DOCSLIB.ORG

Parameter Estimation for Stable Distributions: Spacings-Based and Indirect Inference

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

UNIVERSITY OF CALIFORNIA Santa Barbara Parameter Estimation for Stable Distributions: Spacings-based and Indirect Inference A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Statistics and Applied Probability by Gaoyuan Tian Committee in Charge: Professor S.Rao Jammalamadaka, Chair Professor John Hsu Professor Tomoyuki Ichiba March 2016 The Dissertation of Gaoyuan Tian is approved: Professor John Hsu Professor Tomoyuki Ichiba Professor S.Rao Jammalamadaka, Committee Chairperson March 2016 Parameter Estimation for Stable Distributions: Spacings-based and Indirect Inference Copyright c 2016 by Gaoyuan Tian iii Acknowledgements First and foremost, I would like to express my deepest gratitude to my advisor, Professor S. Rao Jammalamadaka. Only his continuous guidance, support, and care made the research presented in this dissertation possible. I am very grateful to my committee members Professor Tomoyuki Ichiba and Professor John Hsu for their friendly advice and kindly help while serving on my committee. I thank Dr. St´ephaneGuerrier, who as a visiting faculty member in our department, intro- duced me to indirect inference and made many helpful suggestions. During the final phase of my work on indirect inference for stable distributions, I discovered this overlapped with some of the work of Professor David Veredas. I am grateful to him for pointing this out and for his support and suggestions. I would never have been able to finish my dissertation without the guidance of many faculty members, help from friends, as well as support from my family. I wish to thank Professors David Hinkley, Andrew Carter, Jean-Pierre Fouque, and Michael Ludkovski for their wonderful teaching and instruction during my studies here at UCSB. Also my time at UCSB has been made most enjoyable in large part due to the pleasure I had studying and working with many great graduate students, who have become my lifetime friends. I also want to thank them for their help and useful discussions in my study. Finally, I would like to thank my wonderful parents Jun Tian and Haifeng Yuan for all their love and support during all these years. iv Curriculum Vitæ Gaoyuan Tian Education 2015 Doctor of Philosophy in Statistics and Applied Probability, De- partment of Statistics and Applied Probability, University of Cal- ifornia, Santa Barbara 2011 Master of Art in Applied Statistics, University of California, Santa Barbara 2009 Master of Art in Economics,University of California, Santa Bar- bara 2008 Bachelor of Economics in Finance, Renmin University of China, China. Experience 2010-2015 Teaching Assistant, Department of Statistics and Applied Prob- ability, University of California, Santa Barbara 2009-2010 Readership Assistant, Department of Statistics and Applied Prob- ability, University of California, Santa Barbara. v Abstract Parameter Estimation for Stable Distributions: Spacings-based and Indirect Inference Gaoyuan Tian Stable distributions are important family of parametric distributions widely used in signal processing as well as in mathematical finance. Estimation of the parameters of this model, is not quite straightforward due to the fact that there is no closed-form expression for their probability density function. Besides the com- putationally intensive maximum likelihood method where the density has to be evaluated numerically, there are some existing adhoc methods such as the quantile method, and a regression based method. These are introduced in Chapter 2. In this thesis, we introduce two new approaches: One, a spacing based estimation method introduced in Chapter 3 and two, an indirect inference method considered in Chapter 4. Simulation studies show that both these methods are very robust and efficient and do as well or better than the existing methods in most cases. Fi- nally in Chapter 5, we use indirect inference approach to estimate the best fitting income distribution based on limited information that is often available. vi Contents Acknowledgements iv Curriculum Vitæ v Abstract vi List of Figures x List of Tables xi 1 Introduction to Stable Distributions 1 1.1 Definitions ::::::::::::::::::::::::::::::: 2 1.1.1 A Different Parametrization of Stable Laws :::::::: 3 1.2 Basic Properties :::::::::::::::::::::::::::: 5 1.2.1 Densities and Distribution Functions :::::::::::: 5 1.2.2 Tail Probabilities and Moments ::::::::::::::: 7 1.3 Simulation Methods :::::::::::::::::::::::::: 9 2 Existing Estimation Methods 10 2.1 Maximum Likelihood Estimation :::::::::::::::::: 11 2.1.1 The Integral Representations of Zolotarev ::::::::: 12 2.1.2 Fast Fourier Transformation ::::::::::::::::: 13 2.2 Quantile Based Estimation :::::::::::::::::::::: 15 2.3 Characteristic Function based Estimation :::::::::::::: 16 3 Spacing Based Estimation for Stable Distributions 18 3.1 Introduction of Spacings Based Estimation ::::::::::::: 19 3.1.1 Definition ::::::::::::::::::::::::::: 19 3.1.2 Properties ::::::::::::::::::::::::::: 20 3.1.3 Examples ::::::::::::::::::::::::::: 20 3.2 GSE Applied in Stable Distributions :::::::::::::::: 22 vii 3.2.1 Estimating Tail and Skewness Parameters ::::::::: 22 3.2.2 Monto Carlo Studies ::::::::::::::::::::: 25 3.3 Comparison Between Different Methods ::::::::::::::: 26 3.4 Conclusion ::::::::::::::::::::::::::::::: 27 4 Indirect Inference Method Applied to Stable Distributions 28 4.1 Introduction ::::::::::::::::::::::::::::: 28 4.2 Existing Estimation Methods :::::::::::::::::::: 30 4.2.1 Introduction of the Stable Distribution ::::::::::: 30 4.2.2 Available Estimation Methods :::::::::::::::: 31 4.3 Indirect Inference in Stable Distributions :::::::::::::: 33 4.3.1 Indirect Inference Method :::::::::::::::::: 33 4.3.2 Quanitle-based Indirect Inference :::::::::::::: 35 4.3.3 Theoretical Properties :::::::::::::::::::: 37 4.4 Simulation Study ::::::::::::::::::::::::::: 41 4.4.1 Choice of the number of quantiles, m:::::::::::: 41 4.4.2 Weight Matrix :::::::::::::::::::::::: 45 4.4.3 Comparison Between different Methods ::::::::::: 47 4.5 Case Study :::::::::::::::::::::::::::::: 47 4.6 Conclusion ::::::::::::::::::::::::::::::: 52 4.7 Appendix: Influence Function and Robust Property of Quantiles : 53 5 Indirect Inference Applied to Income Distributions 55 5.1 Introduction to Some Inequality Measures ::::::::::::: 56 5.1.1 Lorenz curve ::::::::::::::::::::::::: 56 5.1.2 Gini Index and Other Inequality Measures ::::::::: 57 5.1.3 Some Popular Parametric Income Distributions :::::: 59 5.2 Indirect Inference Method :::::::::::::::::::::: 60 5.2.1 Indirect inference framework ::::::::::::::::: 60 5.2.2 Theoretical Properties :::::::::::::::::::: 60 5.2.3 Goodness of Fit Analysis ::::::::::::::::::: 61 5.2.4 Data :::::::::::::::::::::::::::::: 61 5.3 Simulation Study ::::::::::::::::::::::::::: 63 5.3.1 Numerical Optimization ::::::::::::::::::: 63 5.3.2 Monte Carlo Study :::::::::::::::::::::: 65 5.4 Case Study :::::::::::::::::::::::::::::: 66 5.4.1 Data :::::::::::::::::::::::::::::: 68 5.4.2 Result ::::::::::::::::::::::::::::: 69 5.4.3 Conclusions :::::::::::::::::::::::::: 72 6 Conclusions and Discussion 73 viii A Code 75 A.1 R code of Estimating income distribution by indirect inference :: 75 A.2 Matlab code for spacing based estimation of stable distribution :: 77 Bibliography 80 ix List of Figures 1.1 Pdf of stable distribution of different parameters :::::::::: 6 1.2 Stable noise with different tail parameters ::::::::::::: 8 3.1 Greenwood statistic vs.α::::::::::::::::::::::: 23 3.2 Greenwood statistic vs.(α; β) ::::::::::::::::::::: 24 4.1 Estimation Algorithm ::::::::::::::::::::::::: 37 4.2 MSE ofα ^ by different q ::::::::::::::::::::::: 43 4.3 MSE of β^ by different q ::::::::::::::::::::::: 43 4.4 MSE of scaleσ ^ by different q :::::::::::::::::::: 44 4.5 MSE of locationu ^ by different q ::::::::::::::::::: 44 4.6 Sum of MSE :::::::::::::::::::::::::::::: 45 4.7 Plot of index and return ::::::::::::::::::::::: 49 4.8 histogram ::::::::::::::::::::::::::::::: 50 4.9 QQ plot :::::::::::::::::::::::::::::::: 50 5.1 Lorenz Curve of lognormal and exponential ::::::::::::: 58 5.2 Objective function vs (θ1; θ2) ::::::::::::::::::::: 64 5.3 Objective function vs θ2 ::::::::::::::::::::::: 64 5.4 Boxplot ofµ:::::::::::::::::::::::::::::: ^ 65 5.5 Boxplot ofσ:::::::::::::::::::::::::::::: ^ 66 5.6 GDP growth rate ::::::::::::::::::::::::::: 67 5.7 Lorenz curve 2010 USA vs China :::::::::::::::::: 69 5.8 Lorenz Curve of USA 1981 vs 2010 ::::::::::::::::: 70 5.9 Lorenz Curve of China 1981 vs 2010 ::::::::::::::::: 70 5.10 Income distribution of China 1981 vs 2010 ::::::::::::: 71 5.11 Income distribution of USA 1981 vs 2010 :::::::::::::: 71 x List of Tables 3.1 Mean Square Error: MLE vs GSE :::::::::::::::::: 22 3.2 Mean square error and bias ofα ^ with various sample size ::::: 25 3.3 Mean square error and bias of (^α; β^) with various sample size ::: 26 3.4 Monte Carlo mean and mean square error of (^α; β^) with different methods ::::::::::::::::::::::::::::::::::: 27 4.1 MSE comparison: Identity matrix vs Two-step weight matrix ::: 46 4.2 MSE of different methods :::::::::::::::::::::: 48 4.3 Stable vs Skewed-t :::::::::::::::::::::::::: 52 5.1 Lorenz Curve for some distributions ::::::::::::::::: 57 5.2 Original Data ::::::::::::::::::::::::::::: 62 5.3 Transformed Data ::::::::::::::::::::::::::: 62 5.4 Goodness of Fit Assessment ::::::::::::::::::::: 63 5.5 Income inequality of USA: 1981 v.s 2010 :::::::::::::: 68 5.6 Income inequality of China: 1981 v.s 2010 ::::::::::::: 68 xi
Recommended publications
  • A Random Variable X with Pdf G(X) = Λα Γ(Α) X ≥ 0 Has Gamma

    A Random Variable X with Pdf G(X) = Λα Γ(Α) X ≥ 0 Has Gamma

    DISTRIBUTIONS DERIVED FROM THE NORMAL DISTRIBUTION Definition: A random variable X with pdf λα g(x) = xα−1e−λx x ≥ 0 Γ(α) has gamma distribution with parameters α > 0 and λ > 0. The gamma function Γ(x), is defined as Z ∞ Γ(x) = ux−1e−udu. 0 Properties of the Gamma Function: (i) Γ(x + 1) = xΓ(x) (ii) Γ(n + 1) = n! √ (iii) Γ(1/2) = π. Remarks: 1. Notice that an exponential rv with parameter 1/θ = λ is a special case of a gamma rv. with parameters α = 1 and λ. 2. The sum of n independent identically distributed (iid) exponential rv, with parameter λ has a gamma distribution, with parameters n and λ. 3. The sum of n iid gamma rv with parameters α and λ has gamma distribution with parameters nα and λ. Definition: If Z is a standard normal rv, the distribution of U = Z2 called the chi-square distribution with 1 degree of freedom. 2 The density function of U ∼ χ1 is −1/2 x −x/2 fU (x) = √ e , x > 0. 2π 2 Remark: A χ1 random variable has the same density as a random variable with gamma distribution, with parameters α = 1/2 and λ = 1/2. Definition: If U1,U2,...,Uk are independent chi-square rv-s with 1 degree of freedom, the distribution of V = U1 + U2 + ... + Uk is called the chi-square distribution with k degrees of freedom. 2 Using Remark 3. and the above remark, a χk rv. follows gamma distribution with parameters 2 α = k/2 and λ = 1/2.
  • Stat 5101 Notes: Brand Name Distributions

    Stat 5101 Notes: Brand Name Distributions

    Stat 5101 Notes: Brand Name Distributions Charles J. Geyer February 14, 2003 1 Discrete Uniform Distribution Symbol DiscreteUniform(n). Type Discrete. Rationale Equally likely outcomes. Sample Space The interval 1, 2, ..., n of the integers. Probability Function 1 f(x) = , x = 1, 2, . , n n Moments n + 1 E(X) = 2 n2 − 1 var(X) = 12 2 Uniform Distribution Symbol Uniform(a, b). Type Continuous. Rationale Continuous analog of the discrete uniform distribution. Parameters Real numbers a and b with a < b. Sample Space The interval (a, b) of the real numbers. 1 Probability Density Function 1 f(x) = , a < x < b b − a Moments a + b E(X) = 2 (b − a)2 var(X) = 12 Relation to Other Distributions Beta(1, 1) = Uniform(0, 1). 3 Bernoulli Distribution Symbol Bernoulli(p). Type Discrete. Rationale Any zero-or-one-valued random variable. Parameter Real number 0 ≤ p ≤ 1. Sample Space The two-element set {0, 1}. Probability Function ( p, x = 1 f(x) = 1 − p, x = 0 Moments E(X) = p var(X) = p(1 − p) Addition Rule If X1, ..., Xk are i. i. d. Bernoulli(p) random variables, then X1 + ··· + Xk is a Binomial(k, p) random variable. Relation to Other Distributions Bernoulli(p) = Binomial(1, p). 4 Binomial Distribution Symbol Binomial(n, p). 2 Type Discrete. Rationale Sum of i. i. d. Bernoulli random variables. Parameters Real number 0 ≤ p ≤ 1. Integer n ≥ 1. Sample Space The interval 0, 1, ..., n of the integers. Probability Function n f(x) = px(1 − p)n−x, x = 0, 1, . , n x Moments E(X) = np var(X) = np(1 − p) Addition Rule If X1, ..., Xk are independent random variables, Xi being Binomial(ni, p) distributed, then X1 + ··· + Xk is a Binomial(n1 + ··· + nk, p) random variable.
  • On a Problem Connected with Beta and Gamma Distributions by R

    On a Problem Connected with Beta and Gamma Distributions by R

    ON A PROBLEM CONNECTED WITH BETA AND GAMMA DISTRIBUTIONS BY R. G. LAHA(i) 1. Introduction. The random variable X is said to have a Gamma distribution G(x;0,a)if du for x > 0, (1.1) P(X = x) = G(x;0,a) = JoT(a)" 0 for x ^ 0, where 0 > 0, a > 0. Let X and Y be two independently and identically distributed random variables each having a Gamma distribution of the form (1.1). Then it is well known [1, pp. 243-244], that the random variable W = X¡iX + Y) has a Beta distribution Biw ; a, a) given by 0 for w = 0, (1.2) PiW^w) = Biw;x,x)=\ ) u"-1il-u)'-1du for0<w<l, Ío T(a)r(a) 1 for w > 1. Now we can state the converse problem as follows : Let X and Y be two independently and identically distributed random variables having a common distribution function Fix). Suppose that W = Xj{X + Y) has a Beta distribution of the form (1.2). Then the question is whether £(x) is necessarily a Gamma distribution of the form (1.1). This problem was posed by Mauldon in [9]. He also showed that the converse problem is not true in general and constructed an example of a non-Gamma distribution with this property using the solution of an integral equation which was studied by Goodspeed in [2]. In the present paper we carry out a systematic investigation of this problem. In §2, we derive some general properties possessed by this class of distribution laws Fix).
  • Parametric Models

    Parametric Models

    An Introduction to Event History Analysis Oxford Spring School June 18-20, 2007 Day Two: Regression Models for Survival Data Parametric Models We’ll spend the morning introducing regression-like models for survival data, starting with fully parametric (distribution-based) models. These tend to be very widely used in social sciences, although they receive almost no use outside of that (e.g., in biostatistics). A General Framework (That Is, Some Math) Parametric models are continuous-time models, in that they assume a continuous parametric distribution for the probability of failure over time. A general parametric duration model takes as its starting point the hazard: f(t) h(t) = (1) S(t) As we discussed yesterday, the density is the probability of the event (at T ) occurring within some differentiable time-span: Pr(t ≤ T < t + ∆t) f(t) = lim . (2) ∆t→0 ∆t The survival function is equal to one minus the CDF of the density: S(t) = Pr(T ≥ t) Z t = 1 − f(t) dt 0 = 1 − F (t). (3) This means that another way of thinking of the hazard in continuous time is as a conditional limit: Pr(t ≤ T < t + ∆t|T ≥ t) h(t) = lim (4) ∆t→0 ∆t i.e., the conditional probability of the event as ∆t gets arbitrarily small. 1 A General Parametric Likelihood For a set of observations indexed by i, we can distinguish between those which are censored and those which aren’t... • Uncensored observations (Ci = 1) tell us both about the hazard of the event, and the survival of individuals prior to that event.
  • A Form of Multivariate Gamma Distribution

    A Form of Multivariate Gamma Distribution

    Ann. Inst. Statist. Math. Vol. 44, No. 1, 97-106 (1992) A FORM OF MULTIVARIATE GAMMA DISTRIBUTION A. M. MATHAI 1 AND P. G. MOSCHOPOULOS2 1 Department of Mathematics and Statistics, McOill University, Montreal, Canada H3A 2K6 2Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968-0514, U.S.A. (Received July 30, 1990; revised February 14, 1991) Abstract. Let V,, i = 1,...,k, be independent gamma random variables with shape ai, scale /3, and location parameter %, and consider the partial sums Z1 = V1, Z2 = 171 + V2, . • •, Zk = 171 +. • • + Vk. When the scale parameters are all equal, each partial sum is again distributed as gamma, and hence the joint distribution of the partial sums may be called a multivariate gamma. This distribution, whose marginals are positively correlated has several interesting properties and has potential applications in stochastic processes and reliability. In this paper we study this distribution as a multivariate extension of the three-parameter gamma and give several properties that relate to ratios and conditional distributions of partial sums. The general density, as well as special cases are considered. Key words and phrases: Multivariate gamma model, cumulative sums, mo- ments, cumulants, multiple correlation, exact density, conditional density. 1. Introduction The three-parameter gamma with the density (x _ V)~_I exp (x-7) (1.1) f(x; a, /3, 7) = ~ , x>'7, c~>O, /3>0 stands central in the multivariate gamma distribution of this paper. Multivariate extensions of gamma distributions such that all the marginals are again gamma are the most common in the literature.
  • Estimation of Α-Stable Sub-Gaussian Distributions for Asset Returns

    Estimation of Α-Stable Sub-Gaussian Distributions for Asset Returns

    Estimation of α-Stable Sub-Gaussian Distributions for Asset Returns Sebastian Kring, Svetlozar T. Rachev, Markus Hochst¨ otter¨ , Frank J. Fabozzi Sebastian Kring Institute of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Postfach 6980, 76128, Karlsruhe, Germany E-mail: [email protected] Svetlozar T. Rachev Chair-Professor, Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Postfach 6980, 76128 Karlsruhe, Germany and Department of Statistics and Applied Probability University of California, Santa Barbara CA 93106-3110, USA E-mail: [email protected] Markus Hochst¨ otter¨ Institute of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Postfach 6980, 76128, Karlsruhe, Germany Frank J. Fabozzi Professor in the Practice of Finance School of Management Yale University New Haven, CT USA 1 Abstract Fitting multivariate α-stable distributions to data is still not feasible in higher dimensions since the (non-parametric) spectral measure of the characteristic func- tion is extremely difficult to estimate in dimensions higher than 2. This was shown by Chen and Rachev (1995) and Nolan, Panorska and McCulloch (1996). α-stable sub-Gaussian distributions are a particular (parametric) subclass of the multivariate α-stable distributions. We present and extend a method based on Nolan (2005) to estimate the dispersion matrix of an α-stable sub-Gaussian dis- tribution and estimate the tail index α of the distribution. In particular, we de- velop an estimator for the off-diagonal entries of the dispersion matrix that has statistical properties superior to the normal off-diagonal estimator based on the covariation.
  • C 2013 Johannes Traa

    C 2013 Johannes Traa

    c 2013 Johannes Traa MULTICHANNEL SOURCE SEPARATION AND TRACKING WITH PHASE DIFFERENCES BY RANDOM SAMPLE CONSENSUS BY JOHANNES TRAA THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2013 Urbana, Illinois Adviser: Assistant Professor Paris Smaragdis ABSTRACT Blind audio source separation (BASS) is a fascinating problem that has been tackled from many different angles. The use case of interest in this thesis is that of multiple moving and simultaneously-active speakers in a reverberant room. This is a common situation, for example, in social gatherings. We human beings have the remarkable ability to focus attention on a particular speaker while effectively ignoring the rest. This is referred to as the \cocktail party effect” and has been the holy grail of source separation for many decades. Replicating this feat in real-time with a machine is the goal of BASS. Single-channel methods attempt to identify the individual speakers from a single recording. However, with the advent of hand-held consumer electronics, techniques based on microphone array processing are becoming increasingly popular. Multichannel methods record a sound field from various locations to incorporate spatial information. If the speakers move over time, we need an algorithm capable of tracking their positions in the room. For compact arrays with 1-10 cm of separation between the microphones, this can be accomplished by applying a temporal filter on estimates of the directions-of-arrival (DOA) of the speakers. In this thesis, we review recent work on BSS with inter-channel phase difference (IPD) features and provide extensions to the case of moving speakers.
  • 1 Stable Distributions in Streaming Computations

    1 Stable Distributions in Streaming Computations

    1 Stable Distributions in Streaming Computations Graham Cormode1 and Piotr Indyk2 1 AT&T Labs – Research, 180 Park Avenue, Florham Park NJ, [email protected] 2 Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge MA, [email protected] 1.1 Introduction In many streaming scenarios, we need to measure and quantify the data that is seen. For example, we may want to measure the number of distinct IP ad- dresses seen over the course of a day, compute the difference between incoming and outgoing transactions in a database system or measure the overall activity in a sensor network. More generally, we may want to cluster readings taken over periods of time or in different places to find patterns, or find the most similar signal from those previously observed to a new observation. For these measurements and comparisons to be meaningful, they must be well-defined. Here, we will use the well-known and widely used Lp norms. These encom- pass the familiar Euclidean (root of sum of squares) and Manhattan (sum of absolute values) norms. In the examples mentioned above—IP traffic, database relations and so on—the data can be modeled as a vector. For example, a vector representing IP traffic grouped by destination address can be thought of as a vector of length 232, where the ith entry in the vector corresponds to the amount of traffic to address i. For traffic between (source, destination) pairs, then a vec- tor of length 264 is defined. The number of distinct addresses seen in a stream corresponds to the number of non-zero entries in a vector of counts; the differ- ence in traffic between two time-periods, grouped by address, corresponds to an appropriate computation on the vector formed by subtracting two vectors, and so on.
  • Wrapped Log Kumaraswamy Distribution and Its Applications

    Wrapped Log Kumaraswamy Distribution and Its Applications

    International Journal of Mathematics and Computer Research ISSN: 2320-7167 Volume 06 Issue 10 October-2018, Page no. - 1924-1930 Index Copernicus ICV: 57.55 DOI: 10.31142/ijmcr/v6i10.01 Wrapped Log Kumaraswamy Distribution and its Applications K.K. Jose1, Jisha Varghese2 1,2Department of Statistics, St.Thomas College, Palai,Arunapuram Mahatma Gandhi University, Kottayam, Kerala- 686 574, India ARTICLE INFO ABSTRACT Published Online: A new circular distribution called Wrapped Log Kumaraswamy Distribution (WLKD) is introduced 10 October 2018 in this paper. We obtain explicit form for the probability density function and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for estimation of parameters. The proposed model is also applied to a Corresponding Author: real data set on repair times and it is established that the WLKD is better than log Kumaraswamy K.K. Jose distribution for modeling the data. KEYWORDS: Log Kumaraswamy-Geometric distribution, Wrapped Log Kumaraswamy distribution, Trigonometric moments. 1. Introduction a test, atmospheric temperatures, hydrological data, etc. Also, Kumaraswamy (1980) introduced a two-parameter this distribution could be appropriate in situations where distribution over the support [0, 1], called Kumaraswamy scientists use probability distributions which have infinite distribution (KD), for double bounded random processes for lower and (or) upper bounds to fit data, where as in reality the hydrological applications. The Kumaraswamy distribution is bounds are finite. Gupta and Kirmani (1988) discussed the very similar to the Beta distribution, but has the important connection between non-homogeneous Poisson process advantage of an invertible closed form cumulative (NHPP) and record values.
  • Chapter 2 Parameter Estimation for Wrapped Distributions

    Chapter 2 Parameter Estimation for Wrapped Distributions

    ABSTRACT RAVINDRAN, PALANIKUMAR. Bayesian Analysis of Circular Data Using Wrapped Dis- tributions. (Under the direction of Associate Professor Sujit K. Ghosh). Circular data arise in a number of different areas such as geological, meteorolog- ical, biological and industrial sciences. We cannot use standard statistical techniques to model circular data, due to the circular geometry of the sample space. One of the com- mon methods used to analyze such data is the wrapping approach. Using the wrapping approach, we assume that, by wrapping a probability distribution from the real line onto the circle, we obtain the probability distribution for circular data. This approach creates a vast class of probability distributions that are flexible to account for different features of circular data. However, the likelihood-based inference for such distributions can be very complicated and computationally intensive. The EM algorithm used to compute the MLE is feasible, but is computationally unsatisfactory. Instead, we use Markov Chain Monte Carlo (MCMC) methods with a data augmentation step, to overcome such computational difficul- ties. Given a probability distribution on the circle, we assume that the original distribution was distributed on the real line, and then wrapped onto the circle. If we can unwrap the distribution off the circle and obtain a distribution on the real line, then the standard sta- tistical techniques for data on the real line can be used. Our proposed methods are flexible and computationally efficient to fit a wide class of wrapped distributions. Furthermore, we can easily compute the usual summary statistics. We present extensive simulation studies to validate the performance of our method.
  • Robust Estimation on a Parametric Model Via Testing 3 Estimators

    Robust Estimation on a Parametric Model Via Testing 3 Estimators

    Bernoulli 22(3), 2016, 1617–1670 DOI: 10.3150/15-BEJ706 Robust estimation on a parametric model via testing MATHIEU SART 1Universit´ede Lyon, Universit´eJean Monnet, CNRS UMR 5208 and Institut Camille Jordan, Maison de l’Universit´e, 10 rue Tr´efilerie, CS 82301, 42023 Saint-Etienne Cedex 2, France. E-mail: [email protected] We are interested in the problem of robust parametric estimation of a density from n i.i.d. observations. By using a practice-oriented procedure based on robust tests, we build an estimator for which we establish non-asymptotic risk bounds with respect to the Hellinger distance under mild assumptions on the parametric model. We show that the estimator is robust even for models for which the maximum likelihood method is bound to fail. A numerical simulation illustrates its robustness properties. When the model is true and regular enough, we prove that the estimator is very close to the maximum likelihood one, at least when the number of observations n is large. In particular, it inherits its efficiency. Simulations show that these two estimators are almost equal with large probability, even for small values of n when the model is regular enough and contains the true density. Keywords: parametric estimation; robust estimation; robust tests; T-estimator 1. Introduction We consider n independent and identically distributed random variables X1,...,Xn de- fined on an abstract probability space (Ω, , P) with values in the measure space (X, ,µ). E F We suppose that the distribution of Xi admits a density s with respect to µ and aim at estimating s by using a parametric approach.
  • Wooldridge, Introductory Econometrics, 4Th Ed. Appendix C

    Wooldridge, Introductory Econometrics, 4Th Ed. Appendix C

    Wooldridge, Introductory Econometrics, 4th ed. Appendix C: Fundamentals of mathemati- cal statistics A short review of the principles of mathemati- cal statistics. Econometrics is concerned with statistical inference: learning about the char- acteristics of a population from a sample of the population. The population is a well-defined group of subjects{and it is important to de- fine the population of interest. Are we trying to study the unemployment rate of all labor force participants, or only teenaged workers, or only AHANA workers? Given a population, we may define an economic model that contains parameters of interest{coefficients, or elastic- ities, which express the effects of changes in one variable upon another. Let Y be a random variable (r.v.) representing a population with probability density function (pdf) f(y; θ); with θ a scalar parameter. We assume that we know f;but do not know the value of θ: Let a random sample from the pop- ulation be (Y1; :::; YN ) ; with Yi being an inde- pendent random variable drawn from f(y; θ): We speak of Yi being iid { independently and identically distributed. We often assume that random samples are drawn from the Bernoulli distribution (for instance, that if I pick a stu- dent randomly from my class list, what is the probability that she is female? That probabil- ity is γ; where γ% of the students are female, so P (Yi = 1) = γ and P (Yi = 0) = (1 − γ): For many other applications, we will assume that samples are drawn from the Normal distribu- tion. In that case, the pdf is characterized by two parameters, µ and σ2; expressing the mean and spread of the distribution, respectively.