Mathematical Methods of Statistics

, Volume 27, Issue 2, pp 83–102 | Cite as

Estimating the Index of Increase via Balancing Deterministic and Random Data

  • L. ChenEmail author
  • Y. Davydov
  • N. Gribkova
  • R. Zitikis


We introduce and explore an empirical index of increase that works in both deterministic and random environments, thus allowing to assess monotonicity of functions that are prone to random measurement errors. We prove consistency of the index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results suggest a frequency at which observations should be taken in order to reach any pre-specified level of estimation precision.We illustrate the index using data arising from purely deterministic and error-contaminated functions, which may or may not be monotonic.


index of increase determinism randomness measurement errors smoothing cross validation 

2010 Mathematics Subject Classification

62G05 62G08 62G20 62P15 62P20 62P25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. J. Anscombe, “Graphs in Statistical Analysis”, Amer.Statist. 27, 17–21 (1973).Google Scholar
  2. 2.
    S. Arlot and A. Celisse, “A Survey of Cross-Validation Procedures forModel Selection”, Statist. Surveys 4, 40–79 (2010).CrossRefzbMATHGoogle Scholar
  3. 3.
    M. Bebbington C. D. Lai and R. Zitikis, “Modeling Human Mortality Using Mixtures of Bathtub Shaped Failure Distributions”, J. Theoret. Biology 245, 528–538 (2011).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Bebbington C. D. Lai and R. Zitikis, “ModellingDeceleration in Senescent Mortality”, Math. Population Studies 18, 18–37 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P. J. Bickel F. Götze, and W. R. van Zwet, “Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses”, Statistica Sinica 7, 1–31 (1997).MathSciNetzbMATHGoogle Scholar
  6. 6.
    P. J. Bickel and A. Sakov, “On the Choice of m in the m out of n Bootstrap and Confidence Bounds for Extrema”, Statistica Sinica 18, 967–985 (2008).MathSciNetzbMATHGoogle Scholar
  7. 7.
    H. Bühlmann, “An Economic Premium Principle”, ASTIN Bulletin 11, 52–60 (1980).MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Bühlmann, “The General Economic Premium Principle”, ASTIN Bulletin 14, 13–21 (1984).CrossRefGoogle Scholar
  9. 9.
    A. Celisse, Model Selection via Cross-Validation in Density Estimation, Regression, and Change-Points Detection, UniversitéParis Sud–Paris XI, Paris. HAL Id: tel-00346320. (2008).Google Scholar
  10. 10.
    L. Chen and R. Zitikis, “Measuring and Comparing Student Performance: A New Technique for Assessing Directional Associations”, Education Sciences 7, 1–27 (2017).CrossRefGoogle Scholar
  11. 11.
    A. DasGupta, Asymptotic Theory of Statistics and Probability (Springer, New York, 2008).zbMATHGoogle Scholar
  12. 12.
    A. C. Davison and D. V. Hinkley, Bootstrap Methods and their Application (Cambridge Univ. Press, Cambridge, UK. 1997).CrossRefzbMATHGoogle Scholar
  13. 13.
    Y. Davydov and R. Zitikis, “The Influence of Deterministic Noise on Empirical Measures Generated by Stationary Processes”, Proc. Amer.Math. Soc. 132, 1203–1210 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Y. Davydov and R. Zitikis, “An Index of Monotonicity and its Estimation: A Step beyond Econometric Applications of the Gini Index”, Metron 63 (special issue in memory of Corrado Gini), 351–372 (2005).MathSciNetGoogle Scholar
  15. 15.
    Y. Davydov and R. Zitikis, “Deterministic Noises that can be Statistically Distinguished from the Random Ones”, Statist. Inference for Stochastic Processes 10, 165–179 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Y. Davydov and R. Zitikis, “Quantifying Non-Monotonicity of Functions and the Lack of Positivity in Signed Measures”, Modern Stochastics: Theory and Applications 4, 219–231 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory (Wiley, New York, 1988).zbMATHGoogle Scholar
  18. 18.
    B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman and Hall/CRC, Boca Raton, FL, 1993).CrossRefzbMATHGoogle Scholar
  19. 19.
    M. Egozcue L. Fuentes García, W. K. Wong, and R. Zitikis, “The Covariance Sign of Transformed Random Variables with Applications to Economics and Finance”, IMA J. Management Math. 22, 291–300 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    T. Eichner and A. Wagener, “Multiple Risks and Mean-Variance Preferences”, Operations Research 57, 1142–1154 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    G. Feder R. E. Just and A. Schmitz, “FuturesMarkets and the Theory of the Firmunder PriceUncertainty”, Quarterly J. Economics 94, 317–328 (1980).CrossRefGoogle Scholar
  22. 22.
    M. Friedman and L. J. Savage, “The Utility Analysis of Choices Involving Risk”, J. Political Economy 56, 279–304 (1948).CrossRefGoogle Scholar
  23. 23.
    E. Furman and R. Zitikis, “Weighted Premium Calculation Principles”, Insurance: Math. and Economics 42, 459–465 (2008).MathSciNetzbMATHGoogle Scholar
  24. 24.
    N. V. Gribkova and R. Helmers, “On the Edgeworth Expansion and the M out of N Bootstrap Accuracy for a Studentized TrimmedMean”, Math.Methods Statist. 16, 142–176 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    N. V. Gribkova and R. Helmers, “On the Consistency of the MN Bootstrap Approximation for a Trimmed Mean”, Theory Probab. and Its Appl. 55, 42–53 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    P. Hall, The Bootstrap and Edgeworth Expansion (Springer, New York, 1992).CrossRefzbMATHGoogle Scholar
  27. 27.
    W. Härdle, Smoothing Techniques, with Implementation in S (Springer, New York, 1991).CrossRefzbMATHGoogle Scholar
  28. 28.
    J. D. Hey, “Hedging and the Competitive Labor-Managed Firm under Price Uncertainty”, Amer. Economic Review 71, 753–757 (1981).Google Scholar
  29. 29.
    U. Kamps, “On a Class of Premium Principles Including the Esscher Principle”, Scand. Actuarial J. 1998, 75–80 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    D. Kahneman and A. Tversky, “Prospect Theory of Decisions under Risk”, Econometrica 47, 263–291 (1979).CrossRefzbMATHGoogle Scholar
  31. 31.
    A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis (Dover, New York, 1970).zbMATHGoogle Scholar
  32. 32.
    E. L. Lehmann, “Some Concepts of Dependence”, Ann. Math. Statist. 37, 1137–1153 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    H. Markowitz, “The Utility ofWealth”, J. Political Economy 60, 151–156 (1952).CrossRefGoogle Scholar
  34. 34.
    J. Meyer and L. J. Robison, “Hedging under Output Price Randomness”, Amer. J. Agricultural Economics 70, 268–272 (1988).CrossRefGoogle Scholar
  35. 35.
    I. P. Natanson, Theory of Functions of a Real Variable (Dover, New York, 2016).Google Scholar
  36. 36.
    D. T. Qoyyimi, A Novel Method for Assessing Co-monotonicity: An Interplay between Mathematics and Statistics with Applications, Electronic Thesis and Dissertation Repository Nr. 3322. (2015).Google Scholar
  37. 37.
    D. T. Qoyyimi and R. Zitikis, “Measuring the Lack ofMonotonicity in Functions”, Math. Scientist 39, 107–117 (2014).MathSciNetzbMATHGoogle Scholar
  38. 38.
    D. T. Qoyyimi and R. Zitikis, “Measuring Association via Lack of Co-Monotonicity: the LOC Index and a Problem of Educational Assessment”, DependenceModeling 3, 83–97 (2015).MathSciNetzbMATHGoogle Scholar
  39. 39.
    R Core Team, R: A Language and Environment for Statistical Computing R Foundation for Statistical Computing, Vienna, (2013).Google Scholar
  40. 40.
    C. Sievert C. Parmer T. Hocking S. Chamberlain K. Ram M. Corvellec and P. Despouy, plotly: Create Interactive Web Graphics via’ plotly.js’. (2017).Google Scholar
  41. 41.
    B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman and Hall/CRC, London, 1986).CrossRefzbMATHGoogle Scholar
  42. 42.
    D.W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization, 2nd. ed. (Wiley, New York, 2015).CrossRefzbMATHGoogle Scholar
  43. 43.
    J. Shao and D. Tu, The Jackknife and Bootstrap (Springer, New York, 1995).CrossRefzbMATHGoogle Scholar
  44. 44.
    H.-W. Sinn, “Expected Utility, μ-σ Preferences, and Linear Distribution Classes: A Further Result”, J. Risk and Uncertainty 3, 277–281 (1990).CrossRefGoogle Scholar
  45. 45.
    R. M. Thorndike and T. Thorndike-Christ, Measurement and Evaluation in Psychology and Education, 8th ed. (Prentice Hall, Boston, MA, 2010).Google Scholar
  46. 46.
    A. Tversky and D. Kahneman, “Advances in Prospect Theory: Cumulative Representation of Uncertainty”, J. Risk and Uncertainty 5, 297–323 (1992).CrossRefzbMATHGoogle Scholar
  47. 47.
    S. Wang, “Insurance Pricing and Increased Limits Ratemaking by Proportional Hazards Transforms”, Insurance:Math. and Economics 17, 43–54 (1995).MathSciNetzbMATHGoogle Scholar
  48. 48.
    S. Wang, “An Actuarial Index of the Right-Tail Risk”, NorthAmer. Actuarial J. 2, 88–101 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    L. Wasserman, All of Statistics: a Concise Course in Statistical Inference, 2nd ed. (Springer, New York, 2005).zbMATHGoogle Scholar
  50. 50.
    H. Wickham, ggplot2: Elegant Graphics for Data Analysis (Springer, New York, 2006).zbMATHGoogle Scholar
  51. 51.
    W. K. Wong, “Stochastic Dominance Theory for Location-Scale Family”, J. Appl. Math. and Decision Sci. 2006, 1–10 (2006).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.School of Math. and Statist. Sci.Western UniversityLondonCanada
  2. 2.Chebyshev Lab.St. Petersburg State Univ.St. PetersburgRussia
  3. 3.Faculty Math. and Mech.St. Petersburg State Univ.St. PetersburgRussia

Personalised recommendations