Location parameter explained
, 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:
;
[1] or
;
[2] or
- being defined as resulting from the random variable transformation
, where
is a random variable with a certain, possibly unknown, distribution
[3] (See also
- Additive_noise
).
A direct example of a location parameter is the parameter
of the
normal distribution. To see this, note that the probability density function
of a normal distribution
can have the parameter
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.
Definition[4]
Let
be any probability density function and let
and
be any given constants. Then the function
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
l{F}=\{f(x-\mu):\mu\inR\}
is called the location family with standard probability density function
, where
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. If
is a constant and
W is random
noise with probability density
then
has probability density
and its distribution is therefore part of a location family.
Proofs
For the continuous univariate case, consider a probability density function
f(x|\theta),x\in[a,b]\subsetR
, where
is a vector of parameters. A location parameter
can be added by defining:
g(x|\theta,x0)=f(x-x0|\theta), x\in[a-x0,b-x0]
it can be proved that
is a p.d.f. by verifying if it respects the two conditions
[5]
and
.
integrates to 1 because:
g(x|\theta,x0)dx=
g(x|\theta,x0)dx=
f(x-x0|\theta)dx
now making the variable change
and updating the integration interval accordingly yields:
because
is a p.d.f. by hypothesis.
follows from
sharing the same image of
, which is a p.d.f. so its image is contained in
.
See also
References
- Takeuchi . Kei . A Uniformly Asymptotically Efficient Estimator of a Location Parameter . Journal of the American Statistical Association . 1971 . 66 . 334 . 292–301. 10.1080/01621459.1971.10482258 . 120949417 .
- Huber . Peter J. . Robust estimation of a location parameter . Breakthroughs in Statistics . Springer Series in Statistics . 1992 . 492–518. Springer. 10.1007/978-1-4612-4380-9_35 . 978-0-387-94039-7 .
- Stone . Charles J. . Adaptive Maximum Likelihood Estimators of a Location Parameter . The Annals of Statistics . 1975 . 3 . 2 . 267–284. 10.1214/aos/1176343056 . free .
- Book: Casella, George . Statistical Inference . Berger . Roger . 2001 . 978-0534243128 . 2nd . 116.
- Book: Ross, Sheldon . Introduction to probability models . Academic Press . Amsterdam Boston . 2010 . 978-0-12-375686-2 . 444116127 .