On the boundary crossing problem in memoryless models

Year 2023, , 785 - 794, 30.05.2023

Shangchen Yao Mohammad Khan

https://doi.org/10.15672/hujms.1038176

Cited By: 1

Abstract

The joint Laplace transform of the two sided boundary crossing stopping rule is known for the negative exponential model only under certain conditions. In this paper we eliminate the need for such conditions. Our results also apply to the boundary crossing problem for the geometric models. We further illustrate how the results can be used to obtain the distribution for the multidimensional boundary crossing stopping rules under the memoryless models.

Keywords

Average run length, exponential model, geometric model, SPRT, cusum, sequential analysis, waiting times

References

• [1] V. Abramov and M.K. Khan, A probabilistic analysis of trading the line strategy, Quant. Finance 8 (5), 499-512, 2008.
• [2] F.J. Anscombe and E.S. Page, Sequential tests for binomial and exponential populations, Biometrika 41 (1-2), 252-253, 1954.
• [3] A. Cardenas, S. Radosavac and S. Baras, Performance comparison of detection schemes for MAC layer misbehavior, 26th IEEE International Conference on Computer Communications, 1496-1504, Anchorage, AK, 2007.
• [4] Y.S. Chow, H. Robbins and D. Siegmund, The Theory of Optimal Stopping, Houghton Miffin Company, Boston, 1971.
• [5] Y.S. Chow, H. Robbins and H. Teicher, Moments of randomly stopped sums, Ann. Math. Statist. 36 (3), 789-799, 1965.
• [6] C.P. Cox and T.D. Roseberry, A note on the variance of the distribution of sample number in sequential probability ratio tests, Technometrics 8 (4), 700-704, 1966.
• [7] S.D. De and S. Zacks, Exact calculation of the distributions of the stopping times of two types of truncated SPRT for the mean of the exponential distribution, Methodol. Comput. Appl. Probab. 17 (4), 915-927, 2015.
• [8] M. DeGroot, Optimal Statistical Decisions, McGraw-Hill, Inc., New York, 1970.
• [9] F. Gan, Exact run length distributions for one-sided exponential cusum schemes, Statist. Sinica 2, 297-312, 1992.
• [10] W. Hoeffding, Lower bounds for the expected sample size and the average risk of a sequential procedure, Ann. Math. Statist. 31 (2), 352-368, 1960.
• [11] R.A. Khan, Detecting changes in probabilities of a multi-component process, Sequential Anal. 14 (4), 375-388, 1995.
• [12] R.A. Khan and M.K. Khan, On the use of the SPRT in determining the properties of some cusum procedures, Sequential Anal. 23 (3), 1-24, 2004.
• [13] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Springer-Verlag, Berlin, 2013.
• [14] E.S. Page, Continuous inspection schemes, Biometrika 41 (1-2), 100-115, 1954.
• [15] N.U. Prabhu, Stochastic Storage Systems: Queues, Insurance Risk, Dams, and Data Communication, 2nd ed., Springer, New York, 2012.
• [16] A.N. Shiryaev, Optimal Stopping Rules, Springer, Berlin, 2007.
• [17] D. Siegmund, Sequential Analysis: Tests and Confidence Intervals, Springer-Verlag, New York, 1985.
• [18] W. Stadje, On the SPRT for the mean for an exponential distribution, Statist. Probab. Lett. 5 (6), 389-395, 1987.
• [19] P.W. Starvaggi, and M.K. Khan, On the exact distribution of Wald’s SPRT for the negative exponential model, Sequential Anal. 36 (3), 299-308, 2017.
• [20] A. Wald, Sequential Analysis, John Wiley, New York, 1947.
• [21] S. Yao and M.K. Khan, On the ASN of cusum in multinomial models, Unpublished manuscript, 2022.