© M. M. Choudhury, R. Bhattacharya, S. S. Maiti. Article available under the CC BY-SA 4.0 licence
ARTICLE
ABSTRACT
The Uniformly Minimum Variance Unbiased (UMVU) and the Maximum Likelihood (ML) estimations of R = P(X ≤ Y) and the associated variance are considered for independent discrete random variables X and Y. Assuming a discrete uniform distribution for X and the distribution of Y as a member of the discrete one parameter exponential family of distributions, theoretical expressions of such quantities are derived. Similar expressions are obtained when X and Y interchange their roles and both variables are from the discrete uniform distribution. A simulation study is carried out to compare the estimators numerically. A real application based on demand-supply system data is provided.
KEYWORDS
stress-strength model, uniformly minimum variance unbiased, maximum likelihood
REFERENCES
Ali, M.M, Pal, M, Woo, J, (2005). Inference On P(Y < X) in Generalized Uniform Distributions. Calcutta Statistical Association Bulletin, 57, pp. 35–48.
Belyaev, Y, Lumelskii, Y, (1988). Multidimensional Poisson Walks. Journal of Mathematical Sciences, 40, pp. 162–165.
Barbiero, A, (2013). Inference on Reliability of Stress-Strength Models for Poisson Data. Journal of Quality and Reliability Engineering, 2013, 8 pages.
Ferguson, S. T, (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press.
Hussain, T, Aslam, M, Ahmad, M, (2016). A Two Parameter Discrete Lindley Distribution. Revista Colombiana de Estadistica, 39(1), pp. 45–61.
Ivshin, V, V, Lumelskii, Ya, P, (1995). Statistical estimation problems in "Stress-Strength" models.Perm University Press, Perm, Russia.
Ivshin, V, V, (1996). Unbiased estimation of P(X < Y) and their variances in the case of Uniform and Two-Parameter Exponential distributions. Journal of Mathematical Sciences, 81, pp. 2790–2793.
Kotz, S, Lumelskii, Y, Pensky, M, (2003). The stress-strength model and its generalizations. Singapore: World Scientific.
Lehmann, E. L, Casella, G, (1998). Theory of Point Estimation. New York: Springer.
Maiti, S.S, (1995). Estimation of P(X ?Y) in geometric case. Journal of Indian Statistical Association, 33, pp. 87–91.
Obradovic, M, Jovanovic, M, Milosevic, B, Jevremovic, V, (2015). Estimation of P(X ? Y) for Geometric-Poisson model. Hacettepe Journal of Mathematics and Statistics, 44(4), pp. 949–964.
Rao, C. R, (1973). Linear Statistical Inference and Its Application. JohnWiley & Sons, Inc..
Sathe, Y.S, Dixit, U.J, (2001). Estimation of P(X ? Y) in the negative binomial distribution. Journal of Statistical Planning and Inference, 93, pp. 83–92.