Location parameter

Location parameter

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In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a location family is a class of probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s that is parametrized
Statistical parameter
A statistical parameter is a parameter that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a model....

 by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution. Formally, this means that the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

s or probability mass function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

s in this class have the form
Here, μ is called the location parameter.

In other words, when the function is graphed, the location parameter determines where the origin will be located. If μ is positive, the origin will be shifted to the right, and if μ is negative, it will be shifted to the left.

A location parameter can also be found in families having more than one parameter, such as location-scale families
Location-scale family
In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a...

. In this case, the probability density function or probability mass function will be a special case of the more general form
where μ is the location parameter, θ represents additional parameters, and is a function of the additional parameters.

Additive noise


An alternative way of thinking of location families is through the concept of additive noise. If μ is an unknown constant and W is random noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...

 with probability density , then has probability density and is therefore a location family..

See also

  • Location test
    Location test
    A location test is a statistical hypothesis test that compares the location parameter of a statistical population to a given constant, or that compares the location parameters of two statistical populations to each other...

  • Invariant estimator
    Invariant estimator
    In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities...

  • Location-scale family
    Location-scale family
    In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a...

  • Scale parameter
    Scale parameter
    In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...