We will now focus on the instance when k = 3, and we will denote the three alternatives as a, b, and c. For ease of notation, we will map each voter’s ranking, σi to a tuple (xi, yi, zi) ∈ {−1, 1}3, where xi = 1 if σi(a) > σi(b), and −1 otherwise. Similarly, yi = 1 if σi(b) > σi(c) and −1 otherwise, and zi = 1 if σi(c) > σi(a), and −1 otherwise. Finally, we will let x := (x1, . . . , xn), y := ...

This chapter is concerned with the analysis of statistical models for binary and ordinal outcomes. Binary data arise when a particular response variable of interest yi can take only two values, i.e. yi ∈ {0, 1}, where the index i = 1, . . . , n refers to units in the sample such as individuals, families, firms, and so on. Such dichotomous outcomes are widespread in the social and natural scienc...

Proof of Theorem 4. For wa ∈ [c + ck, rck w−c ], we have the boundary equilibrium. From its definition, yb(ξ) is decreasing in wa (that is, dyb(ξ) dwa < 0) and yb(ξ) = μ + ξ + G−1( r−c−ck r ) when wa = c + ck. The total supply chain profit at the boundary equilibrium is Πtot(yb(ξ), yb(ξ), ξ) = rEmin(μ + ξ + , yb(ξ)) − (c + ck)y(ξ) from Equations (16) and (17). This function is concave and incre...

QL Estimation for Independent Data. For i = 1, . . . ,K, let Yi denote the response variable for the ith individual, and xi = (xi1, . . . , xiv , . . . , xip) ′ be the associated p−dimensional covariate vector. Also, let β be the p−dimensional vector of regression effects of xi on yi. Further suppose that the responses are collected from K independent individuals. It is understandable that if t...

In a recent paper, Chua and Yi introduced the so-called uniform nonsingularity property for a nonlinear transformation on a Euclidean Jordan algebra and showed that it implies the global uniqueness property in the context of symmetric cone complementarity problems. In a related paper, Chua, Lin, and Yi raise the question of converse. In this paper, we show that, for linear transformations, the ...