Abstract
In this paper we consider the problem of estimating the support of a uniform distribution under symmetric additive errors. The maximum likelihood (ML) estimator is of our primary interest, but we also analyze the method of moments (MM) estimator, when it exists. Under some regularity conditions, the ML estimator is consistent and asymptotically efficient. Errors with Student’s t-distribution are shown to be a good choice for robustness issues.
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Notes
As usual, \(z_{\alpha }\) is the \(1-\alpha \) quantile of standard normal distribution.
References
Agterberg, F. P. (1988). Quality of time scales: A statistical appraisal. In D. F. Merriam (Ed.), Current trends in geomathematics. computer applications in the earth sciences (pp. 57–103). Boston: Springer.
Agterberg, F. P. (2014). Geomathematics: Theoretical foundations, applications and future developments. Quantitative geology and geostatistics (Vol. 18). Heidelberg: Springer.
Benšić, M., Hamedović, S., Sabo, K., & Taler, P. (2018). LeArEst: Border and Area Estimation of Data Measured with Additive Error, R package version 1.0.0. https://cran.r-project.org/web/packages/LeArEst/index.html. Accessed 8 Apr 2019.
Benšić, M., & Sabo, K. (2007a). Estimating the width of a uniform distribution when data are measured with additive normal errors with known variance. Computational Statistics and Data Analysis, 51, 4731–4741. https://doi.org/10.1016/j.csda.2006.09.006.
Benšić, M., & Sabo, K. (2007b). Border estimation of a two-dimensional uniform distribution if data are measured with additive error. Statistics-A Journal of Theoretical and Applied Statistics, 41, 311–319. https://doi.org/10.1080/02331880701270564.
Benšić, M., & Sabo, K. (2010). Estimating a uniform distribution when data are measured with a normal additive error with unknown variance. Statistics-A Journal of Theoretical and Applied Statistics, 44, 235–246. https://doi.org/10.1080/02331880903076645.
Benšić, M., & Sabo, K. (2016). Uniform distribution width estimation from data observed with Laplace additive error. Journal of the Korean Statistical Society, 45, 505–517. https://doi.org/10.1016/j.jkss.2016.03.001.
Benšić, M., Taler, P., Hamedović, S., Nyarko, E. K., & Sabo, K. (2017). LeArEst: The length and area estimation from data measured with additive error. R Journal, 9(2), 461–473. https://doi.org/10.32614/RJ-2017-043.
Blázquez, J., García-Berrocal, A., Montalvo, C., & Balbás, M. (2008). The coverage factor in a Flatten–Gaussian distribution. Metrologia, 45, 503–506. https://doi.org/10.1088/0026-1394/45/5/002.
Chan, N. N. (1982). Linear structural relationships with unknown error variances. Biometrika, 69, 277–279. https://doi.org/10.1093/biomet/69.1.277.
Cox, A. V., & Dalrymple, G. B. (1967). Statistical analysis of geomagnetic reversal data and the precision of potassium-argon dating. Journal of Geophysical Research, 72, 2603–2614. https://doi.org/10.1029/JZ072i010p02603.
Cramér, H. (1946). Mathematical methods of statistics. Princeton: Princeton Univerity Press.
Csörgő, S., & Faraway, J. J. (1996). The exact and asymptotic distributions of Cramér-von Mises statistics. Journal of the Royal Statistical Society B, 58, 221–234. https://doi.org/10.1080/03610928308828614.
Davidov, O., & Goldenshluger, A. (2004). Fitting a line segment to noisy data. Journal of Statistical Planing and Inference, 119, 191–206. https://doi.org/10.1016/S0378-3758(02)00409-3.
Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society London, 222, 309–368. https://doi.org/10.1098/rsta.1922.0009.
Fotowicz, P. (2014). Methods for calculating the coverage interval based on the Flatten–Gaussian distribution. Measurement, 55, 272–275. https://doi.org/10.1016/j.measurement.2014.05.006.
Hamedović, S., Benšić, M., Sabo, K., & Taler, P. (2018). Estimating the size of an object captured with error. Central European Journal of Operations Research, 26, 771–781. https://doi.org/10.1007/s10100-017-0504-9.
Lehmann, E. L., & Casella, G. (1998). Theory of point estimation. Berlin: Springer.
Loève, M. (1977). Probability theory I (4th ed.). New York: Springer.
Sabo, K., & Benšić, M. (2009). Border estimation of a disc observed with random errors solved in two steps. Journal of Computational and Applied Mathematics, 229, 16–26. https://doi.org/10.1016/j.cam.2008.10.010.
Schneeweiss, H. (2004). Estimating the endpoint of a uniform distribution under measurement errors. Central European Journal of Operations Research, 12, 221–231.
Tolić, I., Miličević, K., Šuvak, N., & Biondić, I. (2017). Non-linear least squares and maximum likelihood estimation of probability density function of cross-border transmission losses. IEEE Transactions on Power Systems, 33, 2230–2238. https://doi.org/10.1109/TPWRS.2017.2738319.
Werman, M., & Karen, D. (2001). A Bayesian method for fitting parametric and nonparametric models to noisy data. IEEE Transactions of Pattern Analysis and Machine Inteligence, 23, 528–534. https://doi.org/10.1109/34.922710.
Westfall, P. H. (2014). Kurtosis as Peakedness, 1905–2014. R.I.P. The American Statistician, 68, 191–195. https://doi.org/10.1080/00031305.2014.917055.
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This work was supported by the Croatian Science Foundation through research grants IP-2016-06-6545.
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Hamedović, S., Benšić, M. & Sabo, K. Estimating the width of a uniform distribution under symmetric measurement errors. J. Korean Stat. Soc. 49, 822–840 (2020). https://doi.org/10.1007/s42952-019-00035-7
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DOI: https://doi.org/10.1007/s42952-019-00035-7