Abstract
Many statistical data are imprecise due to factors such as measurement errors, computation errors, and lack of information. In such cases, data are better represented by intervals rather than by single numbers. Existing methods for analyzing interval-valued data include regressions in the metric space of intervals and symbolic data analysis, the latter being proposed in a more general setting. However, there has been a lack of literature on the parametric modeling and distribution-based inferences for interval-valued data. In an attempt to fill this gap, we extend the concept of normality for random sets by Lyashenko and propose a Normal hierarchical model for random intervals. In addition, we develop a minimum contrast estimator (MCE) for the model parameters, which is both consistent and asymptotically normal. Simulation studies support our theoretical findings and show very promising results. Finally, we successfully apply our model and MCE to a real data set.
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References
Artstein, Z., Vitale, R. A. (1975). A strong law of large numbers for random compact sets. Annals of Probability, 5, 879–882.
Aumann, R. J. (1965). Integrals and set-valued functions. Journal of Mathematical Analysis and Applications, 12, 1–12.
Billard, L., Diday, E. (2003). From the statistics of data to the statistics of knowledge: symbolic data analysis. Journal of the American Statistical Association, 98, 470–487, review article.
Carvalho, F. A. T., Neto, E. A. L., Tenorio, C. P. (2004). A New Method to Fit a Linear Regression Model for Interval-Valued Data. Lecture Notes in Computer Science, 3238, 295–306.
Diamond, P. (1990). Least squares fitting of compact set-valued data. Journal of Mathematical Analysis and Applications, 147, 531–544.
Gil, M. A., Lubiano, M. A., Montenegro, M., Lopez, M. T. (2002). Least squares fitting of an affine function and strength of association for interval-valued data. Metrika, 56, 97–111.
Heinrich, L. (1993). Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika, 40, 69–74.
Hörmander, L. (1954). Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Arkiv för Matematik, 3, 181–186.
Körner, R. (1995). A Variance of Compact Convex Random Sets. Working paper, Institut für Stochastik, Bernhard-von-Cotta-Str. 2 09599 Freiberg.
Körner, R. (1997). On the variance of fuzzy random variables. Fuzzy Sets and Systems, 92, 83–93.
Körner, R., Näther, W. (1998). Linear regression with random fuzzy variables: extended classical estimates, best linear estimates, least squares estimates. Information Sciences, 109, 95–118.
Lyashenko, N. N. (1979). On limit theorems for sums of independent compact random subsets in the Euclidean space. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, 85, 113–128.
Lyashenko, N. N. (1983). Statistics of random compacts in Euclidean space. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, 98, 115–139.
Matheron, G. (1975). Random sets and integral geometry. New York: Wiley.
Molchanov, I. (2005). Theory of random sets. London: Springer.
Müller, A. (1997). Integral probability metrics and their generating classes of functions. Advances in Applied Probability, 29, 429–443.
Norberg, T. (1984). Convergence and existence of random set distributions. The Annals of Probability, 12(3), 726–732.
Pfanzagl, J. (1969). On the measurability and consistency of minimum contrast estimates. Metrika, 14, 249–272.
Puri, M. L., Ralescu, D. A., Ralescu, S. S. (1986). Gaussian random sets in Banach space. Theory of Probablity and its Applications, 31, 526–529.
Rȧdström, H. (1952). An embedding theorem for spaces of convex sets. Proceedings of the American Mathematical Society, 3, 165–169.
Sriperumbudur, B. K., Fukumizu, K., Gretton, A., Schölkopf, B., Lanckriet, G. R. G. (2012). On the empirical estimation of integral probability metrics. Electronic Journal of Statistics, 6, 1550–1599.
Stoyan, D. (1998). Random sets: models and statistics. International Statistical Review, 66(1), 1–27.
Tanaka, U., Ogata, Y., Stoyan, D. (2008). Parameter estimation and model Selection for Neyman–Scott point processes. Biometrical Journal, 50, 43–57.
Zolotarev, V. M. (1983). Probability metrics. Theory of Probability and its Applications, 28, 278–302.
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The authors thank the editor and the two anonymous referees for their careful reading of the manuscript and valuable comments that improved the old version of the paper.
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Dan Ralescu was supported by a Taft Research Grant.
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Sun, Y., Ralescu, D. A normal hierarchical model and minimum contrast estimation for random intervals. Ann Inst Stat Math 67, 313–333 (2015). https://doi.org/10.1007/s10463-014-0453-1
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DOI: https://doi.org/10.1007/s10463-014-0453-1