Some sidenotes from price theory iii:
Definitions
For distributions (pdf) $f_1(\cdot), f_2(\cdot)$:
Monotone Likelihood Ratio (MLR): let ratio
$$ \phi(l) = \frac{f_1(l)}{f_2(l)}. $$$f_1$ dominate $f_2$ (in MLR sense) iff. $\phi(l)$ is weakly increasing in $l$.
Intuition: Pointwise relative density shifts up
Hazard Rate Dominance (HRD): define the hazard rate (aka conditional failure rate):
$$ h_{f_i}(l) = \frac{f_i(l)}{1 - F_i(l)} \quad i =1, 2. $$$f_1$ dominate $f_2$ (in HRD sense) iff. $h_{f_1}(l)\le h_{f_2}(l), \forall l$.
Intuition: Conditional survival always higher
First Order Stochastic Dominance (FOSD): define the cumulative distribution function:
$$ F_i(l) = \int_{-\infty}^{l} f_i(t)\, \text{d}t \quad i = 1, 2. $$$f_1$ dominates $f_2$ (in FOSD sense) iff. $F_1(l) \leq F_2(l), \forall l$.
Intuition: any non-decreasing utility prefers $f_1$.
Second Order Stochastic Dominance (SOSD): define the second-order distribution function:
$$ F_i^{(2)}(l) = \int_{-\infty}^{l} F_i(t)\, \text{d}t \quad i = 1, 2. $$$f_1$ dominates $f_2$ (in SOSD sense) iff. $F_1^{(2)}(l) \leq F_2^{(2)}(l),\forall l$.
Intuition: Any non-decreasing concave utility prefers $f_1$.
nth Order Stochastic Dominance: define the recursive distribution function:
$$ F_i^{(n)}(l) = \int_{-\infty}^{l} F_i^{(n-1)}(t)\, \text{d}t \quad i = 1, 2, $$with base case $F_i^{(1)}(l) \equiv F_i(l)$.
$f_1$ dominates $f_2$ (in nth-order SD sense) iff. $F_1^{(n)}(l) \leq F_2^{(n)}(l), \forall l$.
Dominance hirgachy:
$$ \text{MLR} \implies \text{HRD} \implies \text{FOSD} \implies \text{SOSD}\implies \cdots \implies \text{nth-order SD} $$The idea is,
- MLR is a local, multiplicative condition on densities
- HRD weakens this to conditional tail behavior
- FOSD weakens further to cumulative mass comparisons
- nth-order SD progressively weakens by averaging/integrating, expanding the class of distributions that can be ranked