Examples of "mlrp"
But not conversely: neither monotone hazard rates nor stochastic dominance imply the MLRP.
If the family of random variables has the MLRP in formula_22, a uniformly most powerful test can easily be determined for the hypotheses formula_28 versus formula_29.
Statistical models often assume that data are generated by a distribution from some family of distributions and seek to determine that distribution. This task is simplified if the family has the Monotone Likelihood Ratio Property (MLRP).
Example: Let formula_9 be an input into a stochastic technology --- worker's effort, for instance --- and formula_31 its output, the likelihood of which is described by a probability density function formula_32 Then the monotone likelihood ratio property (MLRP) of the family formula_7 is expressed as follows: for any formula_34, the fact that formula_35 implies that the ratio formula_36 is increasing in formula_31.
The "Meal, Long Range Patrol" (LRP) is essentially the same as the MCW, but with different accessory packs. The MLRP is designed for troops who may receive limited or no resupply, and weight of the ration is critical. The similar First Strike Ration is along the same lines, but requires no preparation and may be eaten on the go.
For two distributions that satisfy the definition with respect to some argument x, we say they "have the MLRP in "x"." For a family of distributions that all satisfy the definition with respect to some statistic "T"("X"), we say they "have the MLR in "T"("X")."
The MLRP is used to represent a data-generating process that enjoys a straightforward relationship between the magnitude of some observed variable and the distribution it draws from. If formula_4 satisfies the MLRP with respect to formula_5, the higher the observed value formula_2, the more likely it was drawn from distribution formula_7 rather than formula_8. As usual for monotonic relationships, the likelihood ratio's monotonicity comes in handy in statistics, particularly when using maximum-likelihood estimation. Also, distribution families with MLR have a number of well-behaved stochastic properties, such as first-order stochastic dominance and increasing hazard ratios. Unfortunately, as is also usual, the strength of this assumption comes at the price of realism. Many processes in the world do not exhibit a monotonic correspondence between input and output.
Suppose you are working on a project, and you can either work hard or slack off. Call your choice of effort formula_9 and the quality of the resulting project formula_10. If the MLRP holds for the distribution of "q" conditional on your effort formula_9, the higher the quality the more likely you worked hard. Conversely, the lower the quality the more likely you slacked off.