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This thesis is an explorative note on the non-Gaussian innovations of the volatility process. More specifically, the thesis investigates if the decomposition of the Standard Classical Laplace (SCL) distribution to a difference of two exponential is a valid alternative to modelling the asymmetric volatility processes, taking volatility clustering, the leverage effect and asymmetric response in volatility into account. We derive the probability density functions (pdf) and log-likelihood functions for both asymmetric and non-asymmetric case. The pdf exhibit characteristics suitable for modelling the leverage effect, positive kurtosis and fat tails often observed in financial returns, for both the symmetric and asymmetric case. We derive the likelihood function in the non-asymmetric case with regards to the relevant parameters. The likelihood function evaluated concerning past volatility presents a compact solution with the persistence of volatility, whereas, for the other two parameters, the result is complicated. In the asymmetric case, the log-likelihood function is maximized concerning the parameters of interest through simulation. The convergence rate is fast for the AR and lagged volatility parameters, but slower for the asymmetric parameter, which is hard to observe in this setting., This thesis is an explorative note on the non-Gaussian innovations of the volatility process. More specifically, the thesis investigates if the decomposition of the Standard Classical Laplace (SCL) distribution to a difference of two exponential is a valid alternative to modelling the asymmetric volatility processes, taking volatility clustering, the leverage effect and asymmetric response in volatility into account. We derive the probability density functions (pdf) and log-likelihood functions for both asymmetric and non-asymmetric case. The pdf exhibit characteristics suitable for modelling the leverage effect, positive kurtosis and fat tails often observed in financial returns, for both the symmetric and asymmetric case. We derive the likelihood function in the non-asymmetric case with regards to the relevant parameters. The likelihood function evaluated concerning past volatility presents a compact solution with the persistence of volatility, whereas, for the other two parameters, the result is complicated. In the asymmetric case, the log-likelihood function is maximized concerning the parameters of interest through simulation. The convergence rate is fast for the AR and lagged volatility parameters, but slower for the asymmetric parameter, which is hard to observe in this setting.

This thesis is an explorative note on the non-Gaussian innovations of the volatility process. More specifically, the thesis investigates if the decomposition of the Standard Classical Laplace (SCL) distribution to a difference of two exponential is a valid alternative to modelling the asymmetric volatility processes, taking volatility clustering, the leverage effect and asymmetric response in volatility into account. We derive the probability density functions (pdf) and log-likelihood functions for both asymmetric and non-asymmetric case. The pdf exhibit characteristics suitable for modelling the leverage effect, positive kurtosis and fat tails often observed in financial returns, for both the symmetric and asymmetric case. We derive the likelihood function in the non-asymmetric case with regards to the relevant parameters. The likelihood function evaluated concerning past volatility presents a compact solution with the persistence of volatility, whereas, for the other two parameters, the result is complicated. In the asymmetric case, the log-likelihood function is maximized concerning the parameters of interest through simulation. The convergence rate is fast for the AR and lagged volatility parameters, but slower for the asymmetric parameter, which is hard to observe in this setting., This thesis is an explorative note on the non-Gaussian innovations of the volatility process. More specifically, the thesis investigates if the decomposition of the Standard Classical Laplace (SCL) distribution to a difference of two exponential is a valid alternative to modelling the asymmetric volatility processes, taking volatility clustering, the leverage effect and asymmetric response in volatility into account. We derive the probability density functions (pdf) and log-likelihood functions for both asymmetric and non-asymmetric case. The pdf exhibit characteristics suitable for modelling the leverage effect, positive kurtosis and fat tails often observed in financial returns, for both the symmetric and asymmetric case. We derive the likelihood function in the non-asymmetric case with regards to the relevant parameters. The likelihood function evaluated concerning past volatility presents a compact solution with the persistence of volatility, whereas, for the other two parameters, the result is complicated. In the asymmetric case, the log-likelihood function is maximized concerning the parameters of interest through simulation. The convergence rate is fast for the AR and lagged volatility parameters, but slower for the asymmetric parameter, which is hard to observe in this setting.

In 2009, Bitcoin was introduced to the world. Today, ten years later, there are still gaps in the research of how to model the cryptocurrency. In this thesis, the capacities of different volatility models to capture the high volatility of Bitcoin returns are investigated. The models used are GARCH-type models: GARCH(1,1), IGARCH(1,1) and GJR-GARCH(1,1). Jumps, or outliers, are also detected and filtered using two different methods to see how these observations affect the suitability of the models. The volatility of Bitcoin is modelled with other explanatory variables, namely the biggest stock market indices in Europe. The reason for this is to investigate if the volatility of the chosen indices can explain the volatility of Bitcoin. If so, the stock market indices may be used as a tool for forecasting the development of Bitcoin. The volatility of the stock market indices is also modelled with Bitcoin as an explanatory variable. The aim of this analysis is to examine if Bitcoin can be used as a tool for forecasting stock market events. After investigation, there are however no strong evidence that the volatility of Bitcoin can be explained by the volatility of the indices or vice versa. Another conclusion is that the model assumptions are not completely fulfilled regardless of the chosen GARCH-type model when Bitcoin return is used as response variable. The innovations are autocorrelated and seem to follow a Laplace distribution. The modelling of Bitcoin volatility using GARCH-type models must therefore be supplemented, and one possible way of doing this is to filter the jumps. However, neither of the two methods used to detect and filter the jumps are fully satisfying, since, for example, the innovations still are autocorrelated after fitting GARCH-type models to the filtered Bitcoin returns.

This study aims to investigate and model statistical properties of Bitcoin and other major cryptocurrencies. There were recent drastic changes in the level of Bitcoin prices as it moved from $740 in 2014 to $19,187 in 2017, and down to $3,830 in 2018. The current study aims to fill the gap in the analysis of cryptoccurencies, primarily Bitcoin returns statistical process. Specifically, the study selects and estimates a model that traces dynamics of returns using ARMA, and also volatility of the residual from the model. To my knowledge, there is gap in the academic literature for the case of Bitcoin to use such approach. The analysis involves data on past daily prices of Bitcoin, in British Pounds and USD, as well as those of Ethereum and Litecoin, both in Pounds. The results obtained from the study provide evidence that ARMA model in combination with eGARCH volatility model can be used for analysis of statistical process of Bitcoin returns. Moreover, significant statistical differences are identified between Bitcoin prices in UK Pounds and US Dollars. Although there is no evidence that shows the price level has an effect on volatility, significant decline is identified in the volatility of Bitcoin log-returns since 2018 as compared to the previous period. For each considered cryptocurrency, the current study determines the optimal specification of ARMA model, as well as GARCH-type model and optimal statistical distribution of the residual. The set of results can be used to estimate statistical process behind the cryptocurrency historical prices. Moreover, relation between ARMA(p, q) lag order, and type of optimal volatility model and residual distribution is explored, no significant relation is identified.

This study aims to investigate and model statistical properties of Bitcoin and other major cryptocurrencies. There were recent drastic changes in the level of Bitcoin prices as it moved from $740 in 2014 to $19,187 in 2017, and down to $3,830 in 2018. The current study aims to fill the gap in the analysis of cryptoccurencies, primarily Bitcoin returns statistical process. Specifically, the study selects and estimates a model that traces dynamics of returns using ARMA, and also volatility of the residual from the model. To my knowledge, there is gap in the academic literature for the case of Bitcoin to use such approach. The analysis involves data on past daily prices of Bitcoin, in British Pounds and USD, as well as those of Ethereum and Litecoin, both in Pounds. The results obtained from the study provide evidence that ARMA model in combination with eGARCH volatility model can be used for analysis of statistical process of Bitcoin returns. Moreover, significant statistical differences are identified between Bitcoin prices in UK Pounds and US Dollars. Although there is no evidence that shows the price level has an effect on volatility, significant decline is identified in the volatility of Bitcoin log-returns since 2018 as compared to the previous period. For each considered cryptocurrency, the current study determines the optimal specification of ARMA model, as well as GARCH-type model and optimal statistical distribution of the residual. The set of results can be used to estimate statistical process behind the cryptocurrency historical prices. Moreover, relation between ARMA(p, q) lag order, and type of optimal volatility model and residual distribution is explored, no significant relation is identified.

In 2009, Bitcoin was introduced to the world. Today, ten years later, there are still gaps in the research of how to model the cryptocurrency. In this thesis, the capacities of different volatility models to capture the high volatility of Bitcoin returns are investigated. The models used are GARCH-type models: GARCH(1,1), IGARCH(1,1) and GJR-GARCH(1,1). Jumps, or outliers, are also detected and filtered using two different methods to see how these observations affect the suitability of the models. The volatility of Bitcoin is modelled with other explanatory variables, namely the biggest stock market indices in Europe. The reason for this is to investigate if the volatility of the chosen indices can explain the volatility of Bitcoin. If so, the stock market indices may be used as a tool for forecasting the development of Bitcoin. The volatility of the stock market indices is also modelled with Bitcoin as an explanatory variable. The aim of this analysis is to examine if Bitcoin can be used as a tool for forecasting stock market events. After investigation, there are however no strong evidence that the volatility of Bitcoin can be explained by the volatility of the indices or vice versa. Another conclusion is that the model assumptions are not completely fulfilled regardless of the chosen GARCH-type model when Bitcoin return is used as response variable. The innovations are autocorrelated and seem to follow a Laplace distribution. The modelling of Bitcoin volatility using GARCH-type models must therefore be supplemented, and one possible way of doing this is to filter the jumps. However, neither of the two methods used to detect and filter the jumps are fully satisfying, since, for example, the innovations still are autocorrelated after fitting GARCH-type models to the filtered Bitcoin returns.

Historically, the collection of available statistical models for fitting data has been, more or less, restricted to those which are analytically tractable. However, computing power today permits us to use models that, while more complex and often devoid of closed-form distribution or density functions, provide better fits to data. In this thesis, statistical theory, primarily parameter estimation, is developed for three such models: Arnold & Ng's bivariate beta family and its Arnold & Ghosh subfamily of copulas, an 8-parameter family of bivariate Asymmetric Laplace distributions, and a collection of compound random variables. All are distributions whose densities, in general, cannot be written down, but whose realizations can easily be generated via simulation. I apply, and adapt, several methods of likelihood-free statistical inference; including Modified Maximum Likelihood Estimation (MMLE), Approximate Bayesian Computation (ABC), and Markov-Chain Monte-Carlo (MCMC); to achieve various forms of parameter estimates. For each of the models studied, sub-models were identified and cataloged. In doing so, care is taken to assure that a reasonable balance between the dimensionality of the parameter spaces and the flexibility of the resulting models is maintained. Moreover, in one case, a collection of sub-models is formed, each of which permits simple parameter estimation, while one of the models from the collection is chosen to provide the best fit according to a particular metric. This is done to simplify parameter estimation through dimension reduction while maintaining a high level of diversity of available models. In other cases, a more direct approach is taken, where the model is selected based on prior knowledge.

Historically, the collection of available statistical models for fitting data has been, more or less, restricted to those which are analytically tractable. However, computing power today permits us to use models that, while more complex and often devoid of closed-form distribution or density functions, provide better fits to data. In this thesis, statistical theory, primarily parameter estimation, is developed for three such models: Arnold & Ng's bivariate beta family and its Arnold & Ghosh subfamily of copulas, an 8-parameter family of bivariate Asymmetric Laplace distributions, and a collection of compound random variables. All are distributions whose densities, in general, cannot be written down, but whose realizations can easily be generated via simulation. I apply, and adapt, several methods of likelihood-free statistical inference; including Modified Maximum Likelihood Estimation (MMLE), Approximate Bayesian Computation (ABC), and Markov-Chain Monte-Carlo (MCMC); to achieve various forms of parameter estimates. For each of the models studied, sub-models were identified and cataloged. In doing so, care is taken to assure that a reasonable balance between the dimensionality of the parameter spaces and the flexibility of the resulting models is maintained. Moreover, in one case, a collection of sub-models is formed, each of which permits simple parameter estimation, while one of the models from the collection is chosen to provide the best fit according to a particular metric. This is done to simplify parameter estimation through dimension reduction while maintaining a high level of diversity of available models. In other cases, a more direct approach is taken, where the model is selected based on prior knowledge.

In the paper we consider two types of utility functions often used in portfolio allocation problems, i.e. the exponential utility and the quadratic utility. We link the resulting optimal portfolios obtained by maximizing these utility functions to the corresponding optimal portfolios based on the minimum value-at-risk (VaR) approach. This allows us to provide analytic expressions for the risk aversion coefficients as functions of the VaR level. The results are initially derived under the assumption that the vector of asset returns is multivariate normally distributed and they are generalized to the class of elliptically contoured distributions thereafter. We find that the choice of the coefficients of risk aversion depends on the stochastic model used for the data generating process. Finally, we take the parameter uncertainty into account and present confidence intervals for the risk aversion coefficients of the considered utility functions. The theoretical results are validated in an empirical study.

The study of the statistical properties of ocean magnetic anomalies could be very important to obtain new understanding on plate tectonics, especially about the past properties of the lithospheric oceanic plates. Here we analyse in terms of statistical distributions the ocean magnetic anomaly data at the global scale covering the crustal oceanic ages from the present to 180 Ma. Our results show that the marine magnetic anomaly data follow the Laplace statistical distribution. This behaviour is observed at global scale and for any time interval within the last 180 Ma. In addition, the statistical parameters of the Laplace distribution are used to define some properties of the crustal oceanic magnetic field, focusing our investigation on the Cretaceous Normal Superchron. Finally, we also provide a physical explanation of this statistical distribution of marine magnetic anomaly data using a test with synthetic data.

Dette speciale omhandler endimensionale kurvedata, hvor formålet er at estimere en funktionel sammenhæng, der beskriver observationerne. Indledningsvist gennemgår vi hoveddelene i den klassiske tilgang til problemet, som leder til såkaldte splinefunktioner. Vi giver en kort introduktion til de dominerende estimationsmetoder inden for dette område, men går så videre til en alternativ tilgang, med det hovedformål for øje at kunne formulere en komplet statistisk model for splineudglatning. Vi udforsker dualiteten mellem på den ene side inverse differentionaloperatorer og deres tilhørende Green's funktioner og på den anden side Gaussiske processer og deres kovariansfunktioner, og vi viser, at de to tilgange giver samme estimationsresultater. Tager vi et Bayesiansk standpunkt, muliggør dette, at vi kan formulere en komplet statistisk model for splineudglatning. Med det på plads kan vi lave forskellige modifikationer af modelantagelserne. Vi arbejder basalt set med to udvidelser. Først kigger vi på forskellige restriktioner på den funktionelle form af splinefunktionen, og vi viser, hvordan vi kan løse disse betingede problemer. Derefter undersøger vi de såkaldte L1 splines og sigter på at kunne forstå og beskrive disse ved hjælp af Laplace-fordelingen. For at kunne gøre dette gennemgår vi dele af teorien omkring den univariate og også den multivariate Laplace-fordeling. Afslutningsvist konstruerer vi en L1 spline, foreslår en iterativ estimationsprocedure og præsenterer til sidst også nogle nye udvidelser. Specialet er akkompagneret af en \verb+R+-pakke, \verb+L1splines+, som indeholder implementeringerne af de splines, der er udviklet., This thesis considers one-dimensional curve data, where the objective is to estimate a functional relation describing the observations. In the first part, we go through the main parts of the classic approach to the problem, which leads to the theory of splines. We make a brief introduction to the dominating estimation methods within this area, but then moves on to a somewhat differing approach, aiming at providing a full statistical framework for spline smoothing. We investigate the duality between, on the one hand, the theory of inversion of differential operators and their corresponding Green's functions and, on the other hand, Gaussian processes and their covariance functions, and the two approaches are shown to give the same estimation results. Ultimately, this clears the way for us to formulate a full statistical model for spline smoothing taking a Bayesian viewpoint. Having that established, we can make various modifications of the model assumptions. We consider two main extensions. Firstly, we impose different shape restrictions on the functional form of the spline and show how these constrained problems can be solved. Secondly, we consider the so-called L1 splines and aim to understand and describe these using the Laplace distribution. For this, we present some theory on the univariate, and also the multivariate, Laplace distribution. Finally, we propose an iterative estimation method for solving the L1 spline problem, and we present some new extensions. The thesis comes together with an \verb+R+-package, \verb+L1splines+, that contains implementations of all the splines developed.

Dette speciale omhandler endimensionale kurvedata, hvor formålet er at estimere en funktionel sammenhæng, der beskriver observationerne. Indledningsvist gennemgår vi hoveddelene i den klassiske tilgang til problemet, som leder til såkaldte splinefunktioner. Vi giver en kort introduktion til de dominerende estimationsmetoder inden for dette område, men går så videre til en alternativ tilgang, med det hovedformål for øje at kunne formulere en komplet statistisk model for splineudglatning. Vi udforsker dualiteten mellem på den ene side inverse differentionaloperatorer og deres tilhørende Green's funktioner og på den anden side Gaussiske processer og deres kovariansfunktioner, og vi viser, at de to tilgange giver samme estimationsresultater. Tager vi et Bayesiansk standpunkt, muliggør dette, at vi kan formulere en komplet statistisk model for splineudglatning. Med det på plads kan vi lave forskellige modifikationer af modelantagelserne. Vi arbejder basalt set med to udvidelser. Først kigger vi på forskellige restriktioner på den funktionelle form af splinefunktionen, og vi viser, hvordan vi kan løse disse betingede problemer. Derefter undersøger vi de såkaldte L1 splines og sigter på at kunne forstå og beskrive disse ved hjælp af Laplace-fordelingen. For at kunne gøre dette gennemgår vi dele af teorien omkring den univariate og også den multivariate Laplace-fordeling. Afslutningsvist konstruerer vi en L1 spline, foreslår en iterativ estimationsprocedure og præsenterer til sidst også nogle nye udvidelser. Specialet er akkompagneret af en \verb+R+-pakke, \verb+L1splines+, som indeholder implementeringerne af de splines, der er udviklet., This thesis considers one-dimensional curve data, where the objective is to estimate a functional relation describing the observations. In the first part, we go through the main parts of the classic approach to the problem, which leads to the theory of splines. We make a brief introduction to the dominating estimation methods within this area, but then moves on to a somewhat differing approach, aiming at providing a full statistical framework for spline smoothing. We investigate the duality between, on the one hand, the theory of inversion of differential operators and their corresponding Green's functions and, on the other hand, Gaussian processes and their covariance functions, and the two approaches are shown to give the same estimation results. Ultimately, this clears the way for us to formulate a full statistical model for spline smoothing taking a Bayesian viewpoint. Having that established, we can make various modifications of the model assumptions. We consider two main extensions. Firstly, we impose different shape restrictions on the functional form of the spline and show how these constrained problems can be solved. Secondly, we consider the so-called L1 splines and aim to understand and describe these using the Laplace distribution. For this, we present some theory on the univariate, and also the multivariate, Laplace distribution. Finally, we propose an iterative estimation method for solving the L1 spline problem, and we present some new extensions. The thesis comes together with an \verb+R+-package, \verb+L1splines+, that contains implementations of all the splines developed.

This paper investigates the role(s) of high-growth firms (HGFs) in the robust growth-rate distribution. HGFs are identied as firms for which the growth-rate distribution exhibits power-law decay. In contrast to the traditional means of identifying HGFs, a distributional approach eliminates the need to specify an arbitrary growth rate or percentage share. The latter approach is illustrated by the growth-rate distribution for Swedish data on incorporated firms at the aggregate level and at the 2-digit in-dustry level. The empirical results indicate that a power law is sometimes present in the growth-rate distribution and suggest that HGFs are rarer than previously thought., QC 20130215

This paper investigates the role(s) of high-growth firms (HGFs) in the robust growth-rate distribution. HGFs are identied as firms for which the growth-rate distribution exhibits power-law decay. In contrast to the traditional means of identifying HGFs, a distributional approach eliminates the need to specify an arbitrary growth rate or percentage share. The latter approach is illustrated by the growth-rate distribution for Swedish data on incorporated firms at the aggregate level and at the 2-digit in-dustry level. The empirical results indicate that a power law is sometimes present in the growth-rate distribution and suggest that HGFs are rarer than previously thought., QC 20130215

Many estimation problems require a mixture reduction algorithm with which an increasing number of mixture components are reduced to a tractable level. In this technical report a discussion on dierent aspects of mixture reduction is given followed by a presentation of numerical simulation on reduction of mixture densities where the component density belongs to the exponential family of distributions.

This paper investigates the role(s) of high-growth firms (HGFs) in the robust growth-rate distribution. HGFs are identied as firms for which the growth-rate distribution exhibits power-law decay. In contrast to the traditional means of identifying HGFs, a distributional approach eliminates the need to specify an arbitrary growth rate or percentage share. The latter approach is illustrated by the growth-rate distribution for Swedish data on incorporated firms at the aggregate level and at the 2-digit in-dustry level. The empirical results indicate that a power law is sometimes present in the growth-rate distribution and suggest that HGFs are rarer than previously thought., QC 20130215

Many estimation problems require a mixture reduction algorithm with which an increasing number of mixture components are reduced to a tractable level. In this technical report a discussion on dierent aspects of mixture reduction is given followed by a presentation of numerical simulation on reduction of mixture densities where the component density belongs to the exponential family of distributions.

Many estimation problems require a mixture reduction algorithm with which an increasing number of mixture components are reduced to a tractable level. In this technical report a discussion on dierent aspects of mixture reduction is given followed by a presentation of numerical simulation on reduction of mixture densities where the component density belongs to the exponential family of distributions.

Many estimation problems require a mixture reduction algorithm with which an increasing number of mixture components are reduced to a tractable level. In this technical report a discussion on dierent aspects of mixture reduction is given followed by a presentation of numerical simulation on reduction of mixture densities where the component density belongs to the exponential family of distributions.

This paper investigates the role(s) of high-growth firms (HGFs) in the robust growth-rate distribution. HGFs are identied as firms for which the growth-rate distribution exhibits power-law decay. In contrast to the traditional means of identifying HGFs, a distributional approach eliminates the need to specify an arbitrary growth rate or percentage share. The latter approach is illustrated by the growth-rate distribution for Swedish data on incorporated firms at the aggregate level and at the 2-digit in-dustry level. The empirical results indicate that a power law is sometimes present in the growth-rate distribution and suggest that HGFs are rarer than previously thought., QC 20130215

Many estimation problems require a mixture reduction algorithm with which an increasing number of mixture components are reduced to a tractable level. In this technical report a discussion on dierent aspects of mixture reduction is given followed by a presentation of numerical simulation on reduction of mixture densities where the component density belongs to the exponential family of distributions.

This paper investigates the role(s) of high-growth firms (HGFs) in the robust growth-rate distribution. HGFs are identied as firms for which the growth-rate distribution exhibits power-law decay. In contrast to the traditional means of identifying HGFs, a distributional approach eliminates the need to specify an arbitrary growth rate or percentage share. The latter approach is illustrated by the growth-rate distribution for Swedish data on incorporated firms at the aggregate level and at the 2-digit in-dustry level. The empirical results indicate that a power law is sometimes present in the growth-rate distribution and suggest that HGFs are rarer than previously thought., QC 20130215

A broad family of symmetric, thick tailed distributions, the small l- Laplace distributions, is described. These are natural generalizations of the Laplace distribution. A family of random coefficient Autoregressive Moving Average processes with small l-Laplace marginal distributions is constructed and its properties are explored. Extensions to ARIMA processes are considered. The first order autoregression with a standard Laplace marginal distribution is examined in detail and different estimates of the autoregressive parameter are compared theoretically and by simulation.

Due to the stochastic character of the intensities of information flows in high performance information systems, the size of data available for the statistical analysis can be often regarded as random. In the paper general theorem concerning the asymptotic expansions of the distribution function of the statistics based on the sample of random size was proved. Some examples are presented for the cases where the sample size has the negative binomial or discrete Pareto distributions. © ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi,Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors).

In the present paper we prove a general theorem which gives the rates of convergence in distribution of asymptotically normal statistics based on samples of random size. The proof of the theorem uses the rates of convergences in distribution for the random size and for the statistics based on samples of nonrandom size.