scholarly journals Exact boundaries in sequential testing for phase-type distributions

2014 ◽  
Vol 51 (A) ◽  
pp. 347-358
Author(s):  
Hansjörg Albrecher ◽  
Peiman Asadi ◽  
Jevgenijs Ivanovs
Keyword(s):  
Ratio Test ◽  
Type I ◽  
Phase Type Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
Scale Functions ◽  
Phase Type ◽  
Sequential Probability ◽  
Decision Boundaries ◽  
Type Ii Errors

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

scholarly journals Exact boundaries in sequential testing for phase-type distributions

2014 ◽  
Vol 51 (A) ◽  
pp. 347-358
Author(s):  
Hansjörg Albrecher ◽  
Peiman Asadi ◽  
Jevgenijs Ivanovs
Keyword(s):  
Ratio Test ◽  
Type I ◽  
Phase Type Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
Scale Functions ◽  
Phase Type ◽  
Sequential Probability ◽  
Decision Boundaries ◽  
Type Ii Errors

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

scholarly journals Control of Traffic Intensity in Hyperexponential and Mixed Erlang Queueing Systems with a Method Based on SPRT

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Müjgan Zobu ◽  
Vedat Sağlam
Keyword(s):  
Operating Characteristic ◽  
Queueing Systems ◽  
Ratio Test ◽  
Traffic Intensity ◽  
Sample Number ◽  
Average Sample Number ◽  
Phase Type ◽  
Sequential Probability ◽  
Matlab Programming ◽  
Average Sample

The control of traffic intensity is one of the important problems in the study of queueing systems. Rao et al. (1984) developed a method to detect changes in the traffic intensity in queueing systems of the and types based on the Sequential Probability Ratio Test (SPRT). In this paper, SPRT is theoretically investigated for two different phase-type queueing systems which consist of hyperexponential and mixed Erlang. Also, for testing against , Operating Characteristic (OC) and Average Sample Number (ASN) functions are obtained with numerical methods using multipoint derivative equations according to different situations of and type errors. Afterward, numerical illustrations for each model are provided with Matlab programming.

scholarly journals A quintuple law for Markov additive processes with phase-type jumps

2010 ◽  
Vol 47 (2) ◽  
pp. 441-458 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Joint Distribution ◽  
First Passage Times ◽  
Insurance Risk ◽  
Maximal Level ◽  
First Passage ◽  
Additive Process ◽  
Process Map ◽  
Markov Additive Process ◽  
Phase Type ◽  
Passage Times

We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

Accuracy of sequential sampling plans based on Wald's sequential probability ratio test

1983 ◽  
Vol 13 (6) ◽  
pp. 1197-1203 ◽  
Author(s):  
Gary W. Fowler
Keyword(s):  
Monte Carlo ◽  
Sequential Sampling ◽  
Sequential Probability Ratio Test ◽  
Ratio Test ◽  
Type I ◽  
Sample Number ◽  
Sampling Plans ◽  
Probability Ratio ◽  
Sequential Probability ◽  
Relative Errors

Monte Carlo operating characteristic (OC) and average sample number (ASN) functions were compared with Wald's OC and ASN equations for sequential sampling plans based on Wald's sequential probability ratio test (SPRT) using the binomial, negative binomial, normal, and Poisson distributions. This comparison showed that the errors inherent in Wald's equations as a result of "overshooting" the decision boundaries of the SPRT can be large. Relative errors increased for the OC and ASN equations as the difference between the null (θ0)) and alternative (θ1) test parameter values increased. Relative errors also increased for the ASN equation as the probabilities of type I (α) and type II (β) errors increased. For discrete distributions, the relative errors also increased as θ0 increased with θ1/θ0 fixed. Wald's equations, in general, overestimate the true error probabilities and underestimate the true ASN. For the values of θ0, θ1, α, and β used in many sequential sampling plans in forestry, Wald's equations may not be adequate. For those cases where the errors in Wald's equations are important compared with the other errors associated with the sampling plan, two alternative Monte Carlo OC and ASN functions are proposed.

scholarly journals A quintuple law for Markov additive processes with phase-type jumps

2010 ◽  
Vol 47 (02) ◽  
pp. 441-458 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Joint Distribution ◽  
First Passage Times ◽  
Insurance Risk ◽  
Maximal Level ◽  
First Passage ◽  
Additive Process ◽  
Process Map ◽  
Markov Additive Process ◽  
Phase Type ◽  
Passage Times

We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

The use of Wald's sequential probability ratio test to develop composite three-decision sampling plans

1985 ◽  
Vol 15 (2) ◽  
pp. 326-330
Author(s):  
Gary W. Fowler
Keyword(s):  
Monte Carlo ◽  
Negative Binomial ◽  
Sequential Sampling ◽  
Ratio Test ◽  
Sample Number ◽  
Sampling Plans ◽  
Probability Ratio ◽  
Sequential Probability ◽  
Error Probabilities ◽  
Decision Boundaries

Many sequential sampling plans used in forest sampling are composite three-decision plans based on the simultaneous use of two of Wald's sequential probability ratio tests (SPRTs). Wald's operating characteristic (OC) and average sample number (ASN) equations for each SPRT are used to describe the properties of the composite sampling plan. Wald's equations are only approximate because of "overshooting" of the decision boundaries of the SPRTs and the two SPRTs operate simultaneously in the composite plan. Wald's and Monte Carlo OC and ASN functions were developed for (i) two SPRTs used to develop a three-decision composite plan and (ii) the three-decision composite plan based on the negative binomial distribution. Wald's equations, in general, overestimate the true error probabilities and underestimate the true ASN for a given SPRT. Wald's equations are less accurate in describing the properties of the three-decision plan. Monte Carlo functions are more accurate than Wald's functions. Recommendations are made regarding the choice between Wald's and Monte Carlo functions. A Monte Carlo procedure to modify the decision boundaries of the plan to yield actual error probabilities approximately equal to the desired error probabilities is suggested.

An old friend is better than two new ones? Approaches to market research in the context of digital transformation for the antitrust laws enforcement

2020 ◽  
pp. 37-55 ◽  
Author(s):  
A. E. Shastitko ◽  
O. A. Markova
Keyword(s):  
Business Models ◽  
Rapid Development ◽  
Digital Transformation ◽  
Market Definition ◽  
Type I ◽  
Type Ii ◽  
Digital Platforms ◽  
Advantages And Disadvantages ◽  
Pass Through ◽  
Type Ii Errors

Digital transformation has led to changes in business models of traditional players in the existing markets. What is more, new entrants and new markets appeared, in particular platforms and multisided markets. The emergence and rapid development of platforms are caused primarily by the existence of so called indirect network externalities. Regarding to this, a question arises of whether the existing instruments of competition law enforcement and market analysis are still relevant when analyzing markets with digital platforms? This paper aims at discussing advantages and disadvantages of using various tools to define markets with platforms. In particular, we define the features of the SSNIP test when being applyed to markets with platforms. Furthermore, we analyze adjustment in tests for platform market definition in terms of possible type I and type II errors. All in all, it turns out that to reduce the likelihood of type I and type II errors while applying market definition technique to markets with platforms one should consider the type of platform analyzed: transaction platforms without pass-through and non-transaction matching platforms should be tackled as players in a multisided market, whereas non-transaction platforms should be analyzed as players in several interrelated markets. However, if the platform is allowed to adjust prices, there emerges additional challenge that the regulator and companies may manipulate the results of SSNIP test by applying different models of competition.

The Potential of Tax Surprises to Affect Measures of Tax Avoidance and Researchers' Inferences

2018 ◽  
Vol 41 (1) ◽  
pp. 1-30 ◽  
Author(s):  
Chelsea Rae Austin
Keyword(s):  
Stock Options ◽  
Tax Avoidance ◽  
Type I ◽  
Type Ii ◽  
Effective Tax Rate ◽  
Tax Rate ◽  
Tax Savings ◽  
Type Ii Errors

ABSTRACT While not explicitly stated, many tax avoidance studies seek to investigate tax avoidance that is the result of firms' deliberate actions. However, measures of firms' tax avoidance can also be affected by factors outside the firms' control—tax surprises. This study examines potential complications caused by tax surprises when measuring tax avoidance by focusing on one specific type of surprise tax savings—the unanticipated tax benefit from employees' exercise of stock options. Because the cash effective tax rate (ETR) includes the benefits of this tax surprise, the cash ETR mismeasures firms' deliberate tax avoidance. The analyses conducted show this mismeasurement is material and can lead to both Type I and Type II errors in studies of deliberate tax avoidance. Suggestions to aid researchers in mitigating these concerns are also provided.

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