scholarly journals Comparison Results for GARCH Processes

2014 ◽  
Vol 51 (3) ◽  
pp. 685-698
Author(s):  
Fabio Bellini ◽  
Franco Pellerey ◽  
Carlo Sgarra ◽  
Salimeh Yasaei Sekeh
Keyword(s):  
Stochastic Orders ◽  
Stochastic Comparison ◽  
Convex Order ◽  
Comparison Results ◽  
Garch Processes ◽  
Garch Process

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

scholarly journals Comparison Results for GARCH Processes

2014 ◽  
Vol 51 (03) ◽  
pp. 685-698
Author(s):  
Fabio Bellini ◽  
Franco Pellerey ◽  
Carlo Sgarra ◽  
Salimeh Yasaei Sekeh
Keyword(s):  
Stochastic Orders ◽  
Stochastic Comparison ◽  
Convex Order ◽  
Comparison Results ◽  
Garch Processes ◽  
Garch Process

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

The martingale comparison method for Markov processes

2021 ◽  
Vol 58 (1) ◽  
pp. 164-176
Author(s):  
Benedikt Köpfer ◽  
Ludger Rüschendorf
Keyword(s):  
Markov Processes ◽  
Martingale Problem ◽  
Function Class ◽  
Comparison Method ◽  
Stochastic Orders ◽  
Regularity Conditions ◽  
Evolution System ◽  
Basic Step ◽  
Theory Of Evolution ◽  
Comparison Results

AbstractComparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (2) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

scholarly journals Stochastic Epidemic Models in Structured Populations Featuring Dynamic Vaccination and Isolation

2007 ◽  
Vol 44 (03) ◽  
pp. 571-585 ◽  
Author(s):  
Frank Ball ◽  
Philip D. O'Neill ◽  
James Pike
Keyword(s):  
Exposure Period ◽  
Structured Populations ◽  
Infectious Period ◽  
Stochastic Comparison ◽  
Threshold Parameter ◽  
Final Size ◽  
Period Distribution ◽  
Comparison Results ◽  
Stochastic Epidemic ◽  
Large Populations

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

Stochastic comparisons of interfailure times under a relevation replacement policy

2017 ◽  
Vol 54 (1) ◽  
pp. 134-145 ◽  
Author(s):  
Miguel A. Sordo ◽  
Georgios Psarrakos
Keyword(s):  
Stochastic Order ◽  
Replacement Policy ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Failure Times ◽  
Usual Stochastic Order ◽  
Comparison Results ◽  
Excess Wealth Order ◽  
Cumulative Residual

AbstractWe provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

scholarly journals Stochastic Epidemic Models in Structured Populations Featuring Dynamic Vaccination and Isolation

2007 ◽  
Vol 44 (3) ◽  
pp. 571-585 ◽  
Author(s):  
Frank Ball ◽  
Philip D. O'Neill ◽  
James Pike
Keyword(s):  
Exposure Period ◽  
Structured Populations ◽  
Infectious Period ◽  
Stochastic Comparison ◽  
Threshold Parameter ◽  
Final Size ◽  
Period Distribution ◽  
Comparison Results ◽  
Stochastic Epidemic ◽  
Large Populations

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

scholarly journals Trend-transformed independent vectors: construction and stochastic comparison results

Statistics ◽  
2020 ◽  
Vol 54 (4) ◽  
pp. 778-804
Author(s):  
F. G. Badía ◽  
Sophie Mercier ◽  
C. Sangüesa
Keyword(s):  
Stochastic Comparison ◽  
Comparison Results

Stochastic comparisons of random minima and maxima

1997 ◽  
Vol 34 (02) ◽  
pp. 420-425 ◽  
Author(s):  
Moshe Shaked ◽  
Tityik Wong
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Stochastic Comparison ◽  
Stochastic Comparisons ◽  
Comparison Results

Let X 1, X 2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X 1, X 2,…, XN ) and max{X 1, X 2,…, XN }.

Orderings of component level versus system level at active redundancies for coherent systems with dependent components

2021 ◽  
pp. 1-23
Author(s):  
Rongfang Yan ◽  
Junrui Wang ◽  
Bin Lu
Keyword(s):  
Hazard Rate ◽  
System Level ◽  
Stochastic Comparison ◽  
Coherent Systems ◽  
Component Level ◽  
Comparison Results ◽  
Versus System ◽  
Level Versus ◽  
Theoretical Results ◽  
Complete Matching

This paper investigates the issue of stochastic comparison of multi-active redundancies at the component level versus the system level. Based on the assumption that all components are statistically dependent, in the case of complete matching and nonmatching spares, we present some interesting comparison results in the sense of the hazard rate, reversed hazard rate and likelihood ratio orders, respectively. And we also obtain two comparison results between relative agings of resulting systems at the component level and the system level. Several numerical examples are provided to illustrate the theoretical results.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (02) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

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