In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter
, which determines the "location" or shift of the distribution.
In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways: A direct example of a location parameter is the parameter
To see this, note that the probability density function
μ , σ )
of a normal distribution
factored out and be written as: thus fulfilling the first of the definitions given above.
The above definition indicates, in the one-dimensional case, that if
is increased, the probability density or mass function shifts rigidly to the right, maintaining its exact shape.
A location parameter can also be found in families having more than one parameter, such as location–scale families.
In this case, the probability density function or probability mass function will be a special case of the more general form where
is the location parameter, θ represents additional parameters, and
is a function parametrized on the additional parameters.
be any probability density function and let
μ , σ ) =
is a probability density function.
The location family is then defined as follows: Let
be any probability density function.
Then the family of probability density functions
is called the location family with standard probability density function
is called the location parameter for the family.
An alternative way of thinking of location families is through the concept of additive noise.
is a constant and W is random noise with probability density
and its distribution is therefore part of a location family.
For the continuous univariate case, consider a probability density function
A location parameter
can be added by defining: it can be proved that
by verifying if it respects the two conditions[5]
integrates to 1 because: now making the variable change
and updating the integration interval accordingly yields: because