\) are given in (5).
Proof According to Lemmas 2–4, one has
Now applying Taylor series expansions to \(\ln {\tilde{\sigma }}_{s}^{2}\left( x\right) -\ln \sigma _{s}^{2}\left( x\right) \), for \(s=1,2\)
Then one obtains
Since one has
and according to Lemmas 2–4, one has
Hence combining (14), (15) and (16), the proof is completed.
Denote the following processes
As \(\text{ E }\left\{ B_{n_{1},n_{2,}3}^{2}\left( x\right) \right\} =h^{-1}\left\{ n_{1}^{-1}\left( \mu _{1,4}-1\right) +n_{2}^{-1}\left( \mu _{2,4}-1\right) \right\} \int _{-1}^{1}K^{2}\left( u\right) du\), one obtains the following standard Gaussian processes,
where \(\nu _{1,4}=\mu _{1,4}-1\) and \(\nu _{2,4}=\mu _{2,4}-1\).
Another standard Gaussian process is
which is \(\zeta \left( x\right) \) defined in (10).
Lemma 6
The absolute maximum of the process \(\bigtriangleup _{1}\left( x\right) \) follows the same as that of \(\bigtriangleup _{2}\left( x\right) \), and the absolute maximum of the process \(\bigtriangleup _{2}\left( x\right) \) follows the same as that of \(\zeta \left( x\right) \), that is
Proof This lemma can be easily obtained by noting the fact that for \(s=1,2\), the process \(B_{n_{1},n_{2,}3}\left( x\right) ,x\in \left[ h,1-h\right] \) has the same probability law as \(Y_{1,n_{1},1}\left( x\right) -Y_{2,n_{2,}1}\left( x\right) ,x\in \left[ 1,h^{-1}-1\right] \), and the process \(Y_{s,n_{s},1}\left( x\right) ,x\in \left[ h,1-h\right] \) has the same probability law as \(Y_{s,n_{s},2}\left( x\right) ,x\in \left[ 1,h^{-1}-1\right] \).
Proof of Proposition 1
Proposition 1 is a direct corollary of Lemma 5, Lemma 6 and Proposition 3.
Proof of Proposition 2
According to Theorem 2 in Cai et al. (2019) and applying Taylor expansion, one has
which completes the proof.
Proof of Proposition 3
For Gaussian process \(\zeta \left( x\right) \), its correlation function is
which implies that
Define next a Gaussian process \(\varsigma \left( t\right) ,0\le t\le T=T_{n}=h^{-1}-2\),
which is stationary with mean zero and variance one, and covariance function
with \(C=\int _{-1}^{1}K^{\left( 1\right) }\left( u\right) ^{2}du/2\int _{-1}^{1}K^{2}\left( u\right) du\). Hence applying Lemmas 1–6, one has for \( h\rightarrow 0\) or \(T\rightarrow \infty ,\)
where \(a_{T}=\left( 2\log T\right) ^{1/2}\) and \(b_{T}=a_{T}+a_{T}^{-1}\left\{ 2^{-1}\mathrm {log}\left( C_{K}/\left( 4\pi ^{2}\right) \right) \right\} \) . Note that
Hence, applying Slutsky’s Theorem twice, one obtains that
which converges in distribution to the same limit as \(a_{T}\left\{ \sup _{t\in \left[ 0,T\right] }\left| \varsigma \left( t\right) \right| -b_{T}\right\} \). Thus
Hence the proof is completed.
Proof of Theorem 1
According to Proposition 1, as \(n\rightarrow \infty ,\)
where \(a_{h},b_{h}\) and \(v_{n}\) are given in (5). Finally applying Proposition 2, one obtains
Using Slutsky’s Theorem one can substitute \(\ln \frac{{\hat{\sigma }} _{1}^{2}\left( x\right) }{{\hat{\sigma }}_{2}^{2}\left( x\right) }\) for \(\ln \frac{{\tilde{\sigma }}_{1}^{2}\left( x\right) }{{\tilde{\sigma }}_{2}^{2}\left( x\right) }\) in (19). Hence the proof of Theorem 1 is completed.
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Zhong, C., Yang, L. Simultaneous confidence bands for comparing variance functions of two samples based on deterministic designs. Comput Stat 36, 1197–1218 (2021). https://doi.org/10.1007/s00180-020-01043-6
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DOI: https://doi.org/10.1007/s00180-020-01043-6