On the accuracy of inference on heavy-tailed distributions

Article

Novak, S. 2013. On the accuracy of inference on heavy-tailed distributions. Theory of Probability and Its Applications. 58 (3), pp. 509-518. https://doi.org/10.1137/S0040585X97986710
TypeArticle
TitleOn the accuracy of inference on heavy-tailed distributions
AuthorsNovak, S.
Abstract

This paper suggests a simple method of deriving nonparametric lower bounds of the accuracy of statistical inference on heavy-tailed distributions. We present lower bounds of the mean squared error of the tail index, the tail constant, and extreme quantiles estimators. The results show that the normalizing sequences of robust estimators must depend in a specific way on the tail index and the tail constant.

PublisherSociety for Industrial and Applied Mathematics
JournalTheory of Probability and Its Applications
ISSN0040-585X
Publication dates
Print2013
Publication process dates
Deposited14 Oct 2015
Accepted01 Jan 2013
Output statusPublished
Accepted author manuscript
Copyright Statement

Published in Theory of Probability & Its Applications in Volume 58, Issue 3, published by the Society of Industrial and Applied Mathematics (SIAM). Copyright © 2014 Society of Industrial and Applied Mathematics.

Additional information

Published online: 16 September 2014

Digital Object Identifier (DOI)https://doi.org/10.1137/S0040585X97986710
LanguageEnglish
Permalink -

https://repository.mdx.ac.uk/item/85z5x

Log in to edit

Download files

Accepted author manuscript
TVP 2014.pdf

  • 42
    total views
  • 11
    total downloads
  • 2
    views this month
  • 1
    downloads this month

Export as

Related outputs

Compound Poisson approximation
Čekanavičius, V. and Novak, S. 2024. Compound Poisson approximation. New York, NY Chapman & Hall.
On Poisson approximation
Novak, S. 2024. On Poisson approximation. Journal of Theoretical Probability. https://doi.org/10.1007/s10959-023-01310-4
On the T-test
Novak, S. 2022. On the T-test. Statistics & Probability Letters. 189. https://doi.org/10.1016/j.spl.2022.109562
Compound Poisson approximation
Čekanavičius, V. and Novak, S. 2022. Compound Poisson approximation. Probability Surveys. 19, pp. 271-350. https://doi.org/10.1214/22-PS8
Poisson approximation. Addendum
Novak, S. 2021. Poisson approximation. Addendum. Probability Surveys. 18, pp. 272-275. https://doi.org/10.1214/21-PS2
Poisson approximation in terms of the Gini-Kantorovich distance
Novak, S. 2021. Poisson approximation in terms of the Gini-Kantorovich distance. Extremes. 24 (1), pp. 64-87. https://doi.org/10.1007/s10687-020-00392-1
On the T-test
Novak, S. 2020. On the T-test.
On the accuracy of poisson approximation
Novak, S. 2019. On the accuracy of poisson approximation. Extremes. 22 (4), pp. 729-748. https://doi.org/10.1007/s10687-019-00350-6
Poisson approximation
Novak, S. 2019. Poisson approximation. Probability Surveys. 16, pp. 228-276. https://doi.org/10.1214/18-PS318
On the length of the longest head run
Novak, S. 2017. On the length of the longest head run. Statistics & Probability Letters. 30, pp. 111-114. https://doi.org/10.1016/j.spl.2017.06.020
Inference about the Pareto--type distribution
Novak, S. 1992. Inference about the Pareto--type distribution. Transactions of the Eleventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes. Prague 26 - 31 Aug 1990 pp. 251-258
Non-parametric lower bounds and information functions
Novak, S. 2018. Non-parametric lower bounds and information functions. ISNPS-Third Conference of the International Society for Nonparametric Statistics (ISNPS). Avignon, France 11 - 16 Jun 2016 Springer. pp. 69-83 https://doi.org/10.1007/978-3-319-96941-1
On the joint limiting distribution of the first and the second maxima
Novak, S. and Weissman, I. 1998. On the joint limiting distribution of the first and the second maxima. Communications In Statistics. Stochastic Models. 14 (1-2), pp. 311-318. https://doi.org/10.1080/15326349808807473
On blocks and runs estimators of extremal index
Weissman, I. and Novak, S. 1998. On blocks and runs estimators of extremal index. Journal of Statistical Planning and Inference. 66 (2), pp. 281-288. https://doi.org/10.1016/S0378-3758(97)00095-5
Time intervals of constant sojourn of a homogeneous Markov chain in a fixed subset of states
Novak, S. 1988. Time intervals of constant sojourn of a homogeneous Markov chain in a fixed subset of states. Siberian Mathematical Journal. 29 (1), pp. 100-109. https://doi.org/10.1007/BF00975021
Asymptotic properties of the distribution of the length of the longest head run
Novak, S. 1988. Asymptotic properties of the distribution of the length of the longest head run. Thesis Russian Academy of Sciences Institute of Mathematics (Novosibirsk)
Asymptotic expansions in the problem of the longest head run for Markov chain with two states
Novak, S. 1989. Asymptotic expansions in the problem of the longest head run for Markov chain with two states. Trudy Inst. Math. (Novosibirsk). 13, pp. 136-147.
Asymptotics of the distribution of the ratio of sums of random variables
Novak, S. and Utev, S. 1990. Asymptotics of the distribution of the ratio of sums of random variables. Siberian Mathematical Journal. 31 (5), pp. 781-788. https://doi.org/10.1007/BF00974491
Rate of convergence in the limit theorem for the length of the longest head run
Novak, S. 1991. Rate of convergence in the limit theorem for the length of the longest head run. Siberian Mathematical Journal. 32 (3), pp. 444-448. https://doi.org/10.1007/BF00970481
On the distribution of the maximum of random number of random variables
Novak, S. 1991. On the distribution of the maximum of random number of random variables. Theory Probab. Appl.. 36 (4), pp. 714-721.
Longest runs in a sequence of m-dependent random variables
Novak, S. 1992. Longest runs in a sequence of m-dependent random variables. Probability Theory and Related Fields. 91 (3-4), pp. 269-281. https://doi.org/10.1007/BF01192057
On the asymptotic distribution of the number of random variables exceeding a given level
Novak, S. 1993. On the asymptotic distribution of the number of random variables exceeding a given level. Siberian advances in mathematics. 3 (4), pp. 108-122.
Asymptotic expansions for the maximum of random number of random variables
Novak, S. 1994. Asymptotic expansions for the maximum of random number of random variables. Stochastic Processes and their Applications. 51 (2), pp. 297-305. https://doi.org/10.1016/0304-4149(94)90047-7
Poisson approximation for the number of long match patterns in random sequences
Novak, S. 1994. Poisson approximation for the number of long match patterns in random sequences. Theory of Probability and Its Applications. 39 (4), pp. 593-603. https://doi.org/10.1137/1139045
Long match patterns in random sequences
Novak, S. 1995. Long match patterns in random sequences. Siberian Adv. Math.. 5 (3), pp. 128-140.
On extreme values in stationary sequences
Novak, S. 1996. On extreme values in stationary sequences. Siberian Adv. Math.. 6 (3), pp. 68-80.
On the distribution of the ratio of sums of random variables
Novak, S. 1996. On the distribution of the ratio of sums of random variables. Theory of Probability and Its Applications. 41 (3), pp. 479-503. https://doi.org/10.1137/S0040585X97975228
Statistical estimation of the maximal eigenvalue of a matrix
Novak, S. 1996. Statistical estimation of the maximal eigenvalue of a matrix. Russian Math. (Izvestia Vys. Ucheb. Zaved.). 41 (5), pp. 46-49.
On the Erdös-Rènyi maximum of partial sums
Novak, S. 1998. On the Erdös-Rènyi maximum of partial sums. Theory of Probability and Its Applications. 42 (2), pp. 254-270. https://doi.org/10.1137/S0040585X97976118
Limit theorems and estimates of rates of convergence in extreme value theory. – DSc thesis
Novak, S. 2013. Limit theorems and estimates of rates of convergence in extreme value theory. – DSc thesis. Thesis St. Petersburg branch of Steklov Institute of Mathematics, Russian Academy of Sciences, and St. Petersburg University St. Petersburg Department of Steklov Institute of Mathematics
On the limiting distribution of extremes
Novak, S. 1998. On the limiting distribution of extremes. Siberian Adv. Math.. 8 (2), pp. 70-95.
Generalised kernel density estimator
Novak, S. 1999. Generalised kernel density estimator. Theory of Probability and Its Applications. 44 (3), pp. 570-583. https://doi.org/10.1137/S0040585X97977781
On the mode of an unknown probability distribution
Novak, S. 2000. On the mode of an unknown probability distribution. Theory of Probability and Its Applications. 44 (1), pp. 109-113. https://doi.org/10.1137/s0040585x97977392
Measures of financial risk
Novak, S. 2016. Measures of financial risk. in: Longin, F. (ed.) Extreme Events in Finance: A Handbook of Extreme Value Theory and its Applications Wiley. pp. 215-237
On measures of financial risk
Novak, S. 2015. On measures of financial risk. International Conference on Risk Analysis ICRA 6 / RISK 2015. Barcelona, Spain 26 - 29 May 2015 FUNDACIÓN MAPFRE. pp. 541-550
Lower bounds to the accuracy of inference on heavy tails
Novak, S. 2014. Lower bounds to the accuracy of inference on heavy tails. Bernoulli. 20 (2), pp. 979-989. https://doi.org/10.3150/13-BEJ512
On exceedances of high levels
Novak, S. and Xia, A. 2012. On exceedances of high levels. Stochastic Processes and their Applications. 122 (2), pp. 582-599. https://doi.org/10.1016/j.spa.2011.09.003
On limiting cluster size distributions for processes of exceedances for stationary sequences.
Borovkov, K. and Novak, S. 2010. On limiting cluster size distributions for processes of exceedances for stationary sequences. Statistics & Probability Letters. 80 (23-24), pp. 1814-1818. https://doi.org/10.1016/j.spl.2010.08.006
Extreme value methods with applications to finance.
Novak, S. 2011. Extreme value methods with applications to finance. Chapman & Hall / CRC Press.
Lower bounds to the accuracy of sample maximum estimation.
Novak, S. 2009. Lower bounds to the accuracy of sample maximum estimation. Theory of stochastic processes. 15(31) (2), pp. 156-161.
Impossibility of consistent estimation of the distribution function of a sample maximum.
Novak, S. 2010. Impossibility of consistent estimation of the distribution function of a sample maximum. Statistics. 44 (1), pp. 25-30. https://doi.org/10.1080/02331880902986497
A remark concerning value-at-risk
Novak, S. 2010. A remark concerning value-at-risk. International Journal of Theoretical and Applied Finance. 13 (4), pp. 507-515. https://doi.org/10.1142/S0219024910005917
On self-normalised sums.
Novak, S. 2000. On self-normalised sums. Mathematical methods of statistics.. 9 (4), pp. 415-436.
Inference on heavy tails from dependent data.
Novak, S. 2002. Inference on heavy tails from dependent data. Siberian advances in mathematics. 12 (2), pp. 73-96.
On self-normalised sums [supplement]
Novak, S. 2002. On self-normalised sums [supplement]. Mathematical methods of statistics.. 11 (2), pp. 256-258.
Evaluating currency risk in emerging markets.
Novak, S., Dalla, V. and Giraitis, L. 2007. Evaluating currency risk in emerging markets. Acta applicandae mathematicae.. 97 (1-3), pp. 163-175.
Advances in extreme value theory with applications to finance.
Novak, S. 2009. Advances in extreme value theory with applications to finance. in: Keene, J. (ed.) New business and finance research developments. New York Nova Science Publishers. pp. 199-251
Measures of financial risks and market crashes.
Novak, S. 2007. Measures of financial risks and market crashes. Theory of stochastic processes. 13 (1-2), pp. 182-193.
A new characterization of the normal law
Novak, S. 2006. A new characterization of the normal law. Statistics & Probability Letters. 77 (1), pp. 95-98.
On Gebelein's correlation coefficient
Novak, S. 2004. On Gebelein's correlation coefficient. Statistics & Probability Letters. 69 (3), pp. 299-303.
Confidence intervals for a tail index estimator
Novak, S. 2000. Confidence intervals for a tail index estimator. in: Franke, J., Stahl, G. and Hardle, W. (ed.) Measuring risk in complex stochastic systems London Springer.
Long head runs and long match patterns.
Novak, S. and Embrechts, P. 2002. Long head runs and long match patterns. in: Sandmann, K. and Schönbucher, P. (ed.) Advances in finance and stochastics: essays in honour of Dieter Sondermann. London Springer. pp. 57-69
Compound poisson approximation for the distribution of extremes
Novak, S., Barbour, A. and Xia, A. 2002. Compound poisson approximation for the distribution of extremes. Advances in applied probability. 34 (1), pp. 223-240. https://doi.org/10.1239/aap/1019160958
Multilevel clustering of extremes
Novak, S. 2002. Multilevel clustering of extremes. Stochastic Processes and their Applications. 97 (1), pp. 59-75. https://doi.org/10.1016/S0304-4149(01)00123-5
On the accuracy of multivariate compound Poisson approximation
Novak, S. 2003. On the accuracy of multivariate compound Poisson approximation. Statistics & Probability Letters. 62 (1), pp. 35-43. https://doi.org/10.1016/S0167-7152(02)00422-4
On self-normalized sums and student's statistic
Novak, S. 2005. On self-normalized sums and student's statistic. Theory of Probability and Its Applications. 49 (2), pp. 336-344. https://doi.org/10.1137/S0040585X97981081
The magnitude of a market crash can be predicted
Novak, S. and Beirlant, J. 2006. The magnitude of a market crash can be predicted. Journal of Banking and Finance. 30 (2), pp. 453-462.