Variability in Modified Estimators of VaR and ES

Thesis (Master's)--University of Washington, 2016-06

Bayesian estimation of g-and-k distributions using MCMC

In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of the g-and-κ distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach, The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.

Using extreme value theory to measure value-at-risk for daily electricity spot prices

The recent deregulation in electricity markets worldwide has heightened the importance of risk management in energy markets. Assessing Value-at-Risk (VaR) in electricity markets is arguably more difficult than in traditional financial markets because the distinctive features of the former result in a highly unusual distribution of returns-electricity returns are highly volatile, display seasonalities in both their mean and volatility, exhibit leverage effects and clustering in volatility, and feature extreme levels of skewness and kurtosis. With electricity applications in mind, this paper proposes a model that accommodates autoregression and weekly seasonals in both the conditional mean and conditional volatility of returns, as well as leverage effects via an EGARCH specification. In addition, extreme value theory (EVT) is adopted to explicitly model the tails of the return distribution. Compared to a number of other parametric models and simple historical simulation based approaches, the proposed EVT-based model performs well in forecasting out-of-sample VaR. In addition, statistical tests show that the proposed model provides appropriate interval coverage in both unconditional and, more importantly, conditional contexts. Overall, the results are encouraging in suggesting that the proposed EVT-based model is a useful technique in forecasting VaR in electricity markets. (c) 2005 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Convergence analysis of a discrete-time single-unit gradient ICA algorithm

We revisit the one-unit gradient ICA algorithm derived from the kurtosis function. By carefully studying properties of the stationary points of the discrete-time one-unit gradient ICA algorithm, with suitable condition on the learning rate, convergence can be proved. The condition on the learning rate helps alleviate the guesswork that accompanies the problem of choosing suitable learning rate in practical computation. These results may be useful to extract independent source signals on-line.

Non-normally distributed data and non-parametric statistics

Different types of numerical data can be collected in a scientific investigation and the choice of statistical analysis will often depend on the distribution of the data. A basic distinction between variables is whether they are ‘parametric’ or ‘non-parametric’. When a variable is parametric, the data come from a symmetrically shaped distribution known as the ‘Gaussian’ or ‘normal distribution’ whereas non-parametric variables may have a distribution which deviates markedly in shape from normal. This article describes several aspects of the problem of non-normality including: (1) how to test for two common types of deviation from a normal distribution, viz., ‘skew’ and ‘kurtosis’, (2) how to fit the normal distribution to a sample of data, (3) the transformation of non-normally distributed data and scores, and (4) commonly used ‘non-parametric’ statistics which can be used in a variety of circumstances.

Development of a near activity visual questionnaire to assess accommodating intraocular lenses

Purpose: To develop a questionnaire that subjectively assesses near visual function in patients with 'accommodating' intraocular lenses (IOLs). Methods: A literature search of existing vision-related quality-of-life instruments identified all questions relating to near visual tasks. Questions were combined if repeated in multiple instruments. Further relevant questions were added and item interpretation confirmed through multidisciplinary consultation and focus groups. A preliminary 19-item questionnaire was presented to 22 subjects at their 4-week visit post first eye phacoemulsification with 'accommodative' IOL implantation, and again 6 and 12 weeks post-operatively. Rasch Analysis, Frequency of Endorsement, and tests of normality (skew and kurtosis) were used to reduce the instrument. Cronbach's alpha and test-retest reliability (intraclass correlation coefficient, ICC) were determined for the final questionnaire. Construct validity was obtained by Pearson's product moment correlation (PPMC) of questionnaire scores to reading acuity (RA) and to Critical Print Size (CPS) reading speed. Criterion validity was obtained by receiver operating characteristic (ROC) curve analysis and dimensionality of the questionnaire was assessed by factor analysis. Results: Rasch Analysis eliminated nine items due to poor fit statistics. The final items have good separation (2.55), internal consistency (Cronbach's α = 0.97) and test-retest reliability (ICC = 0.66). PPMC of questionnaire scores with RA was 0.33, and with CPS reading speed was 0.08. Area under the ROC curve was 0.88 and Factor Analysis revealed one principal factor. Conclusion: The pilot data indicates the questionnaire to be internally consistent, reliable and a valid instrument that could be useful for assessing near visual function in patients with 'accommodating' IOLS. The questionnaire will now be expanded to include other types of presbyopic correction. © 2007 British Contact Lens Association.

Breaking down the non-normality of daily stock returns

This paper investigates whether the non-normality typically observed in daily stock-market returns could arise because of the joint existence of breaks and GARCH effects. It proposes a data-driven procedure to credibly identify the number and timing of breaks and applies it on the benchmark stock-market indices of 27 OECD countries. The findings suggest that a substantial element of the observed deviations from normality might indeed be due to the co-existence of breaks and GARCH effects. However, the presence of structural changes is found to be the primary reason for the non-normality and not the GARCH effects. Also, there is still some remaining excess kurtosis that is unlikely to be linked to the specification of the conditional volatility or the presence of breaks. Finally, an interesting sideline result implies that GARCH models have limited capacity in forecasting stock-market volatility.

Early warnings of heart rate deterioration

Hospitals can experience difficulty in detecting and responding to early signs of patient deterioration leading to late intensive care referrals, excess mortality and morbidity, and increased hospital costs. Our study aims to explore potential indicators of physiological deterioration by the analysis of vital-signs. The dataset used comprises heart rate (HR) measurements from MIMIC II waveform database, taken from six patients admitted to the Intensive Care Unit (ICU) and diagnosed with severe sepsis. Different indicators were considered: 1) generic early warning indicators used in ecosystems analysis (autocorrelation at-1-lag (ACF1), standard deviation (SD), skewness, kurtosis and heteroskedasticity) and 2) entropy analysis (kernel entropy and multi scale entropy). Our preliminary findings suggest that when a critical transition is approaching, the equilibrium state changes what is visible in the ACF1 and SD values, but also by the analysis of the entropy. Entropy allows to characterize the complexity of the time series during the hospital stay and can be used as an indicator of regime shifts in a patient’s condition. One of the main problems is its dependency of the scale used. Our results demonstrate that different entropy scales should be used depending of the level of entropy verified.

Essays on emissions trading markets

This dissertation is a collection of three economics essays on different aspects of carbon emission trading markets. The first essay analyzes the dynamic optimal emission control strategies of two nations. With a potential to become the largest buyer under the Kyoto Protocol, the US is assumed to be a monopsony, whereas with a large number of tradable permits on hand Russia is assumed to be a monopoly. Optimal costs of emission control programs are estimated for both the countries under four different market scenarios: non-cooperative no trade, US monopsony, Russia monopoly, and cooperative trading. The US monopsony scenario is found to be the most Pareto cost efficient. The Pareto efficient outcome, however, would require the US to make side payments to Russia, which will even out the differences in the cost savings from cooperative behavior. The second essay analyzes the price dynamics of the Chicago Climate Exchange (CCX), a voluntary emissions trading market. By examining the volatility in market returns using AR-GARCH and Markov switching models, the study associates the market price fluctuations with two different political regimes of the US government. Further, the study also identifies a high volatility in the returns few months before the market collapse. Three possible regulatory and market-based forces are identified as probable causes of market volatility and its ultimate collapse. Organizers of other voluntary markets in the US and worldwide may closely watch for these regime switching forces in order to overcome emission market crashes. The third essay compares excess skewness and kurtosis in carbon prices between CCX and EU ETS (European Union Emission Trading Scheme) Phase I and II markets, by examining the tail behavior when market expectations exceed the threshold level. Dynamic extreme value theory is used to find out the mean price exceedence of the threshold levels and estimate the risk loss. The calculated risk measures suggest that CCX and EU ETS Phase I are extremely immature markets for a risk investor, whereas EU ETS Phase II is a more stable market that could develop as a mature carbon market in future years.

An Assessment of the Performances of Several Univariate Tests of Normality

The importance of checking the normality assumption in most statistical procedures especially parametric tests cannot be over emphasized as the validity of the inferences drawn from such procedures usually depend on the validity of this assumption. Numerous methods have been proposed by different authors over the years, some popular and frequently used, others, not so much. This study addresses the performance of eighteen of the available tests for different sample sizes, significance levels, and for a number of symmetric and asymmetric distributions by conducting a Monte-Carlo simulation. The results showed that considerable power is not achieved for symmetric distributions when sample size is less than one hundred and for such distributions, the kurtosis test is most powerful provided the distribution is leptokurtic or platykurtic. The Shapiro-Wilk test remains the most powerful test for asymmetric distributions. We conclude that different tests are suitable under different characteristics of alternative distributions.

Documentation and sedimentology of core PS1021-1 from the Antarctic continental margin off Kap Norvegia

Sedimentological analyses concerning ice rafted debris, grain size distribution, biogenous components, and clay mineral composition of four sediment cores from the Antarctic continental margin off Kapp Norvegia reveal a cyclical pattern of three different sediment facies. These are classified into warm and cold types representing warm and cold climatic periods and a short transition period from cold to warm events. The sedimentological parameters reflect the variations within the cryosphere and the hydrosphere, which are directly influenced by the climatic fluctuations. The unusually high content of carbonaceous planktonic and benthonic foraminifera in these polar sediments, as well as the interfingering of terrigeneous and biogeneous-rich sediments with increasing distance from the continent, might reflect the influence of the Weddell Sea Polynya and the oscillations of polynya, pack-ice and ice shelf extent during the late Pleistocene.

Sedimentology of the northwestern Weddel Sea continental shelf and slope

Geologie cores on two profiles oriented normaly to the continental shelf and slope, have been investigated to reconstruct the Quaternary sedimentary history of the southeast continental border of South Orkney (NW Weddell Sea). The sediments were described macroscopically and their fabric investigated by use of X-radiographs. Laboratory work comprised detailed grain-size analysis, determination of the watercontent, carbonate, organic carbon and sand fraction.composition. Stable oxygen and carbon isotopes have been measured On planktonic foraminifera. Palaeomagnetism, analysis of 230Th-content and detailed comparison of the lithlogic Parameters with the oxygen isotope stages (Martinson curve) were used for stratigraphic classification of the sediments. The sediment cores from the continental slope comprise a maximum age of 300,000 years B. P.. Bottom currents, ice rafting and biogenic input are the main sources of sediment. Based on lithologic parameters a distinction between glacial and interglacial facies is possible. Silty clays without microfossils and few bioturbation characterise the sediments of the glacial facies. Only small amounts of icerafted debris can be recognized. This type of sediment was accumulated during times of lower sea-level and drastically reduced rate of bottom water production. Based on grain-size distribution, bottom current velocities of 0.01 cmls were calculated. Thick sea-ice coverage reduced biogenic production in the surface water, and as consequence benthic communities were depleted. Because of the reduced benthic life, sediments are only slithly bioturbated. At the beginning of the interglacial Stage, the sea-level rised rapidly, and calving rate of icebergs, combined with input of ice-rafted material, increased considerably. Sediments of this transition facies are silty cliiys with a high proportion of coarse ice-rafted debris, but without microfossils. With the onset of bottom water production in connection with shelf ice water, sediments of interglacial facies were formed. They consist of silty clays to clayey silts with considerable content of sand and gravel. Sediments are strongly bioturbated. Based On the sediment caracteristics, current velocities of the bottom water were calculated to be of 0.96 cmls for interglacials. At the southern slope of a NW/SE-striking ridge, bottom water current is channelized, resulting in a drastic increase of current velocities. Current velocities up to 7.5 cm/s lead to formation of residual sediments. While the continental slope has predominantly fine sediments, the South Orkney shelf are mainly sandy silts and silty sands with a high proportion of gravel. These sediments were formed dominantly by ice-rafting during Brunhes- and Matuyama-Epoch. Currents removed the fine fraction of the sediments. Based on microfossil contents it was not possible to differentiate sediments from glacial to interglacial. In the upper Parts of the cores graded sequences truncated by erosion were observed. These sequences were formed during Brunhes-Epoch by strong currents with velocities decreasing periodically from about 7.5 cm/s to about 1 cm/s. Sediments with a high proportion of siliceous microfossils but barren of foraminifera compose the lower part of the shelf cores. These sediments have formed during the warmer Matuyama-Epoch.

Sedimentology on two cores from Scoresby Sund, Greenland

Reconstruction of the postglacial palaeoenvironmental evolution was the main objective of marine geological investigations in the Scorcsby Sund fjord system. For this purpose, samples of marine sediments, taken on RV Polarstern cruises ARK-V/3b and ARK-VII/3b in 1988 and 1990, have been analysed. All investigated fjord sediments are paratills. However, remarkable changes in sediment fabric and composition occur with depth in cores. They are attributable to different modes of sediment deposition. Therefore, a subdivision of the postglacial palaeoenvironmental history into periods of considerably different sedimentary conditions is feasible. The change of sedimentary fades with time is interpreted by deposition under changing climatic conditions during the postglacial. Displacements of cyclonic and anticyclonic centers in the atmosphere change amount of precipitation at the east coast of Greenland. Precipitation strongly influences extension of local ice caps of coastal areas and duration of coverage of the fjords by sea ice. These factors again control the sedimentary regime in the fjord system.

Sedimentology on cores off Filchner Shelf Ice, Weddell Sea, Antarctica

The sediments of 14 box cores and 7 gravity cores, mainly taken directly in front of the Filchner(-Ronne) ice shelf northwest of Berkner Island (Weddell Sea), allowed to distinguish six sediment types. On the one hand,the retreat of the at first grounded and then floated ice from the last glacial maximum is documented. On the other hand,the sediments give an insight into extensive Holocene sediment deposition and remobilization northwest of Berkner Island. The ortho till was deposited directly by the grounded ice sheet and is lacking any marine influence. After floating of the ice shelf, partly very weIl stratified, partly unstratified, non-bioturbated paratill is deposited beneath the ice shelf. Lack of IRD-content in the paratill immediately above the orthotill indicates freezing at the bottom of the ice, at least for a short period after the ice became afloat. The orthotill and paratill contain small amounts of fragmented Tertiary diatoms, which allow the conclusion, that glacial-marine sediments in the accumulation area of the Ronne ice shelf will be eroded and later deposited by ice in the investigation area. Starting of bioturbation and therefore change in sedimentation from paratill to bioturbated paratill,is caused by the retreat of the ice shelf to its actual position. Isostatic uplift of the sea-bed after the Ice Age causes minor water depths with higher current velocities. The fine-fraction is eroding and mean particle-size will increase. Maybe, also isostatic uplift is responsible for repeated great advances of the floated ice shelf as shown in an erosional horizon in some cores containing bioturbated paratill. Postglacial sediment-thicknesses exceed 3 m. Assuming floating of the ice 15.000 YBP, accumulation rates reach nearly 20cm/lOOO years. Following the theories about sediment input in front of wide ice shelves, this was not expected. In the shallower water depths of Berkner Bank, the oscillations of the ice shelf are recorded in the sediments. Sorting and redistribution by high current velocities from beneath the ice up to the calving line, lead to the deposition of the weIl to very weIl sorted sandy till. In front of the calving line the finer fraction will settle down. Remobilization is possible by bioturbation and increasing current-velocity. According to the intensity of mixing of the sandy till with the fine fraction, modified till or muddy till results.