Parameter Estimation on Geometric Distribution of Order k with Different Reward Laws

Year 2021, Volume: 11 Issue: 1, 23 - 47, 30.06.2021

Fatih Şahin , Coşkun Kuş , İsmail Kınacı , Kadir Karakaya , Yunus Akdoğan

https://doi.org/10.37094/adyujsci.873193

Abstract

Let $\ksi1, \ksi2, ....$ be a sequence of independent trials with two possible outcomes, “0” and “1” where “1” represents the success of Type-I, and “0” denotes the success of Type-II. For nonnegative integers kr and kl using a reward scheme, we obtained the distribution of the number of trials (W) until the sum of consecutive rewards of Type-I is equal to or exceeds the
level kr, or the sum of consecutive rewards of Type-II is equal to or exceeds the level kl. The geometric distributed rewards are studied by Eryılmaz et al. in [1]. In this study, the survival function of W is obtained for binary sequence with Bernoulli and exponential rewards as well as geometric rewards. A simulation study is performed to compare the theoretical and simulated probabilities. Proportion estimates are also discussed for distribution of with geometric rewards.

Keywords

Geometric distribution of order k, Bernoulli reward, Exponential reward, Estimation

References

• [1] Eryılmaz, S., Gong, M., Xie, M., Generalized sooner waiting time problems in a sequence of trinary trials, Statistics and probability Letters, 115, 70-78, 2016.
• [2] Philippou, A.N., Georghiou, C., Philippou, G.N., A generalized geometric distribution and some of its properties, Statistics and probability Letters, 1, 171-175, 1983.
• [3] Eryılmaz, S., Geometric distribution of order k with a reward, Statistics and probability Letters, 92, 53-58, 2014.
• [4] Koutras, M.V., Alexandrou, V.A., Sooner waiting time problems in a sequence of trinary trials, Journal Applied Probability, 34, 593-609, 1997.
• [5] Demir, S., Eryılmaz, S., Run statistics in a sequence of arbitrarily dependent binary trials, Statistical Papers, 51(4), 959-973, 2010.
• [6] Mallor, F., Santos, J., Reliability of systems subject to shocks with a stochastic dependence for the damages, Sociedad de Estadistica e Investigacion Opevativa Test, 12(2), 427-444, 2003.
• [7] Khan, M.A., Khalique, A., Abouammoh, A.M., On estimating parameters in a discrete Weibull distribution, IEEE Transactions on Reliability, 38(3), 348-350, 1989.