Location parameter

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 $f(x-x_{0})$; or
• having a cumulative distribution function $F(x-x_{0})$; or
• being defined as resulting from the random variable transformation $x_{0}+X$, where $X$ is a random variable with a certain, possibly unknown, distribution. 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: $$g(x'=x-\mu |\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {x'}{\sigma }}\right)^{2}\right)$$ 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 $$f_{x_{0},\theta }(x)=f_{\theta }(x-x_{0})$$ where $x_{0}$ is the location parameter, θ represents additional parameters, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{\theta }} is a function parametrized on the additional parameters.

Definition

Source:

Let $f(x)$ be any probability density function and let $\mu$ and $\sigma >0$ be any given constants. Then the function

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x|\mu ,\sigma )={\frac {1}{\sigma }}f{\left({\frac {x-\mu }{\sigma }}\right)}}

is a probability density function.

The location family is then defined as follows:

Let $f(x)$ be any probability density function. Then the family of probability density functions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {F}}=\{f(x-\mu ):\mu \in \mathbb {R} \}} is called the location family with standard probability density function $f(x)$, where $\mu$ 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 $x_{0}$ is a constant and W is random noise with probability density Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{W}(w),} then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X=x_{0}+W} has probability density Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 conditions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x|\theta ,x_{0})\geq 0} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{-\infty }^{\infty }g(x|\theta ,x_{0})dx=1} . $g$ integrates to 1 because: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 )dx} now making the variable change Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle u=x-x_{0}} and updating the integration interval accordingly yields: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{a}^{b}f(u|\theta )du=1} because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x | \theta)} is a p.d.f. by hypothesis. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x | \theta, x_0) \ge 0} follows from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} sharing the same image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , which is a p.d.f. so its range is contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1]} .

See also

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

General references

• "1.3.6.4. Location and Scale Parameters". National Institute of Standards and Technology. Retrieved 2025-03-17.