Estimating the width of a uniform distribution under symmetric measurement errors
- PDF / 631,496 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 24 Downloads / 190 Views
Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
Estimating the width of a uniform distribution under symmetric measurement errors S. Hamedovi´c1 · M. Benši´c2 · K. Sabo2 Received: 31 January 2019 / Accepted: 4 November 2019 © Korean Statistical Society 2020
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. Keywords Uniform distribution · Additive error · Maximum likelihood estimator · Robustness
1 Introduction The model we will consider here is of the form X = U + ε, where the random variable U is uniformly distributed over some interval, and the error variable ε is supposed to be absolutely continuous, independent of U . In applications, it is usually assumed that ε is normally distributed. In metrology, the convolution of a uniform and a normal distribution is known as the Flatten–Gaussian distribution. This distribution is the basis for calculating the measurement uncertainty in Blázquez et al. (2008) and Fotowicz (2014), for example. In geology, such a model is used for estimating the age of stage boundaries (see Agterberg 1988, 2014; Cox
B
K. Sabo [email protected] S. Hamedovi´c [email protected] M. Benši´c [email protected]
1
Faculty of Metallurgy and Technology, University of Zenica, Travniˇcka cesta 1, Zenica, Bosnia and Herzegovina
2
Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, Osijek, Croatia
123
Journal of the Korean Statistical Society
and Dalrymple 1967). Recently, this distribution has been identified, among several candidates, as the best one for modelling transmission loss data (see Toli´c et al. 2017). Also, our model appears in fitting a line segment to noisy data, which can be useful in image analysis. The noises in Davidov and Goldenshluger (2004) have normal distributions, while one of the suggested models for the latent variable is uniform. A Bayesian method for fitting models to noisy data is suggested in Werman and Karen (2001), where the uniform prior on parametric models is used, and the noise is additive isotropic Gaussian. This paradigm is shown to have advantage compared to the classical MSE (Mean Square Error) method in a line segment fitting. Parameters in a linear structural relationship were estimated in Chan (1982) with the assumption that the true regressor is uniformly distributed, but observed with a normal error. Some theoretical considerations of the uniform plus error model are given in Benši´c and Sabo (2007a), Schneeweiss (2004) (normal error with known variance), Benši´c and Sabo (2010) (normal error with unknown variance) and Benši´c and Sabo (2016) (Laplace error). It is shown in these papers that the ML estimator
Data Loading...
