Discrete moment problems with distributions known to be unimodal

Abstract

Abstract. Discrete moment problems with given finite supports and unimodal distributions with known mode, are formulated and used to obtain sharp lower and upper bounds for expectations of higher order convex functions of discrete random variables as well as probabilities of the union of events. The bounds are based on the knowledge of some of the power moments of the random variables involved, or the binomial moments of the number of events which occur. The bounding problems are formulated as LP's and dual feasible basis structure theorems as well as the application of the dual method of linear programming provide us with the results. Applications in PERT and reliability are presented.Mathematics subject classification (2000): 60E15, 90B15, 90B25, 90C05.

On the relationship between the discrete and continuous bounding moment problems and their numerical solutions

“…Significant improvements in both of the lower and upper bounds can be obtained if we introduce shape constraints, in addition to the moment matching constraints in the LP of the DMP. We refer to the paper by Subasi et al (2009), where a unimodality constraint restricts the shape of the unknown distribution. The binomial moment problem is used for bounding the probability of the union of events and in addition to unimodality, the knowledge of the mode is assumed.…”

The discrete moment problem with nonconvex shape constraints

“…Perakis and Roels [41] also employ Popescu's framework to provide analytical robust solutions to the newsvendor problem under shape constraints that are better behaved than classical Scarf solutions. For the discrete moment problem, we are aware of only one paper [50] that considers shape constraints. Subasi et al [50] adapt Prékopa's linear programming (LP) methodology to include unimodality, which is modeled by an additional set of linear constraints.…”

“…For the discrete moment problem, we are aware of only one paper [50] that considers shape constraints. Subasi et al [50] adapt Prékopa's linear programming (LP) methodology to include unimodality, which is modeled by an additional set of linear constraints.…”

“…Both Popescu [42] and Subasi et al [50] illustrate how a certain class of constraints can be adapted into existing computational frameworks, SDP-based in the case of Popescu and LP-based in the case of Subasi et al. However, there remains relevant shape constraints that are practical significant and do not naturally fit into these settings.…”

“…(i) LC measures arise naturally in many applications. For example, Subasi et al [50] illustrate how the length of a critical path in a PERT model where individual task times are described by beta distributions has a LC distribution but its other properties (other than moments inferred by the beta distributions) are unknown. Log-concavity has a wide range of applications to statistical modeling and estimation [52], e.g., Duembge et al [17] show how the log-concavity allows the estimation of a distribution based on arbitrarily censored data (which is a common form of data for demand observations).…”