A monotonic likelihood ratio in distributions and

The ratio of the density functions above is increasing in the parameter , so satisfies the **monotone likelihood ratio** property.

- Intuition
- Example: Working hard or slacking off
- Families of distributions satisfying MLR
- List of families
- Hypothesis testing
- Example: Effort and output
- Relation to other statistical properties
- Exponential families
- Most powerful tests: The Karlin–Rubin theorem
- Median unbiased estimation
- Lifetime analysis: Survival analysis and reliability
- Uses
- Economics
- References

In statistics, the **monotone likelihood ratio property** is a property of the ratio of two probability density functions (PDFs). Formally, distributions *ƒ*(*x*) and *g*(*x*) bear the property if

that is, if the ratio is nondecreasing in the argument .

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

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 satisfies the MLRP with respect to , the higher the observed value , the more likely it was drawn from distribution rather than . 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 and the quality of the resulting project . If the MLRP holds for the distribution of *q* conditional on your effort , the higher the quality the more likely you worked hard. Conversely, the lower the quality the more likely you slacked off.

- Choose effort where H means high, L means low
- Observe drawn from . By Bayes' law with a uniform prior,
- Suppose satisfies the MLRP. Rearranging, the probability the worker worked hard is

- which, thanks to the MLRP, is monotonically increasing in (because is decreasing in ). 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 indexed by a parameter taking values in an ordered set is said to have a **monotone likelihood ratio (MLR)** in the statistic if for any ,

- is a non-decreasing function of .

Then we say the family of distributions "has MLR in ".

Family | in which has the MLR |
---|---|

Exponential | observations |

Binomial | observations |

Poisson | observations |

Normal | if known, observations |

If the family of random variables has the MLRP in , a uniformly most powerful test can easily be determined for the hypothesis versus .

Example: Let be an input into a stochastic technology – worker's effort, for instance – and its output, the likelihood of which is described by a probability density function Then the monotone likelihood ratio property (MLRP) of the family is expressed as follows: for any , the fact that implies that the ratio is increasing in .

Monotone likelihoods are used in several areas of statistical theory, including point estimation and hypothesis testing, as well as in probability models.

One-parameter exponential families have monotone likelihood-functions. In particular, the one-dimensional exponential family of probability density functions or probability mass functions with

has a monotone non-decreasing likelihood ratio in the sufficient statistic *T*(*x*), provided that is non-decreasing.

Monotone likelihood functions are used to construct uniformly most powerful tests, according to the Karlin–Rubin theorem.^{ [1] } Consider a scalar measurement having a probability density function parameterized by a scalar parameter *θ*, and define the likelihood ratio . If is monotone non-decreasing, in , for any pair (meaning that the greater is, the more likely is), then the threshold test:

- where is chosen so that

is the UMP test of size *α* for testing

Note that exactly the same test is also UMP for testing

Monotone likelihood-functions are used to construct median-unbiased estimators, using methods specified by Johann Pfanzagl and others.^{ [2] }^{ [3] } One such procedure is an analogue of the Rao–Blackwell procedure for mean-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 of loss functions.^{ [3] }^{(p713)}

If a family of distributions has the monotone likelihood ratio property in ,

- the family has monotone decreasing hazard rates in (but not necessarily in )
- the family exhibits the first-order (and hence second-order) stochastic dominance in , and the best Bayesian update of is increasing in .

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

Let distribution family satisfy MLR in *x*, so that for and :

or equivalently:

Integrating this expression twice, we obtain:

1. To with respect to integrate and rearrange to obtain | 2. From with respect to integrate and rearrange to obtain |

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

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

The MLR is an important condition on the type distribution of agents in mechanism design.^{[ citation needed ]} Most solutions to mechanism design models assume a type distribution to satisfy the MLR to take advantage of a common solution method.^{[ citation needed ]}

In statistics, the **likelihood function** measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.

In statistics, the **likelihood-ratio test** assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

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*k*and a mean parameter*μ*=*kθ*=*α*/*β*.

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In statistics, the **score test** assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the score—evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if the restricted estimator is near the maximum of the likelihood function, the score should not differ from zero by more than sampling error. While the finite sample distributions of score tests are generally unknown, it has an asymptotic χ^{2}-distribution under the null hypothesis as first proved by C. R. Rao in 1948, a fact that can be used to determine statistical significance.

In probability theory and statistics, the **characteristic function** of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

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A **ratio distribution** is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables *X* and *Y*, the distribution of the random variable *Z* that is formed as the ratio *Z* = *X*/*Y* is a *ratio distribution*.

In probability theory and statistics, a **stochastic order** quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable may be neither stochastically greater than, less than nor equal to another random variable . Many different orders exist, which have different applications.

In statistical hypothesis testing, a **uniformly most powerful** (**UMP**) **test** is a hypothesis test which has the **greatest power ** among all possible tests of a given size *α*. For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses.

In probability theory and statistics, the **half-normal distribution** is a special case of the folded normal distribution.

In particle physics, **CLs** represents a statistical method for setting *upper limits* on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values. Although CLs are said to refer to Confidence Levels, "The method's name is ... misleading, as the CLs exclusion region is not a confidence interval." It was first introduced by physicists working at the LEP experiment at CERN and has since been used by many high energy physics experiments. It is a frequentist method in the sense that the properties of the limit are defined by means of error probabilities, however it differs from standard confidence intervals in that the stated confidence level of the interval is not equal to its coverage probability. The reason for this deviation is that standard upper limits based on a most powerful test necessarily produce empty intervals with some fixed probability when the parameter value is zero, and this property is considered undesirable by most physicists and statisticians.

In Monte Carlo Estimation, **exponential tilting (ET)**, **exponential twisting**, or **exponential change of measure (ECM)** is a distribution shifting technique commonly used in rare-event simulation, and rejection and importance sampling in particular. Exponential tilting is also used in Esscher tilting, an indirect Edgeworth approximation technique. The earliest formalization of ECM is often attributed to Esscher with its use in importance sampling being attributed to David Siegmund. ET is known as the Esscher transform in mathematical finance and is used in such contexts as insurance futures pricing.

- ↑ Casella, G.; Berger, R.L. (2008),
*Statistical Inference*, Brooks/Cole. ISBN 0-495-39187-5 (Theorem 8.3.17) - ↑ Pfanzagl, Johann (1979). "On optimal median unbiased estimators in the presence of nuisance parameters".
*Annals of Statistics*.**7**(1): 187–193. doi: 10.1214/aos/1176344563 . - 1 2 Brown, L. D.; Cohen, Arthur; Strawderman, W. E. (1976). "A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications".
*Ann. Statist*.**4**(4): 712–722. doi: 10.1214/aos/1176343543 .

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