<p>Sufficiency of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41096_2025_248_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$|X|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, when <i>X</i> has a certain distribution with a real-valued characterizing parameter <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41096_2025_248_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>, remains to be an intriguing matter of discussion in parametric inference. This work investigates sufficiency of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41096_2025_248_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$|X |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> for continuous distributions with asymmetric support covering 0. Specific attention is given to the Pitman family of distributions and a necessary and sufficient condition for the sufficiency of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41096_2025_248_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$|X|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> is derived.</p>

On the Sufficiency of $|X|$ for Some Families of Continuous Probability Distributions

期刊论文
发表日期：

2025年7月15日

查看全文

Abhik Sinha,
Rahul Bhattacharya

摘要

Sufficiency of $|X|$ $| X |$ , when X has a certain distribution with a real-valued characterizing parameter $\theta$ $θ$ , remains to be an intriguing matter of discussion in parametric inference. This work investigates sufficiency of $|X |$ $| X |$ for continuous distributions with asymmetric support covering 0. Specific attention is given to the Pitman family of distributions and a necessary and sufficient condition for the sufficiency of $|X|$ $| X |$ is derived.

On the Sufficiency of \(|X|\) for Some Families of Continuous Probability Distributions

摘要