Statistical, resampling, simulation, and numerical error ledger — Examples

Given: The stated IID two-point observation law, observed sample $(0,1)$, finite unit space and map $T_2$, bootstrap procedure, and four realized resamples. Prove the conditional bootstrap law and error separation.

The error ledger uses a declared scalar target and explicit successive reference values, with each stage error equal to the difference of adjacent references. (Successive statistical and computational error targets)

The total realized error along the finite reference chain is the sum of its signed stage errors. (Pathwise error decomposition along a reference chain)

Identical distribution and IID families: IID means a common law and mutual independence.

Size-n bootstrap from the empirical law: for a fixed tuple of length $n\ge 1$, draw $n$ indices independently and uniformly from $[n]$, with replacement, and apply the declared measurable statistic to the resampled tuple.

Empirical-law bootstrap draws are conditionally IID: conditional on the fixed observed tuple, the resampled coordinates are IID with common law equal to its empirical measure.

For $B\ge 2$ IID finite-variance replicate outputs, $[B(B-1){]}^{-1}{\sum }_b(Z_b-{\bar{Z}}_B{)}^2$ is unbiased for the variance of their average. (Unbiased variance estimate from independent computation replicates)

The observations are IID with the stated two-point law; the target is their mean $\theta =3\mathrm{/}4$.

The observed ordered tuple is $(0,1)$ in the finite measurable unit space $(\{0,1\},2^{\{0,1\}})$; each entry is one resampling unit and $T_2(x_1,x_2)=(x_1+x_2)\mathrm{/}2$ is measurable on the full power-set sample space.

Four independent repetitions of the stated bootstrap algorithm produce the listed ordered index pairs.

Given: The group, action, statistic, invariant uniform null law, and finite list of sampled transformations in the Example. Prove the exact-orbit and realized finite-$B$ calculations.

A finite-group randomization setup consists of a finite probability space, finite group action, statistic, and null laws invariant under that action. (Finite-group randomization setups)

The inclusive orbit upper-tail rank counts group elements with $T(gx)\ge T(x)$ and divides by $\mathrm{\mid }G\mathrm{\mid }$. (Orbit upper-tail rank p-values)

Under an invariant null, the full-orbit inclusive rank p-value is superuniform. (Finite-group orbit-rank validity)

For a finite scalar reference chain, the total realized error is the sum of its signed adjacent-stage errors. (Pathwise error decomposition along a reference chain)

The operation $a^i{a}^j=a^{i+j(\mathrm{m}\mathrm{o}\mathrm{d}3)}$ is associative with identity $e=a^0$ and inverses $a^{3-i}$; for each $g\in G$, left multiplication $x\mapsto gx$ is a bijection of $G$.

The finite-$B$ formula is the explicitly defined calculation in the Example; no validity property is assumed for it.

Given: The ODE, initial-value law, one-step Euler rule, and independent path simulations specified in the Example. Compute the discretized-target bias and the path-simulation variance.

A scalar error ledger uses successive same-scale references and defines each stage error as the adjacent difference. (Successive statistical and computational error targets)

The total error along a finite scalar reference chain is the sum of its signed stage errors. (Pathwise error decomposition along a reference chain)

For $M\ge 2$ IID finite-variance replicate outputs with variance ${\tau }^2$, the variance of their average is ${\tau }^2\mathrm{/}M$. (Unbiased variance estimate from independent computation replicates)

For real $a<b$, a real function continuous on $[a,b]$ and differentiable on $(a,b)$ satisfies $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$ for some $c\in (a,b)$. (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$)

$e=\exp (1)$ for $\exp (x)={\sum }_{k=0}^{\mathrm{\infty }}{x}^k\mathrm{/}k!$. (The real exponential function and the number $e$ by a power series)

$\exp$ is differentiable with $(\exp {)}^{\mathrm{\prime }}=\exp$. (The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$)

The derivative of a differentiable composition is given by the chain rule. (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$)

IID means common distribution and mutual independence. (Identical distribution and IID families)

A function differentiable at a limit point is continuous there. (A function differentiable at $c$ is continuous at $c$)

Applying measurable maps coordinatewise to independent random variables preserves independence. (Measurable coordinatewise functions preserve independence)

Products of measurable real-valued functions and their real scalar multiples are measurable. (Arithmetic and lattice operations preserve measurability whenever they are defined)

$X_0$ equals $1$ and $2$ with probabilities $1\mathrm{/}2$ each; an ODE solution $x$ is defined and differentiable on an open interval containing $[0,1]$, satisfies $x^{\mathrm{\prime }}=x$, and has initial value $x(0)=X_0$.

The discretization is the explicitly defined one-step rule $X_1^h=X_0+h{X}_0$ at $h=1$; the $M$ simulated initial values are IID copies of $X_0$.

Given: Fixed data and exact integral target $I$, with IID randomized-rule outputs of mean $I$ and finite variance. The observed outputs are $2\mathrm{/}5,1\mathrm{/}2,7\mathrm{/}10$.

For $R\ge 2$ IID replicate outputs, $[R(R-1){]}^{-1}{\sum }_r(Z_r-{\bar{Z}}_R{)}^2$ is unbiased for the variance of their average. (Unbiased variance estimate from independent computation replicates)

A computation stage is recorded as the difference between its exact reference target and its realized output, with all values on the same scalar scale. (Successive statistical and computational error targets)

Given: The vectors, binary64 round-to-nearest ties-to-even arithmetic, unit roundoff $u=2^{-53}$, and the left-to-right evaluation order. Check the exact result, computed result, normal-range hypotheses, and gamma bound.

If $n\ge 1$, $nu<1$, and every exact product $x_j{y}_j$, rounded product $p_j:=\mathrm{fl}(x_j{y}_j)$, and intermediate sum lies in the normal range, the computed left-to-right dot product satisfies $\mathrm{\mid }\mathrm{fl}(x\cdot y)-x\cdot y\mathrm{\mid }\le {\gamma }_n{\sum }_j\mathrm{\mid }{x}_j{y}_j\mathrm{\mid }$, where ${\gamma }_n=nu\mathrm{/}(1-nu)$. (The computed floating-point dot product obeys componentwise and normwise error bounds)

The floating-point model uses relative error at most $u$ only on its stated normal-range operation domain; exact zero is handled separately. (The standard relative floating-point model and unit roundoff)

The final numerical error is $e_{\mathrm{f}\mathrm{p}}=\tilde{r}-r_m$ for the error ledger. (Successive statistical and computational error targets)

The total error along a finite scalar reference chain is the sum of its signed adjacent-stage errors. (Pathwise error decomposition along a reference chain)

In the binade $[2^{53},2^{54})$, a radix-two format with precision $53$ has spacing $2$. Since $2^{53}<{10}^{16}<2^{54}$, ${10}^{16}$ and ${10}^{16}+2$ are adjacent representable values. The lower significand is ${10}^{16}\mathrm{/}2=5000000000000000$, an even integer, whereas the upper significand is odd; therefore the exact midpoint ${10}^{16}+1$ rounds to ${10}^{16}$ under ties-to-even.

Given: A two-point probability space with $U\in \{-1,1\}$ equally likely and $V=U$, interpreted as two successive signed stage errors. Compute the mean-squared sum and compare it with the variance-only sum.

The mean-squared sum of square-integrable components includes twice each pairwise covariance. (MSE of a sum of error components includes covariance terms)

An error ledger records each stage error as the difference of adjacent scalar references. (Successive statistical and computational error targets)