Geometric Process and Its Application

Total Page：16

File Type：pdf, Size：1020Kb

Geometric Process and Its Application Yeh LAM, Department of Statistics and Actuarial Science, The University of Hong Kong and The Northeastern University at Qinhuangdao, 066004, PRC, E-mail: [email protected] Key words: Geometric Process, Renewal Process, Maintenance Problem, Data Analysis Mathematical Subject Classification: 60G55, 62M99, 90B25 Abstract: Lam [1,2] introduced the following geometric process. Deﬁnition. A stochastic process fXn; n = 1; 2; : : :g is a geometric process (GP), if there exists some a > 0 such n− that fa 1Xn; n = 1; 2; : : :g forms a renewal process. The number a is called the ratio of the geometric process. Clearly, a GP is stochastically increasing if the ratio 0 < a ≤ 1; it is stochastically decreasing if the ratio a ≥ 1. A GP will become a renewal process if the ratio a = 1. Thus, the GP is a simple monotone process and is a generalization of renewal process. 2 Let E(X1) = λ and V ar(X1) = σ , then λ E(Xn) = ; an−1 σ2 V ar(Xn) = : a2(n−1) Therefore, a, λ and σ2 are three important parameters in the GP. In this talk, we shall introduce the fundamental probability theory of a GP, then consider the hypothesis testing of the GP and estimation problem of three important parameters in the GP. On the other hand, the application of GP to reliability, especially to the maintenance problem is studied. Furthermore, the application of GP to the analysis of data from a sequence of events with trend is investigated. From real data analysis, it has been shown that on average, the GP model is the best one among four models including a Poisson process model and two nonhomogeneous Poisson process models. See Lam (2005) for a brief review and more references. References [1] Lam, Y. (1988). A Note on the Optimal Replacement Problem, Adv. Appl. Prob., 20, 479-782. [2] Lam, Y. (1988). Geometric Processes and Replacement Problem, Acta Math. Appl. Sinica, 4, 366-377. [3] Lam, Y. (2005). Geometric Process, to appear in The Encyclopedia of Statistical Sciences, 2nd edition, N. Balakrishnan, C. Read, S. Kotz and B. Vidakovic, ed. John Wiley & Sons, Inc., New York. 1.

Copy Link

Recommended publications

Hyperwage Theory
Hyperwage Theory About the cover The cover is a glowing red-hot bark of a tree embedded with the greatest equations of nature that tremendously impacted the thought of mankind and the course of civilization. These are the Maxwell’s electromagnetic field equations, the Navier-Stokes Theorem, Euler’s identity, Newton’s second law of motion, Newton’s law of gravitation, the ideal gas equation of state, Stefan’s law, the Second Law of Thermodynamics, Einstein’s mass-energy equivalence, Einstein’s gravity tensor equation, Lorenz time dilation, Planck’s Law, Heisenberg’s Uncertainty Principle, Schrodinger’s equation, Shannon’s Theorem, and Feynman diagrams in quantum electrodynamics. Concept and design by Thads Bentulan Hyperwage Theory The Misadventures of the Street Strategist Volume 10 Thads Bentulan Street Strategist First Edition 2008 Street Strategist Publications Limited Hyperwage Theory The Misadventures of the Street Strategist Volume 10 by Thads Bentulan ISBN 978-988-17536-9-4 This book is a compilation of articles that first appeared in BusinessWorld, unless otherwise noted. While the author endeavored to credit sources whenever possible, he welcomes corrections for any unacknowledged material. No part of this book may be reproduced, stored, or utilized in any form or by any means, electronic or mechanical, including photocopying, or recording, or by any information storage and retrieval system, including internet websites, without prior written permission. Copyright 2005-2008 by Thads Bentulan [email protected] 08/24/09
[Show full text]
The 7Th Workshop on MARKOV PROCESSES and RELATED TOPICS
The 7th Workshop on MARKOV PROCESSES AND RELATED TOPICS July 19 - 23, 2010 No.6 Lecture room on 3th ﬂoor, Jingshi Building (京京京师师师大大大厦厦厦) Beijing Normal University Organizers: Mu-Fa Chen, Zeng-Hu Li, Feng-Yu Wang Supported by National Natural Science Foundation of China (No. 10721091) Probability Group, Stochastic Research Center School of Mathematical Sciences, Beijing Normal University No.19, Xinjiekouwai St., Beijing 100875, China Phone and Fax: 86-10-58809447 E-mail: [email protected] Website: http://math.bnu.edu.cn/probab/Workshop2010 Schedule 1 July 19 July 20 July 21 July 22 July 23 Chairman Mu-Fa Chen Fu-Zhou Gong Shui Feng Jie Xiong Dayue Chen Opening Shuenn-Jyi Sheu Anyue Chen Leonid Mytnik Zengjing Chen 8:30{8:40 8:30{9:00 8:30{9:00 8:30{9:00 8:30{9:00 M. Fukushima Feng-Yu Wang Jiashan Tang Quansheng Liu Dong Han 8:40{9:30 9:00{9:30 9:00{9:30 9:00{9:30 9:00{9:30 Speaker Tea Break Zhen-Qing Chen Renming Song Xia Chen Hao Wang Zongxia Liang 10:00{10:30 10:00{10:30 10:00{10:30 10:00{10:305 10:00{10:30 Panki Kim Christian Leonard Yutao Ma Xiaowen Zhou Fubao Xi 10:30{11:00 10:30{11:00 10:30{11:00 10:30{11:00 10:30{11:00 Jinghai Shao Xu Zhang Liang-Hui Xia Hui He Liqun Niu 11:00{11:30 11:00{11:20 11:00{11:20 11:00{11:20 11:00{11:20 Lunch Chairman Feng-Yu Wang Shizan Fang Zengjing Chen Zeng-Hu Li Chii-Ruey Hwang Tusheng Zhang Dayue Chen Shizan Fang 14:30{15:00 14:30{15:00 14:30{15:00 14:30{15:00 Ivan Gentil Xiang-Dong Li Alok Goswami Yimin Xiao 15:00{15:30 15:00{15:30 15:00{15:30 15:00{15:30 Speaker Tea Break Fu-Zhou Gong Jinwen
[Show full text]
Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality
Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality Nikolai N. Leonenko School of Mathematics Cardiff University The second conference on Ambit Fields and Related Topics Aarhus August 15 Abstract Superpositions of stationary processes of Ornstein-Uhlenbeck (supOU) type have been introduced by Barndorff-Nielsen. We consider the constructions producing processes with long-range dependence and infinitely divisible marginal distributions. We consider additive functionals of supOU processes that satisfy the property referred to as intermittency. We investigate the properties of multifractal products of supOU processes. We present the general conditions for the Lq convergence of cumulative processes and investigate their q-th order moments and Rényifunctions. These functions are nonlinear, hence displaying the multifractality of the processes. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios. Acknowledgments Joint work with Denis Denisov, University of Manchester, UK Danijel Grahovac, University of Osijek, Croatia Alla Sikorskii, Michigan State University, USA Murad Taqqu, Boston University, USA OU type process I OU Type process is the unique strong solution of the SDE: dX (t) = −λX (t)dt + dZ(λt) where λ > 0, fZ(t)gt≥0 is a (non-decreasing, for this talk) Léviprocess, and an initial condition X0 is taken to be independent of Z(t). Note, in general Zt doesn't have to be a non-decreasing Lévyprocess. I For properties of OU type processes and their generalizations see Mandrekar & Rudiger (2007), Barndorff-Nielsen (2001), Barndorff-Nielsen & Stelzer (2011). I The a.s.
[Show full text]
Program Book
TABLE OF CONTENTS 5 Welcome Message 6 Administrative Program 7 Social Program 9 General Information 13 Program Welcome Message On behalf of the Organizing Committee, I welcome you to Mérida and the 15th Latin American Congress on Probability and Mathematical Statistics. We are truly honored by your presence and sincerely hope that the experience will prove to be professionally rewarding, as well as memorable on a personal level. It goes without saying that your participation is highly significant for promoting the development of this subject matter in our region. Please do not hesitate to ask any member of the staff for assistance during your stay, information regarding academic activities and conference venues, or any other questions you may have. The city of Mérida is renowned within Mexico for its warm hospitality, amid gastronomical, archaeological, and cultural treasures. Do take advantage of any free time to explore the city and its surroundings! Dr. Daniel Hernández Chair of the Organizing Committee, XV CLAPEM Administrative Program REGISTRATION The registration will be at the venue Gamma Mérida El Castellano Hotel. There will be a registration desk at the Lobby. Sunday, December 1, 2019 From 16:00 hrs to 22:00 hrs. Monday, December 2, 2019 From 8:00 to 17:00 hrs. Tuesday, December 3, 2019 From 8:00 to 17:00 hrs Thursday, December 4, 2019 From 8:00 to 16:00 hrs. DO YOU NEED AN INVOICE FOR YOUR REGISTRATION FEE? Please ask for it at the Registration Desk at your arrival. COFFEE BREAKS During the XV CLAPEM there will be coffee break services which will be announced in the program and will be displayed in both venues: Gamma Mérida El Castellano Hotel and CCU, UADY.
[Show full text]
A Geometric-Process Maintenance Model for a Deteriorating System Under a Random Environment Yeh Lam and Yuan Lin Zhang
IEEE TRANSACTIONS ON RELIABILITY, VOL. 52, NO. 1, MARCH 2003 83 A Geometric-Process Maintenance Model for a Deteriorating System Under a Random Environment Yeh Lam and Yuan Lin Zhang Abstract—This paper studies a geometric-process main- Cdf[ ] tenance-model for a deteriorating system under a random pdf[ ] environment. Assume that the number of random shocks, up to Cdf[ ] time , produced by the random environment forms a counting process. Whenever a random shock arrives, the system operating pdf[ ] time is reduced. The successive reductions in the system operating Cdf[ ] time are statistically independent and identically distributed number of system-failures random variables. Assume that the consecutive repair times of an optimal for minimizing the system after failures, form an increasing geometric process; system reward rate under the condition that the system suffers no random shock, the successive operating times of the system after repairs constitute a reduction in the system operating time after # decreasing geometric process. A replacement policy , by which system operating time after repair # , as- the system is replaced at the time of the failure , is adopted. An suming that there is no random shock explicit expression for the average cost rate (long-run average cost the real system operating time after repair # per unit time) is derived. Then, an optimal replacement policy is system repair time after failure # determined analytically. As a particular case, a compound Poisson process model is also studied. system replacement time E[ ] Index Terms—Compound Poisson process, geometric process, random shocks, renewal process, replacement policy. E[ ] E[ ]. 1 ACRONYMS I. INTRODUCTION ACR average cost rate: long-run average cost per unit time T THE initial stage of research in maintenance problems Cdf cumulative distribution function A of a repairable system, a common assumption is “repair is CPPM compound Poisson process model perfect:” a repairable system after repair is “as good as new.” GP geometric process Obviously, this assumption is not always true.
[Show full text]
The Market Price of Risk for Delivery Periods: Arxiv:2002.07561V2 [Q-Fin
The Market Price of Risk for Delivery Periods: Pricing Swaps and Options in Electricity Markets∗ ANNIKA KEMPER MAREN D. SCHMECK ANNA KH.BALCI [email protected] [email protected] [email protected] Center for Mathematical Economics Center for Mathematical Economics Faculty of Mathematics Bielefeld University Bielefeld University Bielefeld University PO Box 100131 PO Box 100131 PO Box 100131 33501 Bielefeld, Germany 33501 Bielefeld, Germany 33501 Bielefeld, Germany 27th April 2020 Abstract In electricity markets, futures contracts typically function as a swap since they deliver the underlying over a period of time. In this paper, we introduce a market price for the delivery periods of electricity swaps, thereby opening an arbitrage-free pricing framework for derivatives based on these contracts. Furthermore, we use a weighted geometric averaging of an artiﬁcial geometric futures price over the corresponding delivery period. Without any need for approximations, this averaging results in geometric swap price dynamics. Our framework allows for including typical features as the Samuelson effect, seasonalities, and stochastic volatility. In particular, we investigate the pricing procedures for electricity swaps and options in line with Arismendi et al. (2016), Schneider and Tavin (2018), and Fanelli and Schmeck (2019). A numerical study highlights the differences between these models depending on the delivery period. arXiv:2002.07561v2 [q-fin.PR] 24 Apr 2020 JEL CLASSIFICATION: G130 · Q400 KEYWORDS: Electricity Swaps · Delivery Period · Market Price of Delivery Risk · Seasonality · Samuelson Effect · Stochastic Volatility · Option Pricing · Heston Model ∗Financial support by the German Research Foundation (DFG) through the Collaborative Research Centre ‘Taming uncertainty and proﬁting from randomness and low regularity in analysis, stochastics and their applications’ is gratefully acknowledged.
[Show full text]
Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method
mathematics Article Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method Alexander Zeifman 1,2,3,* , Yacov Satin 2, Ivan Kovalev 2, Rostislav Razumchik 1,3, Victor Korolev 1,3,4 1 Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, Russia; [email protected] 2 Department of Applied Mathematics, Vologda State University, Lenina 15, 160000 Vologda, Russia; [email protected] (Y.S.); [email protected] (I.K.) 3 Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia; [email protected] 4 Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia * Correspondence: [email protected] Abstract: The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can beneﬁt from methods providing ergodicity bounds because the latter can indicate how to choose the position and the length of the “distant time interval” (in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method—the logarithmic norm method—is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying M/M/2 queue; the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes; and the time-varying Citation: Zeifman, A.; Satin, Y.; Markovian bulk-arrival and bulk-service system with state-dependent control.
[Show full text]
Tukey Max-Stable Processes for Spatial Extremes
Tukey Max-Stable Processes for Spatial Extremes Ganggang Xu1 and Marc G. Genton2 September 11, 2016 Abstract We propose a new type of max-stable process that we call the Tukey max-stable process for spa- tial extremes. It brings additional flexibility to modeling dependence structures among spatial extremes. The statistical properties of the Tukey max-stable process are demonstrated theoreti- cally and numerically. Simulation studies and an application to Swiss rainfall data indicate the effectiveness of the proposed process. Some key words: Brown{Resnick process; Composite likelihood; Extremal coefficient; Extremal- t process; Geometric Gaussian process, Max-stable process; Spatial extreme. Short title: Tukey max-stable processes 1Department of Mathematical Sciences, Binghamton University, State University of New York, Binghamton, New York 13902, USA. E-mail: [email protected] 2CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia. E-mail: [email protected] 1 1 Introduction Max-stable processes are widely used to model spatial extremes because they naturally arise as the limits of pointwise maxima of rescaled stochastic processes (de Haan & Ferreira, 2006) and therefore extend generalized extreme-value distributions to that setting; see the reviews of recent advances in the statistical modeling of spatial extremes by Davison et al. (2012), Cooley et al. (2012), Ribatet (2013) and Davison & Huser (2015). A useful spectral characterisation of simple max-stable processes for which the margins are standardized to unit Fréchet distributions was 1 proposed by de Haan (1984), Penrose (1992) and Schlather (2002). Specifically, let fRjgj=1 be −2 1 the points of a Poisson process on (0; 1) with intensity r dr, and let fWj(s)gj=1 be indepen- dent replicates of a nonnegative stochastic process, W (s), on Rd with continuous sample paths satisfying EfW (s)g = 1 for all s 2 Rd.
[Show full text]
The General Theory of Stochastic Population Processes
THE GENERAL THEORY OF STOCHASTIC POPULATION PROCESSES BY J. E. MOYAL Australian National University, Canberra, Australia (1) 1. Introduction The purpose of the present paper is to lay down the foundations of a general theory of stochastic population processes (see Bartlett [2] for references to previous work on this subject). By population we mean here a collection of individuals, each of which may be found in any one state x of a fixed set of states X. The simplest type of population process is one where there is only one kind of individual and where the total size of the population is always finite with probability unity (finite uni- variate population process). The state of the whole population is characterized by the s~ates, say x 1, ..., xn, of its members, where each of the x~ ranges over X and n = 0,1, 2 .... ; thus we may have for example a biological population whose individuals are charac- terized by their age, weight, location, etc. or a population of stars characterized by their brightness, mass, position, velocity and so on. Such a population is stochastic in the sense that there is defined a probability distribution P on some a-field S of sub- sets of the space ~ of all population states; in w2 we develop the theory of such population probability spaces; the approach is similar to that of Bhabha [4], who, however, restricts himself to the case where X is the real line and P is absolutely continuous. By taking the individual state space X to be arbitrary (i.e., an abstract space), we are able to make the theory completely general, including for example the case of cluster processes, where the members of a given population are clusters, i.e., are them- selves populations, as in Neyman's theory of populations of galaxy clusters (cf.
[Show full text]
A Statistical Derivation of the Significant-Digit
A Statistical Derivation of the Significant-Digit Law* Theodore P. Hill School of Mathematics and Center for Applied Probability Georgia Institute of Technology Atlanta, GA 30332-0160 (e-mail: [email protected]) March 20, 1996 Summary The history, empirical evidence, and classical explanations of the Significant-Digit (or Benford’s) Law are reviewed, followed by a summary of recent invariant-measure char- acterizations and then a new statistical derivation of the law in the form of a CLT-like theorem for significant digits. If distributions are selected at random (in any “unbiased” way), and random samples are then taken from each of these distributions, the signifi- cant digits of the combined sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant-digit phenomenon in many different empirical contexts, and helps justify its recent application to computer design, mathematical modelling, and detection of fraud in accounting data. AMS 1991 subject classification: Primary 60F15, 60G57 Secondary 62P05 Keywords and phrases: First-digit law, Benford’s Law, significant-digit law, scale- invariance, base-invariance, random distributions, random probability measures, random k-samples, mantissa, logarithmic law, mantissa sigma algebra. * Research partly supported by NSF grant DMS 95-03375. 1 THE SIGNIFICANT-DIGIT LAW The Significant-Digit Law of statistical folklore is the empirical observation that in many naturally occurring tables of numerical data, the leading significant digits
[Show full text]
Matrix Analytic Methods with Markov Decision Processes for Hydrological Applications
MATRIX ANALYTIC METHODS WITH MARKOV DECISION PROCESSES FOR HYDROLOGICAL APPLICATIONS Aiden James Fisher December 2014 Thesis submitted for the degree of Doctor of Philosophy in the School of Mathematical Sciences at The University of Adelaide ÕÎêË@ ø@ QK. Acknowledgments I would like to acknowledge the help and support of various people for all the years I was working on this thesis. My supervisors David Green and Andrew Metcalfe for all the work they did mentoring me through out the process of researching, developing and writing this thesis. The countless hours they have spent reading my work and giving me valuable feedback, for supporting me to go to conferences, giving me paid work to support myself, and most importantly, agreeing to take me on as a PhD candidate in the first place. The School of Mathematical Sciences for giving me a scholarship and support for many years. I would especially like to thank all the office staff, Di, Jan, Stephanie, Angela, Cheryl, Christine, Renee, Aleksandra, Sarah, Rachael and Hannah, for all their help with organising offices, workspaces, computers, casual work, payments, the postgraduate workshop, conference travel, or just for having a coffee and a chat. You all do a great job and don't get the recognition you all deserve. All of my fellow postgraduates. In my time as a PhD student I would have shared offices with over 20 postgraduates, but my favourite will always be the \classic line" of G23: Rongmin Lu, Brian Webby, and Jessica Kasza. I would also like to give special thanks to Nick Crouch, Kelli Francis-Staite, Alexander Hanysz, Ross McNaughtan, Susana Soto Rojo, my brotégéStephen Wade, and my guide to postgraduate life Jason Whyte, for being my friends all these years.
[Show full text]
Combinatorial Clustering and the Beta Negative Binomial Process
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPAMI.2014.2318721, IEEE Transactions on Pattern Analysis and Machine Intelligence 1 Combinatorial Clustering and the Beta Negative Binomial Process Tamara Broderick, Lester Mackey, John Paisley, Michael I. Jordan Abstract We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process (NBP) as an inﬁnite- dimensional prior appropriate for such problems. We show that the NBP is conjugate to the beta process, and we characterize the posterior distribution under the beta-negative binomial process (BNBP) and hierarchical models based on the BNBP (the HBNBP). We study the asymptotic properties of the BNBP and develop a three-parameter extension of the BNBP that exhibits power-law behavior. We derive MCMC algorithms for posterior inference under the HBNBP, and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis. Index Terms beta process, admixture, mixed membership, Bayesian, nonparametric, integer latent feature model ✦ 1INTRODUCTION In traditional clustering problems the goal is to induce a set of latent classes and to assign each data point to one and only one class. This problem has been approached within a model-based framework via the use of ﬁnite mixture models, where the mixture components characterize the distributions associated with the classes, and the mixing proportions capture the mutual exclusivity of the classes (Fraley and Raftery, 2002; McLachlan and Basford, 1988).
[Show full text]