A monotonic likelihood ratio in distributions${\displaystyle f(x)}$ and${\displaystyle g(x)}$

The ratio of thedensity functions above is monotone in the parameter${\displaystyle x,}$ so${\displaystyle \frac{f(x)}{g(x)}}$ satisfies themonotone likelihood ratio property.

Instatistics, themonotone likelihood ratio property is a property of the ratio of twoprobability density functions (PDFs). Formally, distributions${\displaystyle f(x)}$ and${\displaystyle g(x)}$ bear the property if

${\displaystyle \text{for every }{x}_2>x_1,\frac{f(x_2)}{g(x_2)}\ge \frac{f(x_1)}{g(x_1)}}$

that is, if the ratio is nondecreasing in the argument${\displaystyle x}$.

If the functions are first-differentiable, the property may sometimes be stated

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{f(x)}{g(x)}\right)\ge 0}$

For two distributions that satisfy the definition with respect to some argument${\displaystyle x,}$ we say they "have the MLRP in${\displaystyle x.}$" For a family of distributions that all satisfy the definition with respect to some statistic${\displaystyle T(X),}$ we say they "have the MLR in${\displaystyle T(X).}$"

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${\displaystyle f(x)}$ satisfies the MLRP with respect to${\displaystyle g(x)}$, the higher the observed value${\displaystyle x}$, the more likely it was drawn from distribution${\displaystyle f}$ rather than${\displaystyle g.}$ As usual for monotonic relationships, the likelihood ratio's monotonicity comes in handy in statistics, particularly when usingmaximum-likelihoodestimation. Also, distribution families with MLR have a number of well-behaved stochastic properties, such asfirst-order stochastic dominance and increasinghazard 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${\displaystyle e}$ and the quality of the resulting project${\displaystyle q.}$ If the MLRP holds for the distribution of${\displaystyle q}$ conditional on your effort${\displaystyle e}$, the higher the quality the more likely you worked hard. Conversely, the lower the quality the more likely you slacked off.

1: Choose effort${\displaystyle e\in \{H,L\}}$ where${\displaystyle H}$ means high effort, and${\displaystyle L}$ means low effort.

2: Observe${\displaystyle q}$ drawn from${\displaystyle f(q|e).}$ ByBayes' law with auniform prior,

${\displaystyle \mathbb{P}[e=H|q]=\frac{f(q|H)}{f(q|H)+f(q|L)}}$

3: Suppose${\displaystyle f(q|e)}$ satisfies the MLRP. Rearranging, the probability the worker worked hard is

${\displaystyle \frac{1}{1+f(q|L)/f(q|H)}}$

which, thanks to the MLRP, is monotonically increasing in${\displaystyle q}$ (because${\displaystyle \frac{f(q|L)}{f(q|H)}}$ is decreasing in${\displaystyle q}$).

Hence if some employer is doing a "performance review" he can infer his employee's behavior from the merits of his work.

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).

A family of density functions${\displaystyle \{f_{\theta }(x)|\theta \in \mathrm{\Theta }\}}$ indexed by a parameter${\displaystyle \theta }$ taking values in an ordered set${\displaystyle \mathrm{\Theta }}$ is said to have amonotone likelihood ratio (MLR) in thestatistic${\displaystyle T(X)}$ if for any

${\displaystyle \frac{f_{{\theta }_2}(X=x_1,x_2,x_3,\dots )}{f_{{\theta }_1}(X=x_1,x_2,x_3,\dots )}}$ is a non-decreasing function of${\displaystyle T(X).}$

Then we say the family of distributions "has MLR in${\displaystyle T(X)}$".

Family	${\displaystyle T(X)}$ in which${\displaystyle f_{\theta }(X)}$ has the MLR
Exponential${\displaystyle [\lambda ]}$	${\displaystyle \sum x_i}$ observations
Binomial${\displaystyle [n,p]}$	${\displaystyle \sum x_i}$ observations
Poisson${\displaystyle [\lambda ]}$	${\displaystyle \sum x_i}$ observations
Normal${\displaystyle [\mu ,\sigma ]}$	if${\displaystyle \sigma }$ known,${\displaystyle \sum x_i}$ observations

If the family of random variables has the MLRP in${\displaystyle T(X),}$ auniformly most powerful test can easily be determined for the hypothesis${\displaystyle H_0:\theta \le {\theta }_0}$ versus${\displaystyle H_1:\theta >{\theta }_0.}$

Example: Let${\displaystyle e}$ be an input into a stochastic technology – worker's effort, for instance – and${\displaystyle y}$ its output, the likelihood of which is described by a probability density function${\displaystyle f(y;e).}$ Then the monotone likelihood ratio property (MLRP) of the family${\displaystyle f}$ is expressed as follows: For any${\displaystyle e_1,e_2,}$ the fact that${\displaystyle e_2>e_1}$ implies that the ratio${\displaystyle \frac{f(y;e_2)}{f(y;e_1)}}$ is increasing in${\displaystyle y.}$

Monotone likelihoods are used in several areas of statistical theory, includingpoint estimation andhypothesis testing, as well as inprobability models.

One-parameterexponential families have monotone likelihood-functions. In particular, the one-dimensional exponential family ofprobability density functions orprobability mass functions with

${\displaystyle f_{\theta }(x)=c(\theta )h(x)\exp (\pi \left(\theta \right)T\left(x\right))}$

has a monotone non-decreasing likelihood ratio in thesufficient statistic${\displaystyle T(x),}$ provided that${\displaystyle \pi (\theta )}$ is non-decreasing.

Monotone likelihood functions are used to constructuniformly most powerful tests, according to theKarlin–Rubin theorem.^[1]Consider a scalar measurement having a probability density function parameterized by a scalar parameter${\displaystyle \theta ,}$ and define the likelihood ratio${\displaystyle \ell (x)=\frac{f_{{\theta }_1}(x)}{f_{{\theta }_0}(x)}.}$ If${\displaystyle \ell (x)}$ is monotone non-decreasing, in${\displaystyle x,}$ for any pair${\displaystyle {\theta }_1\ge {\theta }_0}$ (meaning that the greater${\displaystyle x}$ is, the more likely${\displaystyle H_1}$ is), then the threshold test:

where${\displaystyle x_0}$ is chosen so that${\displaystyle \mathbb{E}\{\phi (X)|{\theta }_0\}=\alpha }$

is the UMP test of size${\displaystyle \alpha }$ for testing${\displaystyle H_0:\theta \le {\theta }_0}$ vs.${\displaystyle H_1:\theta >{\theta }_0.}$

Note that exactly the same test is also UMP for testing${\displaystyle H_0:\theta ={\theta }_0}$ vs.${\displaystyle H_1:\theta >{\theta }_0.}$

Monotone likelihood-functions are used to constructmedian-unbiased estimators, using methods specified byJohann Pfanzagl and others.^[2][3]One such procedure is an analogue of theRao–Blackwell procedure formean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao–Blackwell procedure for mean-unbiased estimation but for a larger class ofloss functions.^[3]: 713

If a family of distributions${\displaystyle f_{\theta }(x)}$ has the monotone likelihood ratio property in${\displaystyle T(X),}$

1. the family has monotone decreasinghazard rates in${\displaystyle \theta }$ (but not necessarily in${\displaystyle T(X)}$)
2. the family exhibits the first-order (and hence second-order)stochastic dominance in${\displaystyle x,}$ and the best Bayesian update of${\displaystyle \theta }$ is increasing in${\displaystyle T(X)}$.

But not conversely: neither monotone hazard rates nor stochastic dominance imply the MLRP.

Let distribution family${\displaystyle f_{\theta }}$ satisfy MLR in${\displaystyle x,}$ so that for${\displaystyle {\theta }_1>{\theta }_0}$ and${\displaystyle x_1>x_0:}$

${\displaystyle \frac{f_{{\theta }_1}(x_1)}{f_{{\theta }_0}(x_1)}\ge \frac{f_{{\theta }_1}(x_0)}{f_{{\theta }_0}(x_0)},}$

or equivalently:

${\displaystyle f_{{\theta }_1}(x_1)f_{{\theta }_0}(x_0)\ge f_{{\theta }_1}(x_0)f_{{\theta }_0}(x_1).}$

Integrating this expression twice, we obtain:

1. To${\displaystyle x_1}$ with respect to${\displaystyle x_0}$ integrate and rearrange to obtain ${\displaystyle \frac{f_{{\theta }_1}(x)}{f_{{\theta }_0}(x)}\ge \frac{F_{{\theta }_1}(x)}{F_{{\theta }_0}(x)}}$		2. From${\displaystyle x_0}$ with respect to${\displaystyle x_1}$ integrate and rearrange to obtain ${\displaystyle \frac{1-F_{{\theta }_1}(x)}{1-F_{{\theta }_0}(x)}\ge \frac{f_{{\theta }_1}(x)}{f_{{\theta }_0}(x)}}$

Combine the two inequalities above to get first-order dominance:

${\displaystyle F_{{\theta }_1}(x)\le F_{{\theta }_0}(x)\mathrm{\forall }x}$

Use only the second inequality above to get a monotone hazard rate:

${\displaystyle \frac{f_{{\theta }_1}(x)}{1-F_{{\theta }_1}(x)}\le \frac{f_{{\theta }_0}(x)}{1-F_{{\theta }_0}(x)}\mathrm{\forall }x}$

The MLR is an important condition on the type distribution of agents inmechanism design andeconomics of information, wherePaul Milgrom defined "favorableness" of signals (in terms of stochastic dominance) as a consequence of MLR.^[4]Most solutions to mechanism design models assume type distributions that satisfy the MLR to take advantage of solution methods that may be easier to apply and interpret.

• Theory of probability distributions
• Statistical hypothesis testing

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