2.2.3. Stochastic Ordering

Stochastic ordering is an important tool to compare continuous random variables. It is well-known that random variable *X*1 is smaller than random variable *X*2 in stochastic ordering (*X*1 ≤*st X*2) if *FX*1 (*x*) ≥ *FX*2 (*x*) for all *x*, and in likelihood ratio order (*X*1 ≤*lr X*2) if *fX*1 (*x*)/ *fX*2 (*x*) decreases with *x*. Using Theorem 1.C.1 and Theorem 2.A.1 of Shaked and Shanthikumar [20], the above stochastic orders hold according to the following implications,

$$X\_1 \le\_{lr} X\_2 \Rightarrow X\_1 \le\_{st} X\_2. \tag{13}$$

The proposition shows that the members of the MPN family can be stochastically ordered according to parameters values.

**Proposition 3.** *Let X*1 ∼ MPN (0, 1, *<sup>α</sup>*1) *and X*2 ∼ MPN (0, 1, *<sup>α</sup>*2)*. If α*1 > *α*2*, then X*1 ≤*lr X*2 *and, therefore, X*1 ≤*st X*2*.*

**Proof.** From the quotient of both densities, it follows that

$$\frac{f\_{\mathbb{X}\_2}(\mathbf{x}; \boldsymbol{\alpha}\_2)}{f\_{\mathbb{X}\_1}(\mathbf{x}; \boldsymbol{\alpha}\_1)} = \frac{\boldsymbol{\alpha}\_2}{\boldsymbol{\alpha}\_1} \left( \frac{2^{\boldsymbol{\alpha}\_1} - 1}{2^{\boldsymbol{\alpha}\_2} - 1} \right) [1 + \Phi(\mathbf{x})]^{\boldsymbol{a}\_2 - \boldsymbol{a}\_1} \boldsymbol{\alpha}$$

is non-decreasing if and only if *μ* (*x*) ≥ 0 for *x* ∈ (−∞, <sup>∞</sup>), where

$$\mu(\mathfrak{x}) = [1 + \Phi(\mathfrak{x})]^{\mathfrak{a}\_2 - \mathfrak{a}\_1}.$$

After some calculations, it is shown that

$$\mu'(\mathbf{x}) = (\alpha\_2 - \alpha\_1)\phi(\mathbf{x})[1 + \Phi(\mathbf{x})]^{a\_2 - a\_1 - 1}.$$

It is straightforward that for *α*1 > *α*2, then *μ* (*x*) < 0 for *x* ∈ (−∞, <sup>∞</sup>). Therefore, *fX*2 (*x*; *<sup>α</sup>*2)/ *fX*1 (*x*; *<sup>α</sup>*1) is decreasing in *x*, and consequently *X*1 ≤*lr X*2. The other implication follows immediately from (13). -
