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{{Short description|Concept in statistics}}{{Multiple issues|{{more citations needed|date=February 2020}}{{disputed|date=July 2021}}}}In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x_0, 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 f(x - x_0);JOURNAL, 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, or
• having a cumulative distribution function F(x - x_0);JOURNAL, 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,weblink or
• being defined as resulting from the random variable transformation x_0 + X, where X is a random variable with a certain, possibly unknown, distributionJOURNAL, 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, (See also Additive_noise).

A direct example of a location parameter is the parameter mu of the normal distribution. To see this, note that the probability density function f(x | mu, sigma) of a normal distribution mathcal{N}(mu,sigma^2) can have the parameter mu factored out and be written as: g(y - mu | sigma) = frac{1}{sigma sqrt{2pi} } e^{-frac{1}{2}left(frac{y}{sigma}right)^2}thus fulfilling the first of the definitions given above.The above definition indicates, in the one-dimensional case, that if x_0 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 x_0 is the location parameter, θ represents additional parameters, and f_theta is a function parametrized on the additional parameters.

DefinitionBOOK, lastCasella, firstGeorge, titleStatistical Inference, last2Berger, first2Roger, year2001, isbn978-0534243128, edition2nd, pages116,

Let f(x) be any probability density function and let mu and sigma > 0 be any given constants. Then the functiong(x| mu, sigma)= frac{1}{sigma}fleft(frac{x-mu}{sigma}right)is a probability density function.The location family is then defined as follows:Let f(x) mathcal{F} = {f(x-mu) : mu in mathbb{R}} f(x), where mu

Additive noise

An alternative way of thinking of location families is through the concept of additive noise. If x_0 is a constant and W is random noise with probability density f_W(w), then X = x_0 + W has probability density f_{x_0}(x) = f_W(x-x_0) 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] subset mathbb{R}, where theta is a vector of parameters. A location parameter x_0 can be added by defining: g(x | theta, x_0) = f(x - x_0 | theta), ; x in [a - x_0, b - x_0]it can be proved that g is a p.d.f. by verifying if it respects the two conditionsBOOK, Ross, Sheldon, Introduction to probability models, Academic Press, Amsterdam Boston, 2010, 978-0-12-375686-2, 444116127, g(x | theta, x_0) ge 0 and int_{-infty}^{infty} g(x | theta, x_0) dx = 1. g integrates to 1 because: int_{-infty}^{infty} g(x | theta, x_0) dx = int_{a - x_0}^{b - x_0} g(x | theta, x_0) dx = int_{a - x_0}^{b - x_0} f(x - x_0 | theta) dxnow making the variable change u = x - x_0 and updating the integration interval accordingly yields: int_{a}^{b} f(u | theta) du = 1because f(x | theta) is a p.d.f. by hypothesis. g(x | theta, x_0) ge 0 follows from g sharing the same image of f, which is a p.d.f. so its image is contained in [0, 1].

See also

• Central tendency
• Location test
• Invariant estimator
• Scale parameter
• Two-moment decision models

References

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