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For a regular model with a suitable ancillary statistic $a$, the displayed expression approximates the conditional density of the maximum-likelihood estimator. The normalizing factor includes $(2\pi {)}^{-p/2}$ for a $p$-dimensional estimator, depending on convention. The observed information is evaluated at the sample's own fitted value. In a pure scale model the factor is proportional to $1/\hat{\theta }$; this Jacobian determinant can make the normalized approximation coincide with the exact conditional density.

For independent observations uniform on $(\theta -1/2,\theta +1/2)$, let $U$ and $V$ be the sample minimum and maximum, $R=V-U$, and $C=(U+V)/2$. For $n\ge 2$ their joint density in $(c,r)$ is $n(n-1)r^{n-2}$ on $0<r<1$, $|c-\theta |<(1-r)/2$. Integrating over $c$ gives $f_R(r)=n(n-1)r^{n-2}(1-r)$, independent of $\theta$, so $R$ is an ancillary statistic. Moreover $C\mid R=r$ is uniform on $(\theta -(1-r)/2,\theta +(1-r)/2)$. Consequentlyis an exact conditional $(1-\alpha )$ confidence interval for $\theta$. The realized range determines the appropriate precision, although the range is not itself evidence about the value of $\theta$.

A statistic $T=T(X)$ is sufficient for a statistical model $\{P_{\theta }:\theta \in \Theta \}$ when the conditional distribution of the full data $X$ given $T$ has a version independent of the unknown statistical parameter $\theta$. Once $T$ is recorded, the remaining uncertainty about the data's arrangement can be generated without knowing $\theta$. ^[ref]

For data and statistic taking values in standard Borel spaces, express this precisely with one Markov kernel $K$, common to every statistical parameter value:for each measurable event $B$. Thus $K(t,\cdot )$ is a regular conditional distribution of $X$ given $T=t$. Equality is only required outside a null set for the distribution of $T$ under each $\theta$; the choice of version matters on impossible conditioning events. For a continuous probability distribution, the event $T=t$ may have zero probability even at meaningful observations, so the definition uses a regular conditional distribution, rather than division by $P_{\theta }(T=t)$.

A sufficient statistic is a property of a specified statistical model, rather than of a formula for summarizing data in isolation. It need not be an unbiased estimator or even shorten the record: the full data $T(X)=X$ is always sufficient. An estimator instead targets a quantity such as a mean. An ancillary statistic has a parameter-independent marginal probability distribution, whereas a sufficient statistic has a parameter-independent conditional full-data law. A minimal sufficient statistic permits the greatest reduction among sufficient statistics. A complete statistic satisfies a separate condition about zero expectations; a complete sufficient statistic must satisfy both conditions.