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On generalized start-up demonstration tests

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Abstract

Start-up demonstration tests and various extensions have been discussed, in which a unit under the test is accepted or rejected according to some criteria. CSTF, CSCF, TSCF and TSTF are most well known start-up demonstration tests. In this paper, two kinds of more general start-up demonstration tests are introduced. CSTF, TSTF, TSCF and CSCF are all special situations of the new tests. For the new generalized start-up demonstration tests, under the assumption of independent and identically distributed trials for each test, the analytic expressions for the expectation, the probability mass function and the distribution of the test length, as well as the probability of acceptance or rejection of the unit are given. All the analyses are based on the finite Markov chain imbedding approach which avoids the complexities of the probability generating function approach and makes the results readily understood and easily extended to the non-i.i.d. cases. Furthermore, an optimal model for generalized start-up demonstration tests is proposed. Finally, a numerical example is presented to make our results more transparent, and it can demonstrate the advantages of the new tests.

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Abbreviations

CS:

consecutive successes;

CSTF:

consecutive successes total failures;

TSTF:

total successes total failures;

CSCF:

consecutive successes consecutive failures;

TSCF:

total successes consecutive failures;

R 1-CS/TS/R 2-CF/TF:

R 1 runs of consecutive successes, total successes, R 2 runs of consecutive failures, total failures;

R 1 :

the number of non-overlapping successful runs required for acceptance;

k C :

the number of consecutive successes in a successful run;

k T :

the number of total successes required for acceptance;

R 2 :

the number of non-overlapping failed runs required for rejection;

d C :

the number of consecutive failures in a failed run;

d T :

the number of total failures required for rejection;

T l :

the test length (e.g. the number of tests) until termination of the test;

α :

the producer’ risk;

β :

the consumer’ risk.

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Authors and Affiliations

  1. School of Management & Economics, Beijing Institute of Technology, Beijing, 100081, P.R. China

    Xian Zhao

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  1. Xian Zhao
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Correspondence to Xian Zhao.

Additional information

This work is supported by the National Natural Science Foundation of China (70901008), the Specialized Research Fund for the Doctoral Program of Higher Education (20091101120053), the Excellent Young Scholars Research Fund of Beijing Institute of Technology (2010YS1304), and the Excellent Talent Culture Foundation of Beijing (2009D013001000040).

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Zhao, X. On generalized start-up demonstration tests. Ann Oper Res 212, 225–239 (2014). https://doi.org/10.1007/s10479-012-1279-y

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