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:

• either as having a probability density function or probability mass function; or

• having a cumulative distribution function; or

• is a random variable with a certain, possibly unknown, distribution. See also .

of the normal distribution thus fulfilling the first of the definitions given above.

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 is the location parameter, θ is a function parametrized on the additional parameters.

Definition

Source:

be any given constants. Then the function

is a probability density function.

The location family is then defined as follows:

is called the location parameter for the family.

Additive noise

An alternative way of thinking of location families is through the concept of additive noise is a constant and W is random noise and its distribution is therefore part of a location family.

Proofs

is a p.d.f. by verifying if it respects the two conditions.

See also

• Central tendency

• Location test

• Invariant estimator

• Scale parameter

• Two-moment decision models

References

General references