Department of Statistics, Lund University
Certain bivariate distributions and random processes connected with maxima and minima
It is well-known that [S(x)]^n and [F(x)]^n are the survival function and the distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]^alpha and [F(x)]^beta, where alpha> 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss generalizations to random process and further extensions.
Derechos de autor 2016 Working Papers in Statistics
Distributions of spatial wave size for random fields
A method of measuring three-dimensional spatial wave size is proposed and statistical distributions of the size characteristics are derived in explicit integral forms for Gaussian sea surfaces. New definitions of wave characteristics such as the crest-height, the length, the size and the wave front location are provided in fully dimensional context. The joint statistical distributions of these wave characteristics are derived using the Rices formulas for expected numbers of local maximum and distance from a local maximum to a level crossing countour. Review of the Rice's method to study crossing distributions will be given.
The Sibuya distribution arises as the distribution of the waiting time for the first success in Bernoulli trials, where the probabilities of success are inversely proportional to the number of a trial. We study a generalization that is obtained as the distribution of the excess random variable N − k given N > k, where N has the Sibuya distribution. We summarize basic facts regarding this distribution and provide several new results and characterizations, shedding more light on its origin and possible applications. In particular, we emphasize the role Sibuya distribution plays in the extreme value theory and point out its invariance property with respect to random thinning operation.
Transmuted distributions and extrema of random number of variables
Recent years have seen an increased interest in the transmuted probability models,which arise from transforming a “base” distribution into its generalized counterpart. While many standard probability distributions were generalized throughout this simple construction, the concept lacked deeper theoretical interpretation. This note demonstrates that the scheme is more than just a simple trick to obtain a new cumulative distribution function. We show that the transmuted distributions can be viewed as the distribution of maxima (or minima) of a random number N of independent and identically distributed variables with the base distribution, where N has a Bernoulli distribution shifted up by one. Consequently, the transmuted models are, in fact, only a special case of extremal distributions defined through a more general N .
We review the generalized Rice formula approach to deriving long-run distributions of a variety of characteristics defined at random events defined on a stochastic process. While the approach stems from the same principle originally introduced by Rice for the level crossing intensity in a random signal, we show how it generalizes to more general contexts. Firstly, we discuss events defined on random surfaces through crossing levels of possibly multivariate valued stochastic fields. Secondly, the dynamics is introduced by adding time argument and introducing the concept of velocity measured at moving surface. Thirdly, extensions beyond the Gaussian model are shown by presenting effective models for sampling from the distribution of a non-Gaussian noise observed at instances of level crossing by a process driven by this noise. The importance of these generalization for engineering applications is illustrated through examples.
Central limit theorems for functionals of large dimensional sample covariance matrix and mean vector in matrix-variate skewed model
In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix variate general skew normal distribution. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of an inverse covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large dimensional asymptotic regime where the dimension p and sample size n approach to infinity such that p/n → c ∈ (0, 1).
A method of measuring three-dimensional spatial wave size is proposed and statistical distributions of the size characteristics are derived in explicit integral forms for Gaussian sea surfaces. New definitions of wave characteristics such as the crest-height, the length, the size and the wave front location are provided in fully dimensional context. The joint statistical distributions of these wave characteristics are expressed in the terms of a directional power spectrum. The proposed approach allows for investigation of the influence of spectral shape and directionality on the joint distributions of the size characteristics. The method is validated by comparing numerical evaluations of the distributions with the observed frequencies extracted from simulations of sea surface. The comparisons demonstrate high accuracy of the obtained distributions.
We show how to minimize the asymptotic variance of multipower estimators using a linear combination of optimal powers. Taking advantage of the lower variance provided by this technique allows to build superior estimators of integrated volatility powers. In particular, we focus on a new efficient quarticity estimator and we show, using simulated data, that we can drastically reduce the mean square error of traditional estimators. The implementation on US stock prices corroborates our theoretical findings and further shows that ecient quarticity noticeably reduces the number of detected jumps, and improves the quality of volatility forecasts.
Sequential Search Algorithm for Estimation of the Number of Classes in a Given Population
Let N be the number of classes in a population to be estimated. Fix any preassigned error probability 0<epsilon< exp(-2) (roughly). We present a sequential search algorithm to estimate the exact value of N, with an error probability of at most epsilon, regardless of the value of N.
Option framing and Markov chain: A descriptive approach in a state-space modeling of customer behavior
In the field of marketing, option framing is a product or service configuration where the consumers customize the package they wish to procure either by adding options to a base model, an initial configuration with a minimum number of essential features, or by subtracting options from a fully-loaded model, a product or service configuration with both essential and all of the optional features. Additive framing is selecting features to augment the base model, while subtractive framing is deselecting features from the fully-loaded model. A focal issue for companies that could possibly offer such products or services with option framing is finding out which process, additive or subtractive framing, is bound to give a final configuration with more features. The scenarios of option framing can be described by a finite Markov chain process. The Markov chain attempts to capture the decision process of the two types of framing through the estimated probabilities of movement from one phase to the other. In each of the decision phases, the key measure is the number of features in the configuration and the transition probabilities. The option framing is used on an actual study, where the empirical results verify the theories favoring subtractive framing, such as differential loss aversion and anchoring-adjustment theories. Separate Markov chains are evaluated for additive and subtractive framing, with the final configurations of the product or service package, along with the corresponding number of options, as main results.
Derechos de autor 2015 Working Papers in Statistics
Laplace distribution models for road topography and roughness
Gaussian models are frequently used for road profiles. However, these models are often only valid for short sections of the road. Here we present a comprehensive approach to describe various aspects of road surface/elevation by using extensions of Gaussian models arising from random gamma distributed variances. These random variances result in the Laplace distribution and thus we refer to the so defined models as Laplace models. The approach is shown to perform well in modelling road topography, road roughness and multi-valued responses of forces and bending moments containing transients. The different Laplace models are presented together with numerical examples and Matlab code for simulation.
Comparing the effects of two treatments on two ordinal outcome variables
When evaluating whether the effect of one treatment is larger than that of another the first step in the comparison is to decide what should be understood by the statement that one patient has achieved a greater effect than has another patient. When the outcome variable is quantitative, measured on a ratio scale, absolute or relative effects are the most commonly used effect measures; however, such effects are usually not meaningful for ordinal outcome variables. In order to answer the question whether one of two treatments acts more effectively on one of two outcome variables and the other treatment more efficiently on the other we shall present a method of comparing the treatment effects of patients that is based on pair-wise comparisons between patients in analogy with many non-parametrical methods. These comparisons use only the ordinal properties of the outcome variables. We shall even define a measure of the difference between the treatment effects and demonstrate how confidence intervals can be constructed.
Third Cumulant for Multivariate Aggregate Claim Models
The third moment cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. It is discussed how this result can be used to obtain a formula for the third cumulant for a classical model of multivariate claims. Two important special cases are considered. In the rst one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results.
Are the Sweden Democrats really Sweden’s largest party? A maximum likelihood ratio test on the simplex
In August 2015 a Swedish newspaper claimed that the Sweden Democrats were the largest political party in Sweden based on the results of single poll. We ask ourselves if this is a correct conclusion, considering the fact that the three largest parties in the poll were of roughly the same size. We analyse the parameter space and identify the subspace where the Sweden Democrats are the largest party. Using this we construct a maximum likelihood ratio test and derive its distribution under the null hypothesis. We finally apply our test to the data and obtaining a p-value between 0.09 and 0.14 are able to refute the claim in the newspaper. Based on the available data one cannot draw the conclusion that the Sweden Democrats are the largest party in Sweden.
On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian vector
In this paper we study the distribution of the product of an inverse Wishart random matrix and a Gaussian random vector. We derive its asymptotic distribution as well as its approximate density function formula which is based on the Gaussian integral and the third order Taylor expansion. Furthermore, we compare obtained asymptotic and approximate density functions with the exact density which is obtained by Bodnar and Okhrin (2011). A good performance of obtained results is documented in the numerical study.
A linear test for the global minimum variance portfolio for small sample and singular covariance
Bodnar and Schmid (2008) derived the distribution of the global minimum variance portfolio weights and obtained the distribution of the test statistics for the general linear hypothesis. Their results are obtained in the case when the number of observations n is bigger or equal than the size of portfolio k. In the present paper, we extend the result by analyzing the portfolio weights in a small sample case of n < k, with the singular covariance matrix. The results are illustrated using actual stock returns. A discussion of practical relevance of the model is presented.
Second-order continuous time moving averages via spectral representation
The spectral representation of a moving average process obtained as a convolution of a kernel with a general noise measure is studied. A proof of the spectral theorem that yields explicit expression for the spectral measure in terms of the noise measure is presented. The main interest is in noise measures generated by second order Lévy motions. For practical considerations, such measures are easily available through independent sampling. On the other hand spectral measures are not since their increments are dependent, with the notable exception of the Gaussian noise case. For this reason the issue of approximating the spectral measure by independent increments of the noise is also addressed. For the purpose of approximating the moving average process through sums of trigonometric functions, the mean square error of discretization of the spectral representation is assessed. For a specified accuracy, the coefficients of approximation are explicitly given. The method is illustrated for moving averages processes driven by Laplace motion.
Tail behavior and dependence structure the APARCH model
The APARCH model is a generalization of the GARCH model that attempts to capture asymmetric responses of returns and of volatility to positive and negative `news shocks' -the phenomenon known as the leverage effect. Despite its potential, the model's mathematical properties have not yet been fully investigated. While the capacity of the model to account for the leverage effect is clear from its defining structure, little is known how the effect is quantifed in terms of the model's parameters. The same applies to the quantifcation of heavy tails and time dependence. Here, in an attempt to fill this void, we study the model in further detail. We obtain sufficient conditions of its existence in different metrics as well as explicit forms of important characteristics: skewness, kurtosis, correlations and leverage. Utilizing these results, we analyze the roles of the parameters and discuss statistical inference. We also propose a natural extension by introducing an additional parameter and discuss how it affects the model. Through theoretical results and a Monte Carlo study we demonstrate that the model can produce heavy-tailed data. We illustrate these properties using S&P500 data as well as country indices for dominant European economies.
The Laplace multi-axial response model for fatigue analysis
This paper introduces means for fatigue damage rates estimation using Laplace distributed multiaxial loads. The model is suitable for description of stresses containing transients of random amplitudes and locations. Explicit formulas for computing the expected value of rainflow damage index as a function of excess kurtosis are given for correlated loads. A Laplace model is used to describe variability of forces and bending moments measured at some location ona cultivator frame. An example of actual cultivator data is used to illustrate the model and demonstrate the accuracy of damage index prediction.
A novel weighted likelihood estimation with empirical Bayes flavor
We propose a novel approach to estimation, where each individual observation in a random sample is used to derive an estimator of an unknown parameter using the maximum likelihood principle. These individual estimators are then combined as a weighted average to produce the final estimator. The weights are chosen to be proportional to the likelihood function evaluated at the estimators based on each observation. The method can be related to a Bayesian approach, where the prior distribution is data driven. In case of estimating a location parameter of a unimodal density, the prior distribution is the empirical distribution of the sample, and converges to the true distribution that generated the data as the sample size increases. We provide several examples illustrating the new method, argue for its consistency, and conduct simulation studies to assess the performance of the estimators. It turns out that this straightforward methodology produces consistent estimators, which seem to be comparable with those obtained by the maximum likelihood method.
AR(1) time series with autoregressive gamma variance for road topography modeling
A non-Gaussian time series with a generalized Laplace marginal distribution is used to model road topography. The model encompasses variability exhibited by a Gaussian AR(1) processwith randomly varying variance that follows a particular autoregressive model that features the gamma distribution as its marginal. A simple estimation method to fit the correlation coefficient of each of two autoregressive components is proposed. The one for the Gaussian AR(1) component is obtained by fitting the frequency of zero crossing, while the autocorrelation coefficient for the gamma autoregressiveprocess is fitted from the autocorrelation of the squared values of the model. The shape parameter of the gamma distribution is fitted using the explicitly given moments of a generalized Laplace distribution. Another general method of model fitting based on the correlation function of the signal is also presented and compared with the zero-crossing method. It is demonstrated that the model has the ability to accurately represent hilliness features of road topography providing a significant improvement over a purely
Bayesian Estimation of the Global Minimum Variance Portfolio
In this paper we consider the estimation of the weights of optimal portfolios from the Bayesian point of view under the assumption that the conditional distribution of the logarithmic returns is normal. Using the standard priors for the mean vector and the covariance matrix, we derive the posterior distributions for the weights of the global minimum variance portfolio. Moreover, we reparameterize the model to allow informative andnon-informative priors directly for the weights of the global minimum variance portfolio. The posterior distributions of the portfolio weights are derived in explicit form for almost all models. The models are compared by using the coverage probabilities of credible intervals. In an empirical study we analyze the posterior densities of the weights of an international portfolio.
We provide clear-cut evidence for economically and statistically significant multivariate jumps (multi-jumps) occurring simultaneously in stock prices by using a novel nonparametric test based on smoothed estimators of integrated variances. Detecting multi-jumps in a panel of liquid stocks is more statistically powerful and economically informative than the detection of univariate jumps in the market index.On the contrary of index jumps, multi-jumps can indeed be associated with sudden and large increases of the variance risk-premium, and possess a statistically significant forecasting power for future volatility and correlations which implies a sizable deterioration in the diversification potential of asset allocation.
Singular Inverse Wishart Distribution with Application to Portfolio Theory
The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order.
Are there house effects in Swedish polls? A compositional approach
We investigate if we can find evidence of house effects in Swedish political opinion polls from 2006 to 2014. In order to respect the inherent restriction of the simplex sample space, we apply compositional data analysis and Aitchison geometry. We use compositional locally weighted regression (C-loess) to estimate the share of voters for each party at given dates, a poll of polls. The estimated shares are compared to the poll(s) for that date using simplicial distances and compositional deviations (inverse perturbation). We can conclude that neither the mean simplicial distance nor the mean compositional deviation are equal for all polling organisations, also when controlling for different sample sizes and limiting the sample to only the last two years.