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For independent samples uniform on $[-\theta ,\theta ]$ with $\theta >0$, the absolute maximum $M$ is a minimal sufficient statistic, since the likelihood is $(2\theta {)}^{-n}{1}_{\{\theta \ge M\}}$. The pair of sample minimum and maximum is a sufficient statistic but is not minimal: simultaneous sign reversal preserves $M$ and generally changes that pair.

For a metric space or pseudometric space $(T,d)$, this is the minimum number of balls of radius $\epsilon >0$, with centers in $T$, needed to cover $T$. The displayed grid bound uses closed balls. For open balls, replacing the radius by a fixed constant factor gives the same entropy estimates. For $T=[0,1]$ and $d(s,t)=c|s-t{|}^H$, a uniform grid gives $N(\epsilon ,T,d)\le 1+\lceil (c/\epsilon {)}^{1/H}\rceil$. These polynomial growth bounds make the Dudley entropy integral finite.

An $\epsilon$-net of a metric space $(T,d)$ is a subset $U\subseteq T$ such that every $x\in T$ has $d(x,u)\le \epsilon$ for some $u\in U$. Its least possible finite size is the metric covering number with closed balls. A maximal set with pairwise distances greater than $\epsilon$ is an $\epsilon$-net.

In a metric space $(M,d)$, the closed ball centred at $x$ of radius $r\ge 0$ is $\{y\in M:d(x,y)\le r\}$. It is a closed set, since the distance to $x$ is a continuous function. Its boundary need not be the whole corresponding distance sphere in an arbitrary metric space.

A subset is $r$-separated if distinct points have distance at least $r$. Such a subset is a packing by points; small balls around them have disjoint interiors at suitable radius. A large separated set supplies many distinguishable alternatives in statistical lower-bound arguments.

For $e^{-\lambda z^2}$, the saddle point descent is the real axis and ascent is the imaginary axis. A descent path from a nonzero endpoint $b$ has $z(s{)}^2=b^2+s$, $s\ge 0$, with the continuous square-root branch from $b$. It ends at positive or negative real infinity according to the sign of $\mathrm{Re}b$. The oriented outward endpoint contribution in steepest descent is $e^{-\lambda b^2}/(2\lambda b)$.

For an integrand $e^{kf(z)}$, a contour along which $\mathrm{Im}f$ is constant and $\mathrm{Re}f$ decreases away from the contributing endpoint or saddle. This suppresses oscillation while localizing the large-parameter integral.

A common-effect synthesis treats study estimates as estimates of one shared effect. For independent approximately unbiased estimates $y_i$ with variances $v_i$, use $\hat{\mu }={\sum }_i{w}_i{y}_i/{\sum }_i{w}_i$ with $w_i=v_i^{-1}$. This model does not accommodate additional between-study heterogeneity.

Systematic displacement of an individual study estimate caused by design, conduct, analysis or selective reporting. Combining estimates can reduce random error without reducing this bias. Bias-adjusted meta-analysis and analyses restricted or stratified by relevant bias domains can assess its impact.

For independent subgroup estimates ${\hat{\mu }}_1,{\hat{\mu }}_2$ with variances $v_1,v_2$, the null contrast has Wald statistic $Z=({\hat{\mu }}_1-{\hat{\mu }}_2)/\sqrt{v_1+v_2}$. If evidence overlaps, include minus twice the covariance in the contrast variance. A nonsignificant comparison is not proof of equivalence.

The reflection-symmetric Newtonian gravitational potential $\Phi =v_0^2\log [(\sqrt{R^2+z^2}+|z|)/R_{\ast }]$ is harmonic off the plane. Its normal-derivative jump is $2v_0^2/R$, so the Poisson equation for Newtonian gravity gives surface density of a disk $S=v_0^2/(2\pi GR)$ and volume mass density $\rho =S\delta (z)$. The midplane circular speed is constant. The scale-free model has infinite total mass and does not use a finite-mass boundary condition at infinity.

The planar Mestel disk has $F(E)\propto e^{-E/v_0^2}$, giving independent centered Gaussian distributions for $v_R,v_{\varphi }$ with velocity dispersions ${\sigma }_R={\sigma }_{\varphi }=v_0$. Its streaming velocity is zero. A maximally prograde stellar distribution instead has mean $\sqrt{2/\pi }{v}_0$ and azimuthal variance $v_0^2(1-2/\pi )$, while all second raw moments remain unchanged.

The prime sum ${\sum }_{p\le y}(\log p)/p=\log y+O(1)$ is the first Mertens theorem. It controls the mean logarithmic size in a truncated Euler-product lower bound.

Social choice theory studies how individual preferences determine a collective decision. A social choice function selects one alternative from a preference profile; strategyproofness asks whether agents can benefit by misrepresenting their preferences.

A mechanism or social choice function is strategyproof if truthful reporting is a weakly dominant strategy: for every true type and every collection of other reports, no misreport yields strictly higher utility or a strictly preferred outcome. This is stronger than merely requiring truthful reports to form a Nash equilibrium.

An auction allocates goods using submitted messages and a specified payment rule. Private-value auctions model each bidder's valuation as its own private information. Equilibrium bids depend on the allocation and payment rules, not only on valuations.

Individual rationality means participation yields at least the outside-option utility, normalized here to zero. A risk-neutral zero-value bidder with nonnegative payments and no positive-valued allocation must have zero expected payment and utility if participation is individually rational.

Incentive compatibility makes the prescribed truthful or type-dependent strategy optimal under the specified solution concept. Dominant-strategy incentive compatibility holds for every other report profile; Bayesian incentive compatibility compares expected utilities over the other agents' types.

A direct revelation mechanism asks agents to report their types directly, then computes allocation and payments from those reports. Being direct does not itself imply strategyproofness; the allocation and payment rules must supply the incentive guarantee.

In a single-parameter mechanism, player $i$ has one private scalar value $v_i$, receives an allocation amount $x_i$, and has quasilinear utility $v_i{x}_i-p_i$. The allocation-payment relation is governed by incentive compatibility and the interim payment identity.