The geometric convergence rate of the classical change-point estimate

2009 ◽  
Vol 79 (2) ◽  
pp. 131-137
Author(s):  
Stergios B. Fotopoulos
Keyword(s):  
Convergence Rate ◽  
Change Point ◽  
Point Estimate ◽  
Geometric Convergence

Geometric convergence of value-iteration in multichain Markov decision problems

1979 ◽  
Vol 11 (01) ◽  
pp. 188-217 ◽  
Author(s):  
P. J. Schweitzer ◽  
A. Federgruen
Keyword(s):  
Convergence Rate ◽  
Iteration Method ◽  
Chain Structure ◽  
Decision Problems ◽  
Value Iteration ◽  
Convergence Factor ◽  
Markov Decision Problems ◽  
Geometric Convergence ◽  
Markov Decision ◽  
Maximal Gain

This paper considers undiscounted Markov decision problems. With no restriction (on either the periodicity or chain structure of the problem) we show that the value iteration method for finding maximal gain policies exhibits a geometric rate of convergence, whenever convergence occurs. In addition, we study the behaviour of the value-iteration operator; we give bounds for the number of steps needed for contraction, describe the ultimate behaviour of the convergence factor and give conditions for the existence of a uniform convergence rate.

Geometric renewal convergence rates from hazard rates

2001 ◽  
Vol 38 (01) ◽  
pp. 180-194 ◽  
Author(s):  
Kenneth S. Berenhaut ◽  
Robert Lund
Keyword(s):  
Convergence Rate ◽  
Hazard Rate ◽  
Convergence Rates ◽  
Hazard Rates ◽  
Finite Support ◽  
Good Convergence ◽  
Renewal Sequence ◽  
Geometric Convergence ◽  
New Better Than Used ◽  
Better Than

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (4) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is a constant, 0 < α < 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

scholarly journals STATISTICAL DETECTION OF SPATIAL PLANT PATTERNS UNDER THE EFFECT OF FOREST USE

2012 ◽  
Vol 05 (06) ◽  
pp. 1250054
Author(s):  
I. LÓPEZ ◽  
T. STANDOVÁR ◽  
J. GARAY ◽  
Z. VARGA ◽  
M. GÁMEZ
Keyword(s):  
Change Point ◽  
Bootstrap Method ◽  
Likelihood Estimation ◽  
Point Estimate ◽  
Line Transect ◽  
Tree Seedlings ◽  
Land System ◽  
Time Period ◽  
Forest Use ◽  
Floristic Data

The analysis of the consequences of land use (in particular forest use) may be considered as a partial step towards an integrated modeling of a land system. In the paper a forest territory is considered, where a gap-cut is made, and after a given time period the eventual change in the spatial distribution of undergrowth plants and tree seedlings is to be detected. Floristic data are collected along a line transect. A method for the detection of the change in the plant distributions along the transect is proposed to see whether this occurs at the geometric frontier of the human intervention. Since in the considered case the distribution of the change-point estimate is not known, as a substitute of its confidence interval, the so-called change-interval is calculated, using an adaptation of the bootstrap method. As an illustration, for a concrete plant species, the maximum likelihood estimation of the change-point and the calculation of the above mentioned change-interval is presented. Finally, the validation of the proposed method against some typical ecological situations is also presented, which provides a justification of the used algorithms.

The geometric convergence rate of a Lindley random walk

1997 ◽  
Vol 34 (3) ◽  
pp. 806-811
Author(s):  
Robert B. Lund
Keyword(s):  
Random Walk ◽  
Convergence Rate ◽  
Large Deviations ◽  
Total Variation ◽  
Short Proof ◽  
Random Variables ◽  
Invariant Distribution ◽  
Initial State ◽  
Geometric Convergence ◽  
Independent And Identically Distributed

Let {Xn} be the Lindley random walk on [0,∞) defined by . Here, {An} is a sequence of independent and identically distributed random variables. When for some r > 1, {Xn} converges at a geometric rate in total variation to an invariant distribution π; i.e. there exists s > 1 such that for every initial state . In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels (1975, 1976): the largest s satisfying the above relationship is and satisfies φ ‘(r0) = 0.

A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (04) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn – 1)Xn +(1 – ß (Zn – 1))Zn – 1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 &lt; c &lt; d &lt; 1. Together with (Zn ), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn – 1 where α is a constant, 0 &lt; α &lt; 1. We show that (Yn ) and (Zn ) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

The geometric convergence rate of a Lindley random walk

1997 ◽  
Vol 34 (03) ◽  
pp. 806-811
Author(s):  
Robert B. Lund
Keyword(s):  
Random Walk ◽  
Convergence Rate ◽  
Large Deviations ◽  
Total Variation ◽  
Short Proof ◽  
Random Variables ◽  
Invariant Distribution ◽  
Initial State ◽  
Geometric Convergence ◽  
Independent And Identically Distributed

Let {Xn } be the Lindley random walk on [0,∞) defined by . Here, {An } is a sequence of independent and identically distributed random variables. When for some r &gt; 1, {Xn } converges at a geometric rate in total variation to an invariant distribution π; i.e. there exists s &gt; 1 such that for every initial state . In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels (1975, 1976): the largest s satisfying the above relationship is and satisfies φ ‘(r 0) = 0.

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