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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
- being defined as resulting from the random variable transformation , where is a random variable with a certain, possibly unknown, distribution (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.
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 is called the location family with standard probability density function , where is called the location parameter for the family.
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.
For the continuous univariate case, consider a probability density function , where is a vector of parameters. A location parameter can be added by defining:
it can be proved that is a p.d.f. by verifying if it respects the two conditions and . integrates to 1 because:
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 .
- Takeuchi, Kei (1971). "A Uniformly Asymptotically Efficient Estimator of a Location Parameter". Journal of the American Statistical Association. 66 (334): 292–301. doi:10.1080/01621459.1971.10482258. S2CID 120949417.
- Huber, Peter J. (1992). "Robust estimation of a location parameter". Breakthroughs in Statistics. Springer Series in Statistics. Springer: 492–518. doi:10.1007/978-1-4612-4380-9_35. ISBN 978-0-387-94039-7.
- Stone, Charles J. (1975). "Adaptive Maximum Likelihood Estimators of a Location Parameter". The Annals of Statistics. 3 (2): 267–284. doi:10.1214/aos/1176343056.
- Casella, George; Berger, Roger (2001). Statistical Inference (2nd ed.). p. 116. ISBN 978-0534243128.
- Ross, Sheldon (2010). Introduction to probability models. Amsterdam Boston: Academic Press. ISBN 978-0-12-375686-2. OCLC 444116127.