# A goodness-of-fit test for marginal distribution of linear random fields with long memory

## Author

Listed:
• Hira L. Koul

() (Michigan State University)

• Nao Mimoto

(University of Akron)

• Donatas Surgailis

(Vilnius University)

## Abstract

Abstract This paper addresses the problem of fitting a known distribution function to the marginal distribution of a stationary long memory moving average random field observed on increasing $$\nu$$ ν -dimensional “cubic” domains when its mean $$\mu$$ μ and scale $$\sigma$$ σ are known or unknown. Using two suitable estimators of $$\mu$$ μ and a classical estimate of $$\sigma$$ σ , a modification of the Kolmogorov–Smirnov statistic is defined based on the residual empirical process and having a Cauchy-type limit distribution, independent of $$\mu ,\sigma$$ μ , σ and the long memory parameter d. Based on this result, a simple goodness-of-fit test for the marginal distribution is constructed, which does not require the estimation of d or any other underlying nuisance parameters. The result is new even for the case of time series, i.e., when $$\nu =1$$ ν = 1 . Findings of a simulation study investigating the finite sample behavior of size and power of the proposed test is also included in this paper.

## Suggested Citation

• Hira L. Koul & Nao Mimoto & Donatas Surgailis, 2016. "A goodness-of-fit test for marginal distribution of linear random fields with long memory," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(2), pages 165-193, February.
• Handle: RePEc:spr:metrik:v:79:y:2016:i:2:d:10.1007_s00184-015-0550-z
DOI: 10.1007/s00184-015-0550-z
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## References listed on IDEAS

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1. Hira Koul & Nao Mimoto & Donatas Surgailis, 2013. "Goodness-of-fit tests for long memory moving average marginal density," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(2), pages 205-224, February.
2. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
3. Giraitis, Liudas & Koul, Hira L. & Surgailis, Donatas, 1996. "Asymptotic normality of regression estimators with long memory errors," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 317-335, September.
4. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2010. "A two-sample test for comparison of long memory parameters," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2118-2136, October.
5. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2009. "Two estimators of the long-run variance: Beyond short memory," Journal of Econometrics, Elsevier, vol. 150(1), pages 56-70, May.
6. Guo, Hongwen & Lim, Chae Young & Meerschaert, Mark M., 2009. "Local Whittle estimator for anisotropic random fields," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 993-1028, May.
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1. repec:eee:spapps:v:127:y:2017:i:8:p:2751-2779 is not listed on IDEAS

### Keywords

Degenerate residual empirical process; Cauchy distribution;

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