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Location parameter
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<h2 id="definition">Definition</h2> <p>Source:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Let f ( x ) {\displaystyle f(x)} be any probability density function and let μ {\displaystyle \mu } and σ > 0 {\displaystyle \sigma >0} be any given constants. Then the function </p><p> g ( x | μ , σ ) = 1 σ f ( x − μ σ ) {\displaystyle g(x|\mu ,\sigma )={\frac {1}{\sigma }}f\left({\frac {x-\mu }{\sigma }}\right)} </p><p>is a probability density function. </p><p> The location family is then defined as follows: </p><p>Let f ( x ) {\displaystyle f(x)} be any probability density function. Then the family of probability density functions F = { f ( x − μ ) : μ ∈ R } {\displaystyle {\mathcal {F}}=\{f(x-\mu ):\mu \in \mathbb {R} \}} is called the location family with standard probability density function f ( x ) {\displaystyle f(x)} , where μ {\displaystyle \mu } is called the location parameter for the family. </p> <h2 id="additive-noise">Additive noise</h2> <p>An alternative way of thinking of location families is through the concept of <a href="/facts/Additive_noise/IE4Gof7T">additive noise</a>. If x 0 {\displaystyle x_{0}} is a constant and <i>W</i> is random <a href="/facts/Noise/aUstHDqU">noise</a> with probability density f W ( w ) , {\displaystyle f_{W}(w),} then X = x 0 + W {\displaystyle X=x_{0}+W} has probability density f x 0 ( x ) = f W ( x − x 0 ) {\displaystyle f_{x_{0}}(x)=f_{W}(x-x_{0})} and its distribution is therefore part of a location family. </p> <h2 id="proofs">Proofs</h2> <p>For the continuous univariate case, consider a probability density function f ( x | θ ) , x ∈ [ a , b ] ⊂ R {\displaystyle f(x|\theta ),x\in [a,b]\subset \mathbb {R} } , where θ {\displaystyle \theta } is a vector of parameters. A location parameter x 0 {\displaystyle x_{0}} can be added by defining: </p> g ( x | θ , x 0 ) = f ( x − x 0 | θ ) , x ∈ [ a + x 0 , b + x 0 ] {\displaystyle g(x|\theta ,x_{0})=f(x-x_{0}|\theta ),\;x\in [a+x_{0},b+x_{0}]} <p>it can be proved that g {\displaystyle g} is a p.d.f. by verifying if it respects the two conditions<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> g ( x | θ , x 0 ) ≥ 0 {\displaystyle g(x|\theta ,x_{0})\geq 0} and ∫ − ∞ ∞ g ( x | θ , x 0 ) d x = 1 {\displaystyle \int _{-\infty }^{\infty }g(x|\theta ,x_{0})dx=1} . g {\displaystyle g} integrates to 1 because: </p> ∫ − ∞ ∞ g ( x | θ , x 0 ) d x = ∫ a + x 0 b + x 0 g ( x | θ , x 0 ) d x = ∫ a + x 0 b + x 0 f ( x − x 0 | θ ) d x {\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} <p>now making the variable change u = x − x 0 {\displaystyle u=x-x_{0}} and updating the integration interval accordingly yields: </p> ∫ a b f ( u | θ ) d u = 1 {\displaystyle \int _{a}^{b}f(u|\theta )du=1} <p>because f ( x | θ ) {\displaystyle f(x|\theta )} is a p.d.f. by hypothesis. g ( x | θ , x 0 ) ≥ 0 {\displaystyle g(x|\theta ,x_{0})\geq 0} follows from g {\displaystyle g} sharing the same image of f {\displaystyle f} , which is a p.d.f. so its range is contained in [ 0 , 1 ] {\displaystyle [0,1]} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Central_tendency/9yhcQ6me">Central tendency</a></li> <li><a href="/facts/Location_test/9botwshp">Location test</a></li> <li><a href="/facts/Invariant_estimator/LQRCCnyM">Invariant estimator</a></li> <li><a href="/facts/Scale_parameter/6lxcyKUf">Scale parameter</a></li> <li><a href="/facts/Two-moment_decision_models/mdbvo0Q9">Two-moment decision models</a></li></ul> <h2 id="general-references">General references</h2> <ul><li><a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm">"1.3.6.4. Location and Scale Parameters"</a>. <i>National Institute of Standards and Technology</i>. Retrieved 2025-03-17.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Huber, Peter J. (1992). "Robust Estimation of a Location Parameter". Breakthroughs in Statistics. Springer Series in Statistics. Springer. pp. 492–518. doi:10.1007/978-1-4612-4380-9_35. ISBN 978-0-387-94039-7. <a href="978-0-387-94039-7" target="_blank">978-0-387-94039-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="https://doi.org/10.1214%2Faos%2F1176343056" target="_blank">https://doi.org/10.1214%2Faos%2F1176343056</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Casella, George; Berger, Roger (2001). Statistical Inference (2nd ed.). Thomson Learning. p. 116. ISBN 978-0534243128. <a href="978-0534243128" target="_blank">978-0534243128</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Ross, Sheldon (2010). Introduction to probability models. Amsterdam Boston: Academic Press. ISBN 978-0-12-375686-2. OCLC 444116127. <a href="978-0-12-375686-2" target="_blank">978-0-12-375686-2</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>