| 1. |
- Erdtman, Elias, 1986-
(författare)
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Change point detection with respect to variance
- 2023
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis examines a simple method for detecting a change with respect to the variance in a sequence of independent normally distributed observations with a constant mean. The method filters out observations with extreme values and divides the sequence into equally large subsequences. For each subsequence, the count of extreme values is translated to a binomial random variable which is tested towards the expected number of extremes. The expected number of extremes comes from prior knowledge of the sequence and a specified probability of how common an extreme value should be. Then specifying the significance level of the goodness-of-fit test yields the number of extreme observations needed to detect a change. The approach is extended to a sequence of independent multivariate normally distributed observations by transforming the sequence to a univariate sequence with the help of the Mahalanobis distance. Thereafter it is possible to apply the same approach as when working with a univariate sequence. Given that a change has occurred, the distribution of the Mahalanobis distance of a multivariate normally distributed random vector with zero mean is shown to approximately follow a gamma distribution. The parameters for the approximated gamma distribution depend only on Σ1−1/2 Σ2Σ1−1/2 with Σ1 and Σ2 being the covariance matrices before and after the change has occurred. In addition to the proposed approach, other statistics such as the largest eigenvalue, the Kullback-Leibler divergence, and the Bhattacharyya distance are considered.
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| 2. |
- Fredriksson, Caroline, et al.
(författare)
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STATISTICAL ANALYSIS OF EXTREME SEA WATER LEVELS AT THE FALSTERBO PENINSULA, SOUTH SWEDEN
- 2016
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Ingår i: Vatten: tidskrift för vattenvård /Journal of Water Management and research. - 0042-2886. ; 2016:2, s. 129-144
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Falsterbo Peninsula on the south coast of Sweden is low-lying and exposed to flooding. In 1872 the extreme storm surge Backafloden caused large devastation on the Swedish, Danish, and German coasts in the South Baltic Sea. For the Falsterbo Peninsula, the peak storm surge level is estimated to have been 240 cm above normal. If a similar event happened today, the consequences would be far worse, as extensive flood prone areas have been developed since 1872. Due to climate change, the mean sea level is expected to rise and increase the flood risk unless preventive measures are taken. This paper discusses the occurrence of extreme sea levels at the Falsterbo Peninsula and design levels for coastal protection. Sea level observations from Skanör, Klagshamn, and Ystad are analysed with General Extreme Value and General Pareto Distribution models to estimate sea levels with return periods of 100–500 years. The estimated return period of the 1872 event, based on these models, resulted in an unrealistically low probability. To better understand the statistical behaviour of extreme storm surges of this magnitude on the Swedish south coast, development of more advanced statistical models will be required.
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| 3. |
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| 4. |
- Hall, P, et al.
(författare)
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Effect of extrapolation on coverage accuracy of prediction intervals computed from Pareto-type data
- 2002
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Ingår i: Annals of Statistics. - : Institute of Mathematical Statistics. - 0090-5364. ; 30:3, s. 875-895
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Tidskriftsartikel (refereegranskat)abstract
- A feature that distinguishes extreme-value contexts from more conventional statistical problems is that in the former we often wish to make predictions well beyond the range of the data. For example, one might have a 10-year sequence of observations of a phenomenon, and wish to make forecasts for the next 20 to 30 years. It is generally unclear how such long ranges of extrapolation affect prediction. In the present paper, and for extremes from a distribution with regularly varying tails at infinity, we address this problem. We approach it in two ways: first, from the viewpoint of predictive inference under a model that is admittedly only approximate, and where the errors of greatest concern are caused by the interaction of long-range extrapolation with model misspecification; second, where the model is accurate but errors arise from a combination of extrapolation and the fact that the method is only approximate. In both settings we show that, in a way which can be defined theoretically and confirmed numerically, one can make predictions exponentially far into the future without committing serious errors.
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| 5. |
- Hall, Peter, et al.
(författare)
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Locating lines among scattered points
- 2006
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Ingår i: Bernoulli. - 1350-7265. ; 12:5, s. 821-839
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Tidskriftsartikel (refereegranskat)abstract
- Consider a process of events on a line L, where, for the most part, the events occur randomly in both time and location. A scatterplot of the pair that represents position on the line, and occurrence time, will resemble a bivariate stochastic point process in a plane, P say. If, however, some of the points on L arise through a more regular phenomenon which travels along the line at an approximately constant speed, creating new points as it goes, then the corresponding points in P will occur roughly in a straight line. It is of interest to locate such lines, and thereby identify, as nearly as possible, the points on L which are associated with the (approximately) constant-velocity process. Such a problem arises in connection with the study of seismic data, where L represents a fault-line and the constant-velocity process there results from the steady diffusion of stress. We suggest methodology for solving this needle-in-a-haystack problem, and discuss its properties. The technique is applied to both simulated and real data. In the latter case it draws particular attention to events occurring along the San Andreas fault, in the vicinity of Parkville, California, on 5 April 1995.
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| 6. |
- Hall, P, et al.
(författare)
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Permutation tests for equality of distributions in high-dimensional settings
- 2002
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Ingår i: Biometrika. - : Oxford University Press (OUP). - 0006-3444 .- 1464-3510. ; 89:2, s. 359-374
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by applications in high-dimensional settings, we suggest a test of the hypothesis H-0 that two sampled distributions are identical. It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite-dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value, and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances.
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| 7. |
- Hall, P, et al.
(författare)
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Prediction regions for bivariate extreme events
- 2004
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Ingår i: Australian & New Zealand Journal of Statistics. - 1467-842X. ; 46:1, s. 99-112
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Tidskriftsartikel (refereegranskat)abstract
- This paper suggests using a mixture of parametric and non-parametric methods to construct prediction regions in bivariate extreme-value problems. The non-parametric part of the technique is used to estimate the dependence function, or copula, and the parametric part is employed to estimate the marginal distributions. A bootstrap calibration argument is suggested for reducing coverage error. This combined approach is compared with a more parametric one, relative to which it has the advantages of being more flexible and simpler to implement. It also enjoys these features relative to predictive likelihood methods. The paper shows how to construct both compact and semi-infinite bivariate prediction regions, and it treats the problem of predicting the value of one component conditional on the other. The methods are illustrated by application to Australian annual maximum temperature data.
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| 8. |
- Hallin, Caroline, et al.
(författare)
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EXTREME VALUE ANALYSIS OF WAVE RUNUP AND DUNE EROSION AT ÄNGELHOLM BEACH, SOUTH SWEDEN : EXTREMVÄRDESANALYS AV VÅGUPPSPOLNING OCH DYNEROSION VID ÄNGELHOLMS STRAND
- 2019
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Ingår i: Vatten: tidskrift för vattenvård /Journal of Water Management and research. - 0042-2886. ; 75:2, s. 227-240
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- I den här studien används extremvärdesanalys för att beräkna våguppspolning och årlig dynerosion med 10, 50, och 100 års återkomsttid. Frekvensanalysen baseras på en 40 år lång tidsserie med simulerad dynerosion och våguppspolning från Ängelholms strand i Skälderviken. Med SMB-metoden beräknas vågor ute till havs som sedan transformeras till kustnära vågor i vågmodellen SWAN. Våguppspolningen beräknas med en formel av Hunt-typ och dynerosionen med en semi-empirisk formel som relaterar erosionsvolymen till vågornas kraft mot dynen. Resultatet visar att dynhöjden längs den största delen av stranden är högre än den mest extrema beräknade våguppspolningen. Dynerna längs stranden skyddar ett bakomliggande bostadsområde mot översvämning och erosion. Beräkningarna visar att dynvolymen idag skyddar mot skador som kan uppkomma vid extrema stormar. Emellertid har inga långtidsförändringar av dynvolymen till följd av stigande havsnivåer eller gradienter i den kustparallella sedimenttransportenbeaktats i denna studie.
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| 9. |
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| 10. |
- Pal, Rakonczai, et al.
(författare)
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On Prediction of Bivariate Extremes
- 2010
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Ingår i: International Journal of Intelligent Technologies and Statistics. - 2218-3434. ; 3:2, s. 115-139
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Tidskriftsartikel (refereegranskat)
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| 11. |
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| 12. |
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| 13. |
- Raimondo, M, et al.
(författare)
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A peaks over threshold model for change-point detection by wavelets
- 2004
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Ingår i: Statistica Sinica. - 1017-0405. ; 14:2, s. 395-412
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Tidskriftsartikel (refereegranskat)abstract
- Newly available wavelet bases on multi-resolution analysis have exciting implications for detection of change-points. By checking the absolute value of wavelet coefficients one call detect discontinuities in ail otherwise smooth curve even in the presence of additive noise. In this paper, we combine wavelet methods and extreme value theory to test the presence of ail arbitrary number of discontinuities in an unknown function observed with noise. Our approach is based on a Peaks Over Threshold modelling of noisy wavelet transforms. Particular features of our method include the estimation of the extreme value index in the tail of the noise distribution. The critical region of our test is, derived using a Generalised Pareto Distribution approximation to normalised sums. Asymptotic results show that our method is powerful in a wide range of medium size wavelet frequencies. We compare our test with competing approaches on simulated examples and illustrate the method on Dow-Jones data.
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| 14. |
- Rootzén, Holger, et al.
(författare)
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Can losses caused by wind storms be predicted from meteorological observations
- 2001
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Ingår i: Scandinavian Actuarial Journal. - : Informa UK Limited. - 0346-1238 .- 1651-2030. ; 2001:2, s. 162-175
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Tidskriftsartikel (refereegranskat)abstract
- This paper contains a study of the extent to which aggregate losses due to severe wind storms can be explained by wind measurements. The analysis is based on 12 years of data for a region, Ska § ne, in southern Sweden. A previous investigation indicated that wind measurements from six recording stations in Ska § ne was insufficient to obtain accurate prediction. The present study instead uses geostrophic winds calculated from pressure readings, at a regular grid of size 50 kilometres over Ska § ne. However, also this meteorological data set is seen to be insufficient for accurate prediction of insurance risk. The results indicate that currently popular methods of evaluating wind storm risks from meteorological data should not be used uncritically by insurers or reinsurers. Nevertheless, wind data does contain some information on insurance. risks. There is a need for further research on how to use this information to improve risk assessment.
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| 15. |
- Rootzén, Holger, et al.
(författare)
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Multivariate generalized Pareto distributions
- 2006
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Ingår i: Bernoulli. - 1350-7265. ; 12:5, s. 917-930
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Tidskriftsartikel (refereegranskat)abstract
- Statistical inference for extremes has been a subject of intensive research over the past couple of decades. One approach is based on modelling exceedances of a random variable over a high threshold with the generalized Pareto (GP) distribution. This has proved to be an important way to apply extreme value theory in practice and is widely used. We introduce a multivariate analogue of the GP distribution and show that it is characterized by each of following two properties: first, exceedances asymptotically have a multivariate GP distribution if and only if maxima asymptotically are extreme value distributed; and second, the multivariate GP distribution is the only one which is preserved under change of exceedance levels. We also discuss a bivariate example and lower-dimensional marginal distributions.
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| 16. |
- Tajvidi, Nader, et al.
(författare)
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A general approach to generate random variates for multivariate copulae
- 2018
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Ingår i: Australian & New Zealand Journal of Statistics. - : Wiley. - 1467-842X .- 1369-1473. ; 60:1, s. 140-155
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Tidskriftsartikel (refereegranskat)abstract
- We suggest two methods for simulating from a multivariate copula in an arbitrary dimension. Although our main emphasis in this paper is on multivariate extreme value distributions, the proposed methods can be applied to any copula. The basic idea is to approximate the (unknown) density of the copula by a distribution that has a piece‐wise constant (histogram) density. This is achieved by partitioning the support of a given copula C into a large number of hyper‐rectangles and using them to generate random variates from an approximation of the copula. We suggest two methods for finding this approximation which correspond to either finding hyper‐rectangles which have equal probability mass with respect to C, or determining a partition using hyper‐squares of equal volume and finding the corresponding probability mass of each hyper‐square. We also discuss how the generated random variates can be used as proposals in a Metropolis–Hastings algorithm, when C is an absolutely continuous distribution function, to generate a sequence of random variates from C. An implementation of the proposed methodologies is provided for the statistical computing and graphics environment R in our package called SimCop.
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| 17. |
- Tajvidi, Nader, et al.
(författare)
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Behavior of Extreme Dependence between Stock Markets when the Regime Shifts
- 2015
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Ingår i: Sri Lankan Journal of Applied Statistics. - : Sri Lanka Journals Online (JOL). - 2424-6271 .- 1391-4987. ; 16:1, s. 21-40
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Tidskriftsartikel (populärvet., debatt m.m.)abstract
- We propose a methodology based on multivariate extreme value theory, to analyze the dependence between markets during the financial crisis. We argue that extreme dependence based on block maximum is a more appropriate measure to study dependence between stock markets, when a regime shifts, than other alternatives. With this methodology, we are able to detect the increase in the extreme dependences between US and other markets during the 2008 financial crisis where traditional approaches fail to do so. In addition, the estimated dependent function allows one to quantify maximum impact of the crisis on each individual market. We then propose the use of a conditional loss distribution as a constructive tool for a stress test analysis in risk management study. Stress test levels with respect to 2008 financial data calculated from the conditional loss distribution are given.
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| 18. |
- Tajvidi, Nader
(författare)
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Characterisation and Some Statistical Aspects of Univariate and Multivariate Generalised Pareto Distributions
- 1996
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Extreme value theory is about the distributions of very large or very small values in a time series or stochastic process. This has numerous applications connected with environmental science, civil engineering, materials science and insurance. A rather recent approach for modelling extreme events is the so called peak over threshold (POT) method. The generalised Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. This thesis consists of three papers. The main focus is on some theoretical and applied statistical issues of univariate and multivariate extreme value modelling. In the first paper we compare the empirical coverage of standard bootstrap and likelihood-based confidence intervals for the parameters and 90\%-quantile of the GPD. By applying a general method of D. N. Lawley, small sample correction factors for likelihood ratio statistics of the parameters and quantiles of the GPD have been calculated. The article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD. In the second paper we give a multivariate analogue of the GPD and consider estimation of parameters in some specific bivariate generalised Pareto distributions (BGPD's). We generalise two of existing bivariate extreme value distributions and study maximum likelihood estimation of parameters in the corresponding BGPD's. The procedure is illustrated with an application to a bivariate series of wind data. The main interest in the thesis has been on practicality of the methods so when a new method has been developed, it's performance has been studied with the help of both real life data and simulations. In the third paper we use three previous articles as examples to illustrate difficulties which might arise in application of the theory and methods which may be used to solve them. A common theme in these articles is univariate and multivariate generalised Pareto distributions. However, the discussed problems are of a rather general nature and demonstrate some typical tasks in applied statistical research. We also discuss a general approach to design and implementation of statistical computations.
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| 19. |
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| 20. |
- Tajvidi, Nader
(författare)
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On approximating dependence function and its derivatives
- 2024
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Ingår i: Extremes. - 1572-915X. ; 27:3, s. 479-503
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Tidskriftsartikel (refereegranskat)abstract
- Bivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions.
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