Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes
Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Martsynyuk, Yu. V. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
nonparametric test for change in the mean “no-change in the mean” null hypothesis “at most one change in the mean” alternative hypothesis |
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| Übergeordnetes Werk: |
Enthalten in: Mathematical methods of statistics - New York, NY : Allerton Press, 2007, 25(2016), 3 vom: Juli, Seite 219-232 |
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| Übergeordnetes Werk: |
volume:25 ; year:2016 ; number:3 ; month:07 ; pages:219-232 |
| Links: |
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| DOI / URN: |
10.3103/S1066530716030054 |
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| Katalog-ID: |
SPR02405853X |
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| 245 | 1 | 0 | |a Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes |
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| 520 | |a Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. | ||
| 650 | 4 | |a nonparametric test for change in the mean |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a “no-change in the mean” null hypothesis |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a “at most one change in the mean” alternative hypothesis |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a sup-functional of weighted tied-down partial sums processes |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a standard Wiener process |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Brownian bridge |7 (dpeaa)DE-He213 | |
| 773 | 0 | 8 | |i Enthalten in |t Mathematical methods of statistics |d New York, NY : Allerton Press, 2007 |g 25(2016), 3 vom: Juli, Seite 219-232 |w (DE-627)535184913 |w (DE-600)2375497-7 |x 1934-8045 |7 nnns |
| 773 | 1 | 8 | |g volume:25 |g year:2016 |g number:3 |g month:07 |g pages:219-232 |
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10.3103/S1066530716030054 doi (DE-627)SPR02405853X (SPR)S1066530716030054-e DE-627 ger DE-627 rakwb eng 510 ASE Martsynyuk, Yu. V. verfasserin aut Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. nonparametric test for change in the mean (dpeaa)DE-He213 “no-change in the mean” null hypothesis (dpeaa)DE-He213 “at most one change in the mean” alternative hypothesis (dpeaa)DE-He213 sup-functional of weighted tied-down partial sums processes (dpeaa)DE-He213 standard Wiener process (dpeaa)DE-He213 Brownian bridge (dpeaa)DE-He213 Enthalten in Mathematical methods of statistics New York, NY : Allerton Press, 2007 25(2016), 3 vom: Juli, Seite 219-232 (DE-627)535184913 (DE-600)2375497-7 1934-8045 nnns volume:25 year:2016 number:3 month:07 pages:219-232 https://dx.doi.org/10.3103/S1066530716030054 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2016 3 07 219-232 |
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10.3103/S1066530716030054 doi (DE-627)SPR02405853X (SPR)S1066530716030054-e DE-627 ger DE-627 rakwb eng 510 ASE Martsynyuk, Yu. V. verfasserin aut Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. nonparametric test for change in the mean (dpeaa)DE-He213 “no-change in the mean” null hypothesis (dpeaa)DE-He213 “at most one change in the mean” alternative hypothesis (dpeaa)DE-He213 sup-functional of weighted tied-down partial sums processes (dpeaa)DE-He213 standard Wiener process (dpeaa)DE-He213 Brownian bridge (dpeaa)DE-He213 Enthalten in Mathematical methods of statistics New York, NY : Allerton Press, 2007 25(2016), 3 vom: Juli, Seite 219-232 (DE-627)535184913 (DE-600)2375497-7 1934-8045 nnns volume:25 year:2016 number:3 month:07 pages:219-232 https://dx.doi.org/10.3103/S1066530716030054 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2016 3 07 219-232 |
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10.3103/S1066530716030054 doi (DE-627)SPR02405853X (SPR)S1066530716030054-e DE-627 ger DE-627 rakwb eng 510 ASE Martsynyuk, Yu. V. verfasserin aut Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. nonparametric test for change in the mean (dpeaa)DE-He213 “no-change in the mean” null hypothesis (dpeaa)DE-He213 “at most one change in the mean” alternative hypothesis (dpeaa)DE-He213 sup-functional of weighted tied-down partial sums processes (dpeaa)DE-He213 standard Wiener process (dpeaa)DE-He213 Brownian bridge (dpeaa)DE-He213 Enthalten in Mathematical methods of statistics New York, NY : Allerton Press, 2007 25(2016), 3 vom: Juli, Seite 219-232 (DE-627)535184913 (DE-600)2375497-7 1934-8045 nnns volume:25 year:2016 number:3 month:07 pages:219-232 https://dx.doi.org/10.3103/S1066530716030054 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2016 3 07 219-232 |
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10.3103/S1066530716030054 doi (DE-627)SPR02405853X (SPR)S1066530716030054-e DE-627 ger DE-627 rakwb eng 510 ASE Martsynyuk, Yu. V. verfasserin aut Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. nonparametric test for change in the mean (dpeaa)DE-He213 “no-change in the mean” null hypothesis (dpeaa)DE-He213 “at most one change in the mean” alternative hypothesis (dpeaa)DE-He213 sup-functional of weighted tied-down partial sums processes (dpeaa)DE-He213 standard Wiener process (dpeaa)DE-He213 Brownian bridge (dpeaa)DE-He213 Enthalten in Mathematical methods of statistics New York, NY : Allerton Press, 2007 25(2016), 3 vom: Juli, Seite 219-232 (DE-627)535184913 (DE-600)2375497-7 1934-8045 nnns volume:25 year:2016 number:3 month:07 pages:219-232 https://dx.doi.org/10.3103/S1066530716030054 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2016 3 07 219-232 |
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10.3103/S1066530716030054 doi (DE-627)SPR02405853X (SPR)S1066530716030054-e DE-627 ger DE-627 rakwb eng 510 ASE Martsynyuk, Yu. V. verfasserin aut Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. nonparametric test for change in the mean (dpeaa)DE-He213 “no-change in the mean” null hypothesis (dpeaa)DE-He213 “at most one change in the mean” alternative hypothesis (dpeaa)DE-He213 sup-functional of weighted tied-down partial sums processes (dpeaa)DE-He213 standard Wiener process (dpeaa)DE-He213 Brownian bridge (dpeaa)DE-He213 Enthalten in Mathematical methods of statistics New York, NY : Allerton Press, 2007 25(2016), 3 vom: Juli, Seite 219-232 (DE-627)535184913 (DE-600)2375497-7 1934-8045 nnns volume:25 year:2016 number:3 month:07 pages:219-232 https://dx.doi.org/10.3103/S1066530716030054 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 25 2016 3 07 219-232 |
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Enthalten in Mathematical methods of statistics 25(2016), 3 vom: Juli, Seite 219-232 volume:25 year:2016 number:3 month:07 pages:219-232 |
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nonparametric test for change in the mean “no-change in the mean” null hypothesis “at most one change in the mean” alternative hypothesis sup-functional of weighted tied-down partial sums processes standard Wiener process Brownian bridge |
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V.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2016</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. 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Martsynyuk, Yu. V. |
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Martsynyuk, Yu. V. ddc 510 misc nonparametric test for change in the mean misc “no-change in the mean” null hypothesis misc “at most one change in the mean” alternative hypothesis misc sup-functional of weighted tied-down partial sums processes misc standard Wiener process misc Brownian bridge Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes |
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510 ASE Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes nonparametric test for change in the mean (dpeaa)DE-He213 “no-change in the mean” null hypothesis (dpeaa)DE-He213 “at most one change in the mean” alternative hypothesis (dpeaa)DE-He213 sup-functional of weighted tied-down partial sums processes (dpeaa)DE-He213 standard Wiener process (dpeaa)DE-He213 Brownian bridge (dpeaa)DE-He213 |
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ddc 510 misc nonparametric test for change in the mean misc “no-change in the mean” null hypothesis misc “at most one change in the mean” alternative hypothesis misc sup-functional of weighted tied-down partial sums processes misc standard Wiener process misc Brownian bridge |
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ddc 510 misc nonparametric test for change in the mean misc “no-change in the mean” null hypothesis misc “at most one change in the mean” alternative hypothesis misc sup-functional of weighted tied-down partial sums processes misc standard Wiener process misc Brownian bridge |
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ddc 510 misc nonparametric test for change in the mean misc “no-change in the mean” null hypothesis misc “at most one change in the mean” alternative hypothesis misc sup-functional of weighted tied-down partial sums processes misc standard Wiener process misc Brownian bridge |
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Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes |
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testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes |
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Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes |
| abstract |
Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. |
| abstractGer |
Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. |
| abstract_unstemmed |
Abstract Let X1,..., Xn, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H0: X1,..., Xn is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H0 versus the “at most one change (AMOC) in the mean” alternative hypothesis HA: there is an integer k*, 1 ≤ k* < n, such that EX1 = · · · = $ EX_{k} $* ≠ $ EX_{k*+1} $ = ··· = EXn. A natural way of testing for H0 versus HA is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations $ Δ_{k} $ = |Sk/k - (Sn - Sk)/(n - k)|, where Sk:= X1 + ··· + Xk. Asymptotic distributions of these test statistics under H0 as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Zn(t)|, where {Zn(t), 0 ≤ t ≤ 1}n≥1 are the tied-down partial sums processes such that Zn(t):=(S⌈(n+1)t⌉−[(n+1)t]Sn/n)/n%${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n %$ if 0 ≤ t < 1, and Zn(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H0 versus HA via sup-functionals of appropriately weighted |Zn(t)|. Simply considering $ max_{1ɤk<n} $ Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Zn(t)|/(t(1 - t)) under H0 and assuming also that E|X|r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Zn(t)| has not been considered before. |
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| title_short |
Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes |
| url |
https://dx.doi.org/10.3103/S1066530716030054 |
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| score |
7.3987207 |