Approaching Mean-Variance Efficiency for Large Portfolios
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A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess
S S symmetry Article A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess Guillermo Martínez-Flórez 1, Víctor Leiva 2,* , Emilio Gómez-Déniz 3 and Carolina Marchant 4 1 Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 14014, Colombia; [email protected] 2 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, 2362807 Valparaíso, Chile 3 Facultad de Economía, Empresa y Turismo, Universidad de Las Palmas de Gran Canaria and TIDES Institute, 35001 Canarias, Spain; [email protected] 4 Facultad de Ciencias Básicas, Universidad Católica del Maule, 3466706 Talca, Chile; [email protected] * Correspondence: [email protected] or [email protected] Received: 30 June 2020; Accepted: 19 August 2020; Published: 1 September 2020 Abstract: In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal. Keywords: beta distribution; centered skew-normal distribution; maximum-likelihood methods; Monte Carlo simulations; proportions; R software; rates; zero/one inflated data 1. -
Random Processes
Chapter 6 Random Processes Random Process • A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t). • For a fixed (sample path): a random process is a time varying function, e.g., a signal. – For fixed t: a random process is a random variable. • If one scans all possible outcomes of the underlying random experiment, we shall get an ensemble of signals. • Random Process can be continuous or discrete • Real random process also called stochastic process – Example: Noise source (Noise can often be modeled as a Gaussian random process. An Ensemble of Signals Remember: RV maps Events à Constants RP maps Events à f(t) RP: Discrete and Continuous The set of all possible sample functions {v(t, E i)} is called the ensemble and defines the random process v(t) that describes the noise source. Sample functions of a binary random process. RP Characterization • Random variables x 1 , x 2 , . , x n represent amplitudes of sample functions at t 5 t 1 , t 2 , . , t n . – A random process can, therefore, be viewed as a collection of an infinite number of random variables: RP Characterization – First Order • CDF • PDF • Mean • Mean-Square Statistics of a Random Process RP Characterization – Second Order • The first order does not provide sufficient information as to how rapidly the RP is changing as a function of timeà We use second order estimation RP Characterization – Second Order • The first order does not provide sufficient information as to how rapidly the RP is changing as a function -
“Mean”? a Review of Interpreting and Calculating Different Types of Means and Standard Deviations
pharmaceutics Review What Does It “Mean”? A Review of Interpreting and Calculating Different Types of Means and Standard Deviations Marilyn N. Martinez 1,* and Mary J. Bartholomew 2 1 Office of New Animal Drug Evaluation, Center for Veterinary Medicine, US FDA, Rockville, MD 20855, USA 2 Office of Surveillance and Compliance, Center for Veterinary Medicine, US FDA, Rockville, MD 20855, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-240-3-402-0635 Academic Editors: Arlene McDowell and Neal Davies Received: 17 January 2017; Accepted: 5 April 2017; Published: 13 April 2017 Abstract: Typically, investigations are conducted with the goal of generating inferences about a population (humans or animal). Since it is not feasible to evaluate the entire population, the study is conducted using a randomly selected subset of that population. With the goal of using the results generated from that sample to provide inferences about the true population, it is important to consider the properties of the population distribution and how well they are represented by the sample (the subset of values). Consistent with that study objective, it is necessary to identify and use the most appropriate set of summary statistics to describe the study results. Inherent in that choice is the need to identify the specific question being asked and the assumptions associated with the data analysis. The estimate of a “mean” value is an example of a summary statistic that is sometimes reported without adequate consideration as to its implications or the underlying assumptions associated with the data being evaluated. When ignoring these critical considerations, the method of calculating the variance may be inconsistent with the type of mean being reported. -
Basic Statistics and Monte-Carlo Method -2
Applied Statistical Mechanics Lecture Note - 10 Basic Statistics and Monte-Carlo Method -2 고려대학교 화공생명공학과 강정원 Table of Contents 1. General Monte Carlo Method 2. Variance Reduction Techniques 3. Metropolis Monte Carlo Simulation 1.1 Introduction Monte Carlo Method Any method that uses random numbers Random sampling the population Application • Science and engineering • Management and finance For given subject, various techniques and error analysis will be presented Subject : evaluation of definite integral b I = ρ(x)dx a 1.1 Introduction Monte Carlo method can be used to compute integral of any dimension d (d-fold integrals) Error comparison of d-fold integrals Simpson’s rule,… E ∝ N −1/ d − Monte Carlo method E ∝ N 1/ 2 purely statistical, not rely on the dimension ! Monte Carlo method WINS, when d >> 3 1.2 Hit-or-Miss Method Evaluation of a definite integral b I = ρ(x)dx a h X X X ≥ ρ X h (x) for any x X Probability that a random point reside inside X O the area O O I N' O O r = ≈ O O (b − a)h N a b N : Total number of points N’ : points that reside inside the region N' I ≈ (b − a)h N 1.2 Hit-or-Miss Method Start Set N : large integer N’ = 0 h X X X X X Choose a point x in [a,b] = − + X Loop x (b a)u1 a O N times O O y = hu O O Choose a point y in [0,h] 2 O O a b if [x,y] reside inside then N’ = N’+1 I = (b-a) h (N’/N) End 1.2 Hit-or-Miss Method Error Analysis of the Hit-or-Miss Method It is important to know how accurate the result of simulations are The rule of 3σ’s Identifying Random Variable N = 1 X X n N n=1 From -
Monte Carlo Simulation by Marco Liu, Cqf
OPTIMAL NUMBER OF TRIALS FOR MONTE CARLO SIMULATION BY MARCO LIU, CQF This article presents a way to estimate the number of trials required several hours. It would be beneficial to know what level of precision, for a desired confidence interval in the context of a Monte Carlo or confidence interval, we could achieve for a certain number of simulation. iterations in advance of running the program. Without the confidence interval, the time commitment for a trial-and-error process would Traditional valuation approaches such as Option Pricing Method be significant. In this article, we will present a simple method to (“OPM”) or Probability-Weighted Expected Return Method estimate the number of simulations needed for a desired level of (“PWERM”) may not be adequate in providing fair value estimation precision when running a Monte Carlo simulation. for financial instruments that require distribution assumptions on multiple input parameters. In such cases, a numerical method, Monte Carlo simulation for instance, is often used. The Monte Carlo 95% of simulation is a computerized algorithmic procedure that outputs a area wide range of values – typically unknown probability distribution – by simulating one or multiple input parameters via known probability distributions. Statistically, 95% of the area under a normal distribution curve is described as being plus or minus 1.96 standard This technique is often used to find fair value for financial deviations from the mean. For 90%, the z-statistic is 1.64. instruments for which probabilistic distributions are unknown. The simulation procedure is typically repeated multiple times and the average of the results is taken. -
The Multivariate Normal Distribution
Multivariate normal distribution Linear combinations and quadratic forms Marginal and conditional distributions The multivariate normal distribution Patrick Breheny September 2 Patrick Breheny University of Iowa Likelihood Theory (BIOS 7110) 1 / 31 Multivariate normal distribution Linear algebra background Linear combinations and quadratic forms Definition Marginal and conditional distributions Density and MGF Introduction • Today we will introduce the multivariate normal distribution and attempt to discuss its properties in a fairly thorough manner • The multivariate normal distribution is by far the most important multivariate distribution in statistics • It’s important for all the reasons that the one-dimensional Gaussian distribution is important, but even more so in higher dimensions because many distributions that are useful in one dimension do not easily extend to the multivariate case Patrick Breheny University of Iowa Likelihood Theory (BIOS 7110) 2 / 31 Multivariate normal distribution Linear algebra background Linear combinations and quadratic forms Definition Marginal and conditional distributions Density and MGF Inverse • Before we get to the multivariate normal distribution, let’s review some important results from linear algebra that we will use throughout the course, starting with inverses • Definition: The inverse of an n × n matrix A, denoted A−1, −1 −1 is the matrix satisfying AA = A A = In, where In is the n × n identity matrix. • Note: We’re sort of getting ahead of ourselves by saying that −1 −1 A is “the” matrix satisfying -
Mean and Covariance Matrix of a Multivariate Normal Distribution with One Doubly-Truncated Component
Technical report from Automatic Control at Linköpings universitet Mean and covariance matrix of a multivariate normal distribution with one doubly-truncated component Henri Nurminen, Rafael Rui, Tohid Ardeshiri, Alexandre Bazanella, Fredrik Gustafsson Division of Automatic Control E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] 7th July 2016 Report no.: LiTH-ISY-R-3092 Address: Department of Electrical Engineering Linköpings universitet SE-581 83 Linköping, Sweden WWW: http://www.control.isy.liu.se AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications. Abstract This technical report gives analytical formulas for the mean and covariance matrix of a multivariate normal distribution with one component truncated from both below and above. Keywords: doubly-truncated multivariate normal distribution, mean, co- variance matrix Mean and covariance matrix of a multivariate normal distribution with one doubly-truncated component Henri Nurminen,∗ Rafael Rui,y Tohid Ardeshiri, Alexandre Bazanella,y and Fredrik Gustafsson 2016-09-02 Abstract This technical report gives analytical formulas for the mean and co- variance matrix of a multivariate normal distribution with one component truncated from both below and above. The result is used in the compu- tation of the moments of a mixture of such distributions in [1]. 1 Introduction In this technical report doubly-truncated multivariate normal distribution (DTMND) is a multivariate normal distribution, where one component is truncated from both below and above. In this report we present and derive the analytical for- mulas for the mean and covariance matrix of a DTMND. -
FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM
FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (1) Markus K. Brunnermeier LECTURE 06: MEAN-VARIANCE ANALYSIS & CAPM FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (2) Overview 1. Introduction: Simple CAPM with quadratic utility functions (from beta-state price equation) 2. Traditional Derivation of CAPM – Demand: Portfolio Theory for given – Aggregation: Fund Separation Theorem prices/returns – Equilibrium: CAPM 3. Modern Derivation of CAPM – Projections – Pricing Kernel and Expectation Kernel 4. Testing CAPM 5. Practical Issues – Black-Litterman FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (3) Recall State-price Beta model Recall: 퐸 푅ℎ − 푅푓 = 훽ℎ퐸 푅∗ − 푅푓 cov 푅∗,푅ℎ Where 훽ℎ ≔ var 푅∗ very general – but what is 푅∗ in reality? FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (4) Simple CAPM with Quadratic Expected Utility 1. All agents are identical 휕1푢 • Expected utility 푈 푥0, 푥1 = 푠 휋푠푢 푥0, 푥푠 ⇒ 푚 = 퐸 휕0푢 2 • Quadratic 푢 푥0, 푥1 = 푣0 푥0 − 푥1 − 훼 • ⇒ 휕1푢 = −2 푥1,1 − 훼 , … , −2 푥푆,1 − 훼 • Excess return cov 푚, 푅ℎ 푅푓cov 휕 푢, 푅ℎ 퐸 푅ℎ − 푅푓 = − = − 1 퐸 푚 퐸 휕0푢 푅푓cov −2 푥 − 훼 , 푅ℎ 2cov 푥 , 푅ℎ = − 1 = 푅푓 1 퐸 휕0푢 퐸 휕0푢 • Also holds for market portfolio ℎ 푓 ℎ 퐸 푅 − 푅 cov 푥1, 푅 푚푘푡 푓 = 푚푘푡 퐸 푅 − 푅 cov 푥1, 푅 FIN501 Asset Pricing Lecture 06 Mean-Variance & CAPM (5) Simple CAPM with Quadratic Expected Utility ℎ 푓 ℎ 퐸 푅 − 푅 cov 푥1, 푅 푚푘푡 푓 = 푚푘푡 퐸 푅 − 푅 cov 푥1, 푅 2. Homogenous agents + Exchange economy 푚 ⇒ 푥1 = aggr. -
Descriptive Statistics
Statistics: Descriptive Statistics When we are given a large data set, it is necessary to describe the data in some way. The raw data is just too large and meaningless on its own to make sense of. We will sue the following exam scores data throughout: x1, x2, x3, x4, x5, x6, x7, x8, x9, x10 45, 67, 87, 21, 43, 98, 28, 23, 28, 75 Summary Statistics We can use the raw data to calculate summary statistics so that we have some idea what the data looks like and how sprad out it is. Max, Min and Range The maximum value of the dataset and the minimum value of the dataset are very simple measures. The range of the data is difference between the maximum and minimum value. Range = Max Value − Min Value = 98 − 21 = 77 Mean, Median and Mode The mean, median and mode are measures of central tendency of the data (i.e. where is the center of the data). Mean (µ) The mean is sum of all values divided by how many values there are N 1 X 45 + 67 + 87 + 21 + 43 + 98 + 28 + 23 + 28 + 75 xi = = 51.5 N i=1 10 Median The median is the middle data point when the dataset is arranged in order from smallest to largest. If there are two middle values then we take the average of the two values. Using the data above check that the median is: 44 Mode The mode is the value in the dataset that appears most. Using the data above check that the mode is: 28 Standard Deviation The standard deviation (σ) measures how spread out the data is. -
The Multivariate Normal Distribution
The Multivariate Normal Distribution Why should we consider the multivariate normal distribution? It would seem that applied problems are so complex that it would only be interesting from a mathematical perspective. 1. It is mathematically tractable for a large number of problems, and, therefore, progress towards answers to statistical questions can be provided, even if only approximately so. 2. Because it is tractable for so many problems, it provides insight into techniques based upon other distributions or even non-parametric techniques. For this, it is often a benchmark against which other methods are judged. 3. For some problems it serves as a reasonable model of the data. In other instances, transfor- mations can be applied to the set of responses to have the set conform well to multivariate normality. 4. The sampling distribution of many (multivariate) statistics are normal, regardless of the parent distribution (Multivariate Central Limit Theorems). Thus, for large sample sizes, we may be able to make use of results from the multivariate normal distribution to answer our statistical questions, even when the parent distribution is not multivariate normal. Consider first the univariate normal distribution with parameters µ (the mean) and σ (the variance) for the random variable x, 2 1 − 1 (x−µ) f(x)=√ e 2 σ2 (1) 2πσ2 for −∞ <x<∞, −∞ <µ<∞,andσ2 > 0. Now rewrite the exponent (x − µ)2/σ2 using the linear algebra formulation of (x − µ)(σ2)−1(x − µ). This formulation matches that for the generalized or Mahalanobis squared distance (x − µ)Σ−1(x − µ), where both x and µ are vectors. -
Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM)
Eco 525: Financial Economics I LectureLecture 05:05: MeanMean--VarianceVariance AnalysisAnalysis && CapitalCapital AssetAsset PricingPricing ModelModel ((CAPMCAPM)) Prof. Markus K. Brunnermeier 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-1 Eco 525: Financial Economics I OverviewOverview •• SimpleSimple CAPMCAPM withwith quadraticquadratic utilityutility functionsfunctions (derived from state-price beta model) • Mean-variance preferences – Portfolio Theory – CAPM (intuition) •CAPM – Projections – Pricing Kernel and Expectation Kernel 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-2 Eco 525: Financial Economics I RecallRecall StateState--priceprice BetaBeta modelmodel Recall:Recall: E[RE[Rh]] -- RRf == ββh E[RE[R*-- RRf]] where βh := Cov[R*,Rh] / Var[R*] veryvery generalgeneral –– butbut whatwhat isis RR* inin realityreality?? 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-3 Eco 525: Financial Economics I SimpleSimple CAPMCAPM withwith QuadraticQuadratic ExpectedExpected UtilityUtility 1. All agents are identical • Expected utility U(x0, x1) = ∑s πs u(x0, xs) ⇒ m= ∂1u / E[∂0u] 2 • Quadratic u(x0,x1)=v0(x0) - (x1- α) ⇒ ∂1u = [-2(x1,1- α),…, -2(xS,1- α)] •E[Rh] – Rf = - Cov[m,Rh] / E[m] f h = -R Cov[∂1u, R ] / E[∂0u] f h = -R Cov[-2(x1- α), R ] / E[∂0u] f h = R 2Cov[x1,R ] / E[∂0u] • Also holds for market portfolio m f f m •E[R] – R = R 2Cov[x1,R ]/E[∂0u] ⇒ 16:14 Lecture 05 Mean-Variance Analysis and CAPM Slide 05-4 Eco 525: Financial Economics I SimpleSimple CAPMCAPM withwith QuadraticQuadratic ExpectedExpected -
Comparing TL-Moments, L-Moments and Conventional Moments of Dagum Distribution by Simulated Data
Revista Colombiana de Estadística Junio 2013, volumen 36, no. 1, pp. 79 a 93 Comparing TL-Moments, L-Moments and Conventional Moments of Dagum Distribution by Simulated data Comparación de momentos TL, momentos L y momentos convencionales de la distribución Dagum mediante datos simulados Mirza Naveed Shahzad1;a, Zahid Asghar2;b 1Department of Statistics, University of Gujrat, Gujrat, Pakistan 2Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan Abstract Modeling income, wage, wealth, expenditure and various other social variables have always been an issue of great concern. The Dagum distribu- tion is considered quite handy to model such type of variables. Our focus in this study is to derive the L-moments and TL-moments of this distribution in closed form. Using L & TL-moments estimators we estimate the scale parameter which represents the inequality of the income distribution from the mean income. Comparing L-moments, TL-moments and conventional moments, we observe that the TL-moment estimator has lessbias and root mean square errors than those of L and conventional estimators considered in this study. We also find that the TL-moments have smaller root mean square errors for the coefficients of variation, skewness and kurtosis. These results hold for all sample sizes we have considered in our Monte Carlo simu- lation study. Key words: Dagum distribution, L-moments, Method of moments, Para- meter estimation, TL-moments. Resumen La modelación de ingresos, salarios, riqueza, gastos y muchas otras varia- bles de tipo social han sido siempre un tema de gran interés. La distribución Dagum es considerada para modelar este tipo de variables.