1 / 31

Testing extreme value conditions  an overview and recent approaches

Testing extreme value conditions  an overview and recent approaches. Isabel Fraga Alves CEAUL & DEIO University Lisbon, Portugal Cláudia Neves UIMA & DM University Aveiro, Portugal. Contents. Introduction Preliminaries and notation Testing extremes Parametric Approaches

kerry
Download Presentation

Testing extreme value conditions  an overview and recent approaches

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Testing extreme value conditionsan overview and recent approaches Isabel Fraga AlvesCEAUL & DEIO University Lisbon, PortugalCláudia NevesUIMA & DM University Aveiro, Portugal

  2. Contents • Introduction • Preliminaries and notation • Testing extremes • Parametric Approaches • Annual Maxima (AM) • Peaks Over Threshold (POT) • Largest Observations (LO) • Semi-Parametric Approaches • Testing EV Conditions • PORT approach  Three Tests • A case study  S&P500 data

  3. Introduction • In analysis of extreme large (or small) values it is of relevant importance the model assumptions on the right (or left) tail of the underlying distribution function (d.f.) F to the sample data. • We focus on the problem of extreme large values. By an obvious transformation, the problem of extreme small values is analogous. • Statistical inference about rare events can clearly be deduced only from those observations which are extreme in some sense: • classical Gumbel method of block of annual maxima (AM) • peaks-over-threshold (POT) methods • peaks-over-random-threshold (PORT) methods. • Statistical inference is clearly improved if one make an a priori statistical choice about the more appropriate tail decay for the underlying df: light tails with finite right endpoint orpolynomial exponential • This is supported by Extreme Value Theory (EVT).

  4. Theory and Extreme Values Analysis • Extreme Values Analysis Models for Extreme Values, not central values; modelling the tail of the underlying distribution • Problem:How to make inference beyond the sample data? • One Answer: use techniques based on EVT in such a way that it is possible to make statistical inference about rare events, using only a limited amount of data! • Notation:

  5. Basic Theory – distribution of the Maximum • Gnedenko (1943) • Then [GEV- Generalized Extreme Value] von Mises-Jenkinson Representation

  6. Extreme Value Distributions (maxima) • The GEV(g) incorporates the 3 types:[Fisher-Tippett] • Fréchet: limit for heavy tailed distributions • Weibull: limit for short tailed distributions with • Gumbel: limit for exponential tailed distributions

  7. Parametric aprochesFitting GEV(g) to Anual Maxima (AM) – GUMBEL METHOD • Inclusion of location l and scale dparametersin GEV(g) df tail index (shape) g Block 1 Block 2 Block 3 Block 4 Block 5

  8. Testing problem in GEV(g) The shape parameter g determines the weight of the tail Choice between Gumbel, Weibull or Fréchet or • Van Montfort (1970) • Bardsley (1977) • Otten and Van Montfort (1978) • Tiago de Oliveira (1981) • Gomes (1982) • Tiago de Oliveira (1984) • Tiago de Oliveira and Gomes (1984) • Hosking (1984) • Marohn (1994) • Wang, Cooke, and Li (1996) • Marohn (2000)

  9. Heavy Tail • Pareto: bounded support • Beta: • Exponential: Exponential tail Generalized Pareto distribution GP(g) • GP(g) df includes the models:

  10. Excesses over high thresholds – POT ( Peaks Over Thresholds ) • Balkema-de Haan’74+Pickands’75 u

  11. Testing problem in GP(g) The shape parameter g determines the weight of the tail Choice between Exponential, Beta or Pareto or • Fitting GPdf to data • Castillo and Hadi (1997) • Goodness-of-fit tests for GPdf model • Choulakian and Stephens (2001) • Goodness-of-fit problem heavy tailed Pareto-type dfs • Beirlant, de Wet and Goegebeur (2006) • Fitting GPdf to data • Castillo and Hadi (1997) • Goodness-of-fit tests for GPdf model • Choulakian and Stephens (2001) • Goodness-of-fit problem heavy tailed Pareto-type dfs • Beirlant, de Wet and Goegebeur (2006) • Van Montfort and Witter (1985) • Gomes and Van Montfort (1986) • Brilhante (2004) • Marohn (2000) AM & POT

  12. LO (Larger Observations) k largest observations of the sample: are modeledbyjoint pdf GEV(g) - extremal process

  13. Testing problem in GEV(g) GEV(g)-extremal process The shape parameter g determines the weight of the tail Choice between Gumbel, Weibull or Fréchet or • Gomes and Alpuim (1986) • Gomes (1989) LO & AM • Goodness-of-fit tests • Gomes (1987)

  14. Semi-Parametric Approach– Upper Order Statistics upper intermediate o.s.

  15. Peaks Over Random Threshold - PORT Excesses Over Random Threshold

  16. Testing Problem: Max-Domains of Attraction The shape parameter g determines the weight of the tail Choice between Domains of Attraction or • Galambos (1982) • Castillo, Galambos and Sarabia (1989) • Hasofer and Wang (1992) • Falk (1995) • Fraga Alves and Gomes (1996) • Fraga Alves (1999) • Marohn (1998a,b) • Segers and Teugels (2000) • PORT approach • Neves, Picek and Fraga Alves(2006) • Neves and Fraga Alves (2006)

  17. Testing EV conditions upper intermediate o.s. • Adapted Goodness-of-fit tests • (Kolmogorov-Smirnov & Cramér-von Mises type) • Dietrich, de Haan and Husler (2002) • Drees, de Haan and Li (2006)

  18. PORT approach  Three Tests for Largest Observations Excesses over the Random Threshold • Define the r-Moment of Excesses

  19. NPFA test statistic:Ratio between the Maximum and the Mean of Excesses Neves, Picek & FragaAlves ‘06 • The distribution does NOT depend on the location and scale • Motivation: different behaviour of the ratio between the maximum and the mean for light and heavy tails

  20. Gt test statistic:Greenwood-type Statistic (Neves & FragaAlves ‘06) • The distribution does NOT depend on the location and scale • Motivation: based on the statistic Greenwood ’46

  21. HW - test statistic:Hasofer and Wang Statistic (Hasofer & Wang ’92; Neves & FragaAlves ‘06) • The distribution does NOT depend on the location and scale • Motivation: based on goodness-of-fit statistic Shapiro-Wilk ’65

  22. Gumbel quantile NPFA - Test at asymptotic level a under H0 + extra second order conditions on the upper tail of F + extra conditions on convergence rate of k to infinity Reject H0(light tails) in favour of H1 (bilateral) if: Reject H0(light tails) in favour of H1 (heavy tails) if: Reject H0(light tails) in favour of H1 (short tails) if:

  23. e - Normal quantile Reject H0(light tails) in favour of H1 (bilateral) if: Reject H0(light tails) in favour of H1 (heavy tails) if: Reject H0(light tails) in favour of H1 (short tails) if: Gt & HW - Tests at asymptotic level a under H0 + extra second order conditions on the upper tail of F + extra conditions on convergence rate of k to infinity

  24. Exact Properties of NPFA, GT & HW - Tests An extensive simulation study concerning the proposed procedures, allows us to conclude that: • The Gt-test is shown to good advantage when testing the presence of heavy-tailed distributions is in demand. • While the Gt-test barely detects small negative values of g, the HW-test is the most powerful test under study concerning alternatives in the Weibulldomain of attraction. • Since the NPFA- test based on the very simple Tn-statistic tends to be a conservative test and yet detains a reasonable power, this test proves to be a valuablecomplement to the remainder procedures.

  25. Financial data: stock index log-returns • EVT offers a powerful framework to characterize financial market crashes and booms. • The exact distribution of financial returns remains an open question. • Heavy tails are consistent with a variety of financial theories. • In financial studies, the following question is relevant: • are return distributions symmetric in the tails? • Differences in the behavior of extreme positive and negative tail movements within the same market constitute a point of investigation. • The aforementioned tests can be seen as a first test for symmetry between the positive and negative tails of thelog-returns of some stock index.

  26. S&P500: left and right tails of stock index log-returns • S&P500 data: n=6985 observations series of closing prices, {Si , i = 1, … , n} of S&P500 stock index taken from 4 January, 1960 up to Friday, 16 October, 1987 (the last trading day before the crash of Black Monday, October 19, 1987 ), from which we use the daily log-returns (assumed to be stationary and weakly dependent). • Study left tail of the distribution of the returns: negative log-returns, i.e., Li := - log (Si+1/ Si) , i = 1,…, n -1. • Study right tail of the distribution of the returns: positive log-returns, defined as Xi := log (Si+1/ Si)= -Li , i = 1,…, n -1.

  27. S&P500: percentage log-returnsXi := log (Si+1/ Si )

  28. Sample paths of the statistics T*, R* and W*,plotted against k = 5, … , 1200, applied to S&P500: negative log-returns Li := - log (Si+1/ Si) NPFA-test Gt-test HW-test

  29. Sample paths of the statistics T*, R* and W*,plotted against k = 5, … , 1200, applied to S&P500: positive log-returnsXi := log (Si+1/ Si) Gt-test NPFA-test HW-test

  30. S&P500: left and right tails of stock index log-returns • NPFA, HW and Gt testing procedures under the PORT approach yielded the sample paths plots presented. • This analysis suggests the consideration of the Fréchet and Gumbeldomains of attraction, respectively, for the left and righttails of the returns distribution. • This may have the following interpretation: in this stock index the crashes are much more likely than large gain values.

  31. Main References • Neves, C., Picek, J. and Fraga Alves, M.I. (2006). Contribution of the maximum to the sum of excesses for testing max-domains of attraction. JSPI, 136, 4, 1281-1301. • Neves, C. and Fraga Alves, M.I. (2006). Semi-parametric Approach to Hasofer-Wang and Greenwood Statistics in Extremes. To appear in TEST.

More Related