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.
In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the monotone likelihood ratio property is a property of the ratio of two
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
s (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'')."
Intuition
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
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
estimation
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
. Also, distribution families with MLR have a number of well-behaved stochastic properties, such as
first-order stochastic dominance and increasing
hazard ratio
In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions characterised by two distinct levels of a treatment variable of interest. For example, in a clinical study of a drug, the treated populati ...
s. 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.
Example: Working hard or slacking off
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.
Families of distributions satisfying MLR
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
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
if for any
,
:
is a non-decreasing function of
.
Then we say the family of distributions "has MLR in
".
List of families
Hypothesis testing
If the family of random variables has the MLRP in
, a
uniformly most powerful test
In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 - \beta among all possible tests of a given size ''α''. For example, according to the Neyman–Pearson lemma, the likelih ...
can easily be determined for the hypothesis
versus
.
Example: Effort and output
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
.
Relation to other statistical properties
Monotone likelihoods are used in several areas of statistical theory, including
point estimation
In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown popula ...
and
hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
, as well as in
probability model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...
s.
Exponential families
One-parameter
exponential families
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
have monotone likelihood-functions. In particular, the one-dimensional exponential family of
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
s or
probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
s with
:
has a monotone non-decreasing likelihood ratio in the
sufficient statistic
In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the pa ...
''T''(''x''), provided that
is non-decreasing.
Uniformly most powerful tests: The Karlin–Rubin theorem
Monotone likelihood functions are used to construct
uniformly most powerful test
In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 - \beta among all possible tests of a given size ''α''. For example, according to the Neyman–Pearson lemma, the likelih ...
s, according to the
Karlin–Rubin theorem
In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 - \beta among all possible tests of a given size ''α''. For example, according to the Neyman–Pearson lemma, the likelih ...
. 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
Median unbiased estimation
Monotone likelihood-functions are used to construct
median-unbiased estimator
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic ...
s, using methods specified by
Johann Pfanzagl
Johann Richard Pfanzagl (2 July 1928 – 4 June 2019) was an Austrian mathematician known for his research in mathematical statistics.
Life and career
Pfanzagl studied from 1946 to 1951 at the University of Vienna and received his doctorate t ...
and others.
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 function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
s.
Lifetime analysis: Survival analysis and reliability
If a family of distributions
has the monotone likelihood ratio property in
,
# the family has monotone decreasing
hazard rate
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analys ...
s 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.
Proofs
Let distribution family
satisfy MLR in ''x'', so that for
and
:
:
or equivalently:
:
Integrating this expression twice, we obtain:
First-order stochastic dominance
Combine the two inequalities above to get first-order dominance:
:
Monotone hazard rate
Use only the second inequality above to get a monotone hazard rate:
:
Uses
Economics
The MLR is an important condition on the type distribution of agents in
mechanism design
Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts a ...
. Most solutions to mechanism design models assume a type distribution to satisfy the MLR to take advantage of a common solution method.
References
{{DEFAULTSORT:Monotone Likelihood Ratio Property
Theory of probability distributions
Statistical hypothesis testing