Calibration compares predicted survival probabilities with observed survival experience at relevant horizons or over time. Censoring must be handled, for example by suitable survival estimates or inverse-probability weighting. Calibration should be assessed on validated predictions, because the fitting sample can hide optimism from variable selection and tuning. It is distinct from the risk-ranking information measured by a concordance index.

An accelerated failure time model changes the time scale of a common event-time distribution. In a log-location-scale representation, $T_z=e^{\mu (z)}{e}^{\sigma X}$ with common $X$ and positive scale $\sigma$. An additive change in $\mu (z)$ multiplies all time quantiles by the same factor. A larger time multiplier indicates longer survival, whereas a larger hazard multiplier in a proportional hazards model indicates faster failure.

The contact angle is measured through a liquid between its interface and a solid substrate at their contact line. In a gently sloping two-dimensional film on a horizontal substrate, its small-angle value is the limiting magnitude of the height slope. A zero contact angle means a tangential approach to the substrate; it does not imply bounded curvature, as the limiting zero-flux thermocapillary film profiles illustrate.

Explicit breaking changes the action so that the former supercharges are not conserved symmetries of the full Hamiltonian. Spontaneous breaking preserves the action and algebra but has a vacuum that some supercharge does not annihilate. Energy positivity in global supersymmetry then implies positive vacuum energy density, and a Goldstino accompanies the broken symmetry. That exact-algebra energy argument does not apply to an arbitrarily explicitly broken Hamiltonian. The physical particle spectrum in a broken vacuum need not assemble into the finite equal-mass supermultiplets of an invariant vacuum.

With one canonical ungauged chiral superfield and $\kappa \ne 0$, the auxiliary expectation is nonzero at every scalar value. The potential is flat and positive, so supersymmetry is broken without an isolated minimum. The fermion is a massless Goldstino and the scalar is massless at tree level. Equality of these zero masses does not imply an unbroken supersymmetric vacuum.

A sufficient statistic is complete when every integrable function of it with expectation zero for every parameter is almost surely zero for every parameter. Full regular one-parameter exponential families with natural parameter ranging over an open interval provide a standard completeness result. Completeness makes an unbiased function of a sufficient statistic unique, giving the Lehmann–Scheffé theorem.

For $\mathcal{L}y=-y^{\prime \prime }-2a{y}^{\prime }$, setting $y=e^{-ax}v$ gives $\mathcal{L}y=e^{-ax}(-v^{\prime \prime }+a^{2}v)$. Dirichlet conditions are unchanged. The weighted self-adjoint form is $-e^{-2ax}(e^{2ax}{y}^{\prime }{)}^{\prime }$, with weight $e^{2ax}$. On the interval $(0,L)$, eigenvalues are $(n\pi /L{)}^2+a^2$ and eigenfunctions are $e^{-ax}\sin (n\pi x/L)$.

Adding ${\partial }_{\rho }{B}^{\rho \mu \nu }$ with $B^{\rho \mu \nu }=-B^{\mu \rho \nu }$ to a stress-energy tensor preserves its divergence identically, since commuting partial derivatives annihilates the antisymmetric pair. The total four-momentum changes only by a spatial boundary term. Improvements can make local densities symmetric or gauge-invariant; equality of integrated charges requires the chosen boundary conditions. Equation-of-motion terms can also relate representatives on shell.

A stream tube is a narrow tube whose side boundary consists of streamlines. In steady flow no fluid crosses its sides, so continuity makes its mass flux $A\rho u$ constant. If flow follows a magnetic field with constant magnetohydrodynamic mass loading, its normal area obeys $A|\mathbf{B}|=\mathrm{constant}$.

The small Witt design is a Steiner system $S(5,6,12)$ with 132 blocks. A six-set duad-syntheme duality constructs it on two disjoint six-sets: two whole-half blocks, forty-five blocks of each of the types $(2,4)$ and $(4,2)$, and forty corresponding-partition blocks of type $(3,3)$. The internal-duad/cross-syntheme incidence rule proves that each five-set lies in exactly one of these blocks.

A scale family rescales a fixed distribution by a positive parameter. The standardized variable $Y/\sigma$ has density $f$, independent of $\sigma$. This simplifies moments, simulations and the Jeffreys prior for a scale parameter.

A statistical path is a one-dimensional family of probability measures inside a statistical model, passing through a specified baseline $P$. A differentiability in quadratic mean condition gives its score function, the first-order direction of the family in square-root-density coordinates. The path must respect all restrictions of the statistical model, including normalization and any fixed moments.

A regression model specifies features of the conditional distribution of a response given predictor covariates, often a conditional mean or a full parametric density. Linear regression specifies a linear conditional mean; logistic regression specifies a logit-linear conditional probability for a binary response. The conditional comparison represented by a coefficient depends on which covariates are included.

The number of free dimensions remaining after specified constraints often determines a reference sampling distribution. In a full-rank normal linear model with $n$ observations and $p$ mean coefficients, the residual space has dimension $n-p$. A regular nested likelihood-ratio test instead uses the difference in free statistical parameter dimensions. Boundary constraints may invalidate this regular calibration.

A scoring rule turns a probability distribution forecast and an observed outcome into a number. For example, a forecast of rain with probability $0.7$ should express uncertainty more usefully than a bare prediction of rain. A scoring rule lets us compare such forecasts after observations arrive, and its choice determines whether exaggerating confidence pays. A proper scoring rule rewards an honest account of uncertainty in expected value.

There are two sign conventions. A reward $S(P,x)$ is maximized; a loss function $L(P,x)$ is minimized. The conversion $L=-S$ reverses every comparison. Here the linear probability score is a reward, while the binary Brier score and the negative logarithmic scoring rule are losses. Calling all three a "score" without specifying the direction would hide the central difference.

Throughout the binary examples, the random variable $Y$ has a Bernoulli distribution with actual success probability $q=0.7$, and the report $p\in [0,1]$ assigns probability $p$ to $Y=1$ and $1-p$ to $Y=0$. The value $q=0.7$ is stipulated by this statistical model; it is not inferred from data or a claim that a forecasting system is calibrated. In practice the true predictive distribution is unknown, and the relevant $q$ is the conditional probability of success given the information available when the forecast is issued. Propriety evaluates incentives for reporting that predictive distribution; it does not supply the unknown probabilities.

Probability sampling chooses observational units by a specified random mechanism with known inclusion probabilities. This separates design-based representativeness from the convenience of obtaining particular observations. Simple random sampling, stratified sampling and cluster sampling are different designs with different precision and cost properties.

Statistical estimation uses observed data to infer numerical properties of a population or model. A point estimator gives a value, while a confidence interval describes uncertainty with a specified coverage property. Maximum likelihood estimation is one approach; bias, variance, identifiability and the sampling or selection mechanism all affect an estimate's interpretation.

A statistical functional assigns a target value to each probability measure in a statistical model. Examples include an expected value, a quantile, and an integral of a power of a probability density function. Its behavior along statistical paths determines whether first-order influence-function representers are available.

Statistical efficiency compares the precision of procedures estimating the same estimand. For unbiased estimators, relative efficiency is commonly a ratio of their variances; lower variance means more information from the same sample size. An efficient estimator attains a relevant information lower bound. A baseline-adjusted linear regression can improve precision in a randomized controlled trial by removing prognostic variation.

An estimand is the population quantity an analysis seeks to learn, as distinct from an estimator calculated from a sample. In a randomized controlled trial, the causal effect of treatment assignment and the effect of actual treatment receipt are different estimands when adherence is imperfect.