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Discrete approximations of continuous and mixed measures on a compact interval

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When considering a mixed probability measure on [0, 1] with atoms in zero and one, the frontier points have to be treated (and “weighted”) differently and separately with respect to the interior points. In order to avoid the troublesome consequences of a mixed model, an easily interpretable discretization of [0, 1] on uniformly spaced atoms is here proposed. This “homogeneous support” is used to define two discrete models, a parametric and a nonparametric one. An application to real data on recovery risk of the Bank of Italy’s loans is here considered to exemplify both discretization and proposed models. Finally, a simulation study is performed to analyze the behavior of the nonparametric proposal in terms of goodness-of-fit.

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  1. Altman EI, Resti A, Sironi A (2005) Loss given default: a review of the literature. In: Altman E, Resti A, Sironi A (eds) The next challenge in credit risk management. Riskbooks, London

  2. Banca d’Italia (2001) Principali risultati della rilevazione sull’attività di recupero dei crediti. Bollettino di vigilanza 12

  3. Basel Comittee on Banking Supervision (2004) International capital measurement and capital standards: a revised framework. Bank for international Settlements

  4. Calabrese R, Zenga M (2008) Measuring loan recovery rate: methodology and empirical evidence. Stat Appl 6(2): 193–214

  5. Calabrese R, Zenga M (2010) Bank loan recovery rates: measuring and nonparametric density estimation. J Bank Financ 34(5): 903–911

  6. Chen SX (1999) Beta kernel estimators for density functions. Comput Stat Data Anal 31(2): 131–145

  7. Cochran WG (1954) Some methods for strengthening the common χ2 tests. Biometrics 10(4): 417–451

  8. Cramer H (1946) Mathematical methods of statistics. Princeton University Press, New Jersey

  9. Cressie N, Read TRC (1984) Multinomial goodness-of-fit tests. J Roy Stat Soc B (Methodological) 46(3): 440–464

  10. Efromovich S (1999) Nonparametric curve estimation: methods, theory and applications. Springer-Verlag, New York

  11. Fu JC, Wang L (2002) A random-discretization based Monte Carlo sampling method and its applications. Methodol Comput Appl Probab 4(1): 5–25

  12. Gupton GM, Finger CC, Bhatia M (1997) CreditMetricsTM—technical document. J. P. Morgan & Co, New York

  13. Horn SD (1977) Goodness-of-fit tests for discrete data: a review and an application to a health impairment scale. Biometrics 33(1): 237–248

  14. Johnson NL, Kotz S, Kotz S (1970) Continuous univariate distributions, vol 2. Wiley, New York

  15. Johnson NL, Kotz S, Kemp AW (1992) Univariate discrete distributions. Wiley, New York

  16. Kendall MG (1952) The advanced theory of statistics, vol 1. Griffin, London

  17. Liu H, Hussain F, Tan CL, Dash M (2002) Discretization: an enabling technique. Data Min Knowl Discov 6(4): 393–423

  18. McLachlan GJ, Peel D (2000) Finite mixture models. Wiley, New York

  19. Qureshi T, Zighed DA (2009) A decision boundary based discretization technique using resampling. World Acad Sci Eng Technol 49: 820–825

  20. Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, London

  21. Yang Y, Webb GI (2009) Discretization for naive-bayes learning: managing discretization bias and variance. Machine Learn 74(1): 39–74

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Correspondence to Antonio Punzo.

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Punzo, A., Zini, A. Discrete approximations of continuous and mixed measures on a compact interval. Stat Papers 53, 563–575 (2012). https://doi.org/10.1007/s00362-011-0365-6

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  • Discretization
  • Beta distribution
  • Mixture models
  • Kernel smoothing