General restrictions on tail probabilities

Abstract

When limited information on the distribution of a positive random variable X (continuous or discrete) is known (e.g., mode, mean, variance), the tail probability P(X 2 t) cannot be chosen independently. In this paper supremum and infimum for P(X 2 t) will be calculated over the set of positive random variables with unique mode, mean and/or variance given.

A simulation optimisation approach for inventory management decision support based on incomplete information

“…These results can be found in Appendix B (De Schepper and Heijnen 1995). However, in inventory management, it is also more interesting to know, given an expected stock-out probability the company wants to face, what the inventory level should be at least or at most.…”

“…This section describes the method (Heijnen and Goovaerts 1989;De Schepper and Heijnen 1995) to calculate upper and lower bounds on the number of stock-out units and the stock-out probability, when only the first and second moment and the mode of the demand distribution are known.…”

“…The transformation described in this section can be used when µ 1 ≤ m and the best upper bound for the tail probability if d > m has to be derived or the best lower De Schepper and Heijnen 1995). Denoting this distribution as F , a new distribution function H will be defined as…”

“…A stop-loss premium limits the risk X of an insurance company to a certain amount d. Furthermore, several authors deduced bounds on tail probabilities (De Schepper and Heijnen 1995).…”

Tight tail probability bounds for distribution-free decision making

“…This is particularly true for OR applications that consider uncertain parameters that are known to be nonnegative, such as inventory management, service operations, appointment scheduling and pricing mechanisms. We remark that a tight tail probability bound under knowledge of the mean, variance and a bounded support was derived by De Schepper and Heijnen (1995). Next to restricting the support, a second potential improvement of Chebychev's inequality concerns robustness for outliers.…”