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An unbiased estimator that is a function of a complete sufficient statistic is the unique uniformly minimum-variance unbiased estimator, up to almost sure equality. Rao-Blackwellization supplies variance reduction, and completeness supplies uniqueness. The theorem applies to a conditional statistical experiment as well, when unbiasedness, sufficiency and completeness are all evaluated under that conditional family.

For an MA(1) process, the recursion ${\lambda }_t=\theta /(1+{\theta }^2-\theta {\lambda }_{t-1})$ has candidate limits $\theta$ and $1/\theta$. The limit in $[-1,1]$ is $\theta$ when $|\theta |\le 1$ and $1/\theta$ otherwise. The limiting innovation variance is ${\sigma }^2\max (1,{\theta }^2)$, expressing the same covariance law through an invertible reciprocal representation.

Fix $k\ge 1$, ${\gamma }_0>1/(k+1)$, and ${\gamma }_0\le \gamma \le 1-\omega (n)/\log n$ with $\omega (n)\to \infty$. A binomial random graph has one giant component of the displayed order and all other graph components are tree components of orders at most $k$, with high probability. The expected value and variance of the isolated vertex count concentrate it around $n^{1-\gamma }\to \infty$. A Cayley formula upper bound for connected sets excludes component sizes $k+1$ through a small fixed fraction of $n$; the empty-graph cut bound excludes the remaining sizes up to $n/2$. An extra-edge count excludes cyclic small graph components. The expected value of the number of vertices in small nonisolated tree components is $O(nd{e}^{-2d})=o(n{e}^{-d})$, where $d=np$. The Markov inequality completes the count of vertices outside the unique large graph component.

The log odds ratio is the natural logarithm of an odds ratio. For two independent binomial groups with positive event and non-event counts $(a,b)$ and $(c,d)$, its estimate is $\log (ad/(bc))$, with first-order variance $1/a+1/b+1/c+1/d$ by the delta method. Zero cells need a method that handles boundary data; the displayed normal approximation is not valid there.

The logarithm of a risk ratio is zero at equal risks, negative when the first risk is smaller, and positive when it is larger. For independent binomial event counts $d_1,d_0$ from samples $n_1,n_0$, its usual delta-method variance estimate is $1/d_1-1/n_1+1/d_0-1/n_0$. This approximation requires nonzero counts; zero-event arms need another procedure. Log estimates are convenient for inverse-variance meta-analysis, and exponentiation returns to the risk-ratio scale.

For independent two-by-two tables with counts $a_i,b_i,c_i,d_i$ and total $n_i$, the Mantel–Haenszel pooled odds ratio is the displayed ratio of sums. When all $b_i{c}_i>0$, it is also the weighted arithmetic mean of study odds ratios with weights $b_i{c}_i/n_i$. These differ from inverse estimated variance weights for log odds ratios. Both constructions can represent a fixed-effect meta-analysis, but they should not be silently interchanged when interpreting a supplied plot.

For a fixed set $L$ of $\lfloor \sqrt{n}\rfloor$ vertices, let ${\tau }_L$ be the first step at which they all belong to one graph component. In the uniform random graph process, ${\tau }_L$ lies between $n(\log n-\omega )/4$ and $n(\log n+\omega )/4$ with high probability for every $\omega \to \infty$. Below that scale the marked isolated vertex count has divergent expected value and small relative variance. Above it the logarithmic-regime giant component misses $o(\sqrt{n})$ vertices; exchangeability and a union bound show that none are marked. The label coupling then transfers the two assertions to the process.

With a fully observed covariate $X$, response $Y$, and an independent response indicator with probability $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic variance $[\mathrm{Var}(m)+E\{\mathrm{Var}(Y\mid X)\}/\rho ]/N$, whereas the complete-case mean has variance $\mathrm{Var}(Y)/(N\rho )$. The difference is $(1-\rho )\mathrm{Var}(m)/(N\rho )$. Thus missing completely at random does not justify discarding predictive observed covariates on efficiency grounds.

With one initial ancestor and independent offspring counts of finite mean and variance, conditioning on the current generation gives the displayed recurrences. For $\mu \ne 1$ the variance is ${\sigma }^2{\mu }^{n-1}({\mu }^n-1)/(\mu -1)$ for positive generation index. At $\mu =1$ it is $n{\sigma }^2$. At zero offspring mean the distribution is concentrated at zero and later generations vanish. These formulas are for the Galton-Watson process, not arbitrary dependent reproduction.

A meta-analysis combines statistical evidence from several studies addressing a common estimand. It may assume a shared effect or use a random-effects meta-analysis to represent between-study variation. Inverse-variance weighted means combine independent effect estimates when their variances are known or estimated; differences in study validity and effect modifiers require substantive assessment, not just weighting.

For $\mu =\int hf$ and $0<A=\int |h|f<\infty$, the proposal $q_{\ast }=|h|f/A$ minimizes the variance of the ordinary unbiased importance sampling estimator. The Cauchy-Schwarz inequality gives $\int h^2{f}^2/q\ge A^2$, so the minimum variance for $n$ independent samples is $(A^2-{\mu }^2)/n$. The proposal need only cover the nonzero integrand.

For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu =\mathbb{E}{N}_t$ and ${\sigma }^2=\mathrm{var}{N}_t$, thenThe expected value sees only the net growth rate $B-D$, whereas the variance also sees the total event rate $B+D$.

Use the pooled estimate and total variance from Rubin's rules. A basic large-complete-sample scalar interval is $\overline{Q}\pm t_{\nu ,1-\alpha /2}\sqrt{T}$, with $\nu =(M-1)[1+\overline{U}/((1+M^{-1})B){]}^2$ for $B>0$. Small complete-data samples need an appropriate degrees-of-freedom adjustment; the normal limit applies as the available degrees of freedom become large.

A normal approximation replaces a sampling distribution by a normal distribution, often through a central limit theorem. In a Wald test, approximate statistical power is determined by the standardized difference under the alternative. This yields Neyman allocation for fixed treatment-specific variances.

For a normal linear model with coefficient vector $\eta$ and residual variance ${\sigma }^2$, a normal-inverse-gamma prior has $\eta \mid {\sigma }^2\sim N(m,{\sigma }^{2}V)$ and ${\sigma }^2\sim \mathrm{IG}(a,b)$, with $V$ a positive-definite matrix and $a,b>0$. The inverse-gamma density is proportional to $({\sigma }^2{)}^{-a-1}{e}^{-b/{\sigma }^2}$. Multiplying by the normal linear model likelihood function and completing the square preserves the family, giving an analytic posterior and evidence integral.