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AUTHOR(S):

Wayan Somayasa, Asrul Sani, Yulius Bara Pasolon

 

TITLE

A Most Powerful Test for the Adequateness of an Asymptotic Spatial Regression Model when the Observation is Disturbed by the Set-Indexed Brownian Sheet

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ABSTRACT

In this work we establish an optimal test for checking the appropriateness of a spatial regression model. In the study of model check for regression, the correctness of an assumed model is investigated by the partial sums of the residuals. In this work an inverted procedure is proposed in that we firstly embed the observation into a partial sums process to get the corresponding asymptotic regression model. Instead of considering the residuals of the model, we derive the Cameron-Martin density of the observation. For simple hypotheses under H0 as well as under H1 we derive the Neyman-Person test based on the ratio of the densities under H0 and H1. Interestingly, the rejection region can be exactly computed as an integral with respect to the partial sums process of the observation. An application of the procedure to a real data is also discussed.

KEYWORDS

Most powerful test, set-indexed Brownian sheet, Neyman-Pearson test, Cameron-Martin density, reproducing kernel Hilbert space

REFERENCES

[1] K.S. Alexander and R. Pyke, A uniform central limit theorem for set-indexed partial-sum processes with finite variance, The Annals of Probability, 14, 1986, pp. 582–597. [1] K.S. Alexander and R. Pyke, A uniform central limit theorem for set-indexed partial-sum processes with finite variance, The Annals of Probability, 14, 1986, pp. 582–597. 

[2] S.F. Arnold, Asymptotic validity of F tests for odinary linear model and the multiple correlation model, Journal of the American Statistical Association, 75 (372), 1980, pp. 890–894. 

[3] S.F. Arnold, The Theory of Linear Models and Multivarite Analysis, John Wiley & Sons, Inc., New York, 1981. 

[4] H. Bauer, Measure and Integration Theory, Walter de Gruyter, Berlin, 2001. 

[5] P. Billingsley, Convergence of Probability Measures (2nd Edition), John Wiley & Sons, Inc., New York, 1999. 

[6] W. Bischoff, A functional central limit theorem for regression models, Ann. Stat. 6, 1998, pp. 1398–1410. 

[7] W. Bischoff, The structure of residual partial sums limit processes of linear regression models, Theory of Stochastic Processes, 2, 2002, pp. 23– 28. 

[8] W. Bischoff and A. Gegg, The CameronMartin theorem for (p−)Slepian processes, Preprint, Catholic University EichstaettIngolstadt,Germany, 2014. 

[9] W. Bischoff and W. Somayasa, The limit of the partial sums process of spatial least squares residuals, J. Multivariate Analysis, 100, 2009, pp. 2167–2177. 

[10] P.Gaenssler, On recent development in the theory of set-indexed processes (A unified approach to empirical and partial-sum processes) in Asymptotic Statistics, Springer, Berlin, 1993. 

[11] A. Gegg, Moving Windows zum Testen auf Change-Points (Sequentielle und beste Tests), Ph.D. Dissertation, Catholic University Eichstaett-Ingolstadt, Germany, 2013. 

[12] E.L. Lehmann and J.P. Romano, Testing Statitical Hypotheses, 3rd edn., Springer, New York, 2005. 

[13] M. Lifshits, Lectures on Gaussian Processes, Springer Briefs in Mathematics, Springer, Berlin, 2012. 

[14] I.B. MacNeill, Properties of partial sums of polynomial regression residuals with applications to test for change of regression at unknown times, Ann. Statist. 6, 1978, pp. 422–433. 

[15] I.B. MacNeill, Limit processes for sequences partial sums of regression residuals, Ann. Probab. 6, 1978, pp. 695–698. 

[16] I.B. MacNeill and V.K. Jandhyala, Change-point methods for spatial data, Multivariate Environmental Statistics eds. by G.P. Patil and C.R. Rao, Elsevier Science Publishers B.V., 1993, pp. 298– 306. 

[17] W.J. Pyke, A uniform central limit theorem for partial sum processes indexed by sets, Ann.Probab. 79, 1983, pp. 219–240. 

[18] W. Somayasa, On set-indexed residual partial sum limit process of spatial linear regression models, J. Indones. Math. Soc. 17(2), 2011, pp. 73–83. 

[19] W. Somayasa, The partial sums of the least squares residuals of spatial observations sampled according to a probability measure”, J. Indones. Math. Soc. 19(1), 2013, pp. 23–40. 

[20] W. Somayasa, Asymptotic statistical model building based on the partial sums of the residuals of the observations with an application to mining industry, Proceedings of the 2 nd International Conference on Mathematical, Computational and Statistical Sciences (MCSS), 2014, pp. 136–145. 

[21] W. Somayasa, Ruslan, E. Cahyono, L.O. Engkoimani, Cheking adequateness of spatial regressions using set-indexed partial sums technique, Fareast Journal of Mathematical Sciences, 96(8), 2015, pp. 933–966. 

[22] M. Tahir, Prediction of the amount of nickel deposit based on the results of drilling bores on several points (case study: south mining region of PT. Aneka Tambang Tbk., Pomalaa, Southeast Sulawesi), research report, Halu Oleo University, Kendari, 2010. 

[23] L. Xie and I.B. MacNeill, Spatial residual processes and boundary detection, South African Statist. J. 4, 2006, pp. 33–53.

Cite this paper

Wayan Somayasa, Asrul Sani, Yulius Bara Pasolon. (2016) A Most Powerful Test for the Adequateness of an Asymptotic Spatial Regression Model when the Observation is Disturbed by the Set-Indexed Brownian Sheet. International Journal of Mathematical and Computational Methods, 1, 214-220

 

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