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For the positive-shape generalized Pareto distribution with shape $\psi >0$ and scale $\sigma >0$, the expected Fisher information matrix for one observation isThe orthogonalization equation for two statistical parameters gives ${\partial }_{\psi }\sigma =-\sigma /(1+\psi )$ at fixed new nuisance coordinate. Thus $\eta =\sigma (1+\psi )$ is a global interest-respecting reparametrization. In coordinates $(\psi ,\eta )$, the information is diagonal with entries $(1+\psi {)}^{-2}$ and $[{\eta }^2(1+2\psi ){]}^{-1}$.

Conditioning a Pareto distribution on exceeding a threshold changes its minimum while preserving its shape parameter. For $u\ge \beta$ and $x\ge u$, the definition of conditional probability givesEquivalently, for $r\ge 1$,The ratio to the threshold has the same Pareto distribution at every admissible threshold.^[ref] For the chosen $\alpha =2$, $\beta =10$, only $1/4$ of outcomes exceed 20, but among those outcomes the probability of exceeding 40 is again $1/4$. When $\alpha >1$, the same identity gives conditional expectation $\mathbb{E}[X\mid X\ge u]=\alpha u/(\alpha -1)$.

This multiplicative scaling differs from the additive memoryless property of an exponential distribution. Indeed $Y=\log (X/\beta )$ has exponential distribution of rate $\alpha$, because $\mathbb{P}(Y>y)=e^{-\alpha y}$ for $y\ge 0$. Thus logarithms turn multiplication into addition and explain the scaling property. Conversely, if $U$ has uniform distribution on $(0,1)$, inverse transform sampling gives $X=\beta (1-U{)}^{-1/\alpha }$.

The excess $Z=X-u$ conditional on $X\ge u$ has survival function $(1+z/u{)}^{-\alpha }$ for $z\ge 0$. In the generalized Pareto distribution convention $\mathbb{P}(Z>z)=(1+\xi z/\sigma {)}^{-1/\xi }$, this means $\xi =1/\alpha$ and $\sigma =u/\alpha$. Here the excess starts at zero; the original Pareto distribution starts at $\beta >0$.

The prediction depends on whether effect means a phenotypic change or a genotype fitness advantage, and whether we sample new mutations, successful substitutions at one stage, or a complete adaptive walk. These distributions differ even in simple models. An adaptive substitution is a change that spreads because it increases genotype fitness in the relevant genetic background and environment; its effect on a measured phenotypic trait is a separate quantity.

Fisher's geometric model represents a phenotype as a point displaced from an optimum. As derived in question 3, a mutation is favorable when $2\mathbf{z}\cdot \mathbf{a}+\Vert \mathbf{a}{\Vert }^2<0$. If its length is $r$ and the current distance is $d$, favorable directions require $\cos \theta <-r/(2d)$. Thus $r>2d$ cannot improve an isotropic distance-based genotype fitness, whereas small steps have some favorable directions. With random directions in a high-dimensional phenotypic trait space, large steps are particularly unlikely to improve all the selected components together. This is a probability-of-benefit argument, not yet a prediction for what fixes: extremely small advantageous effects are easily lost through genetic drift.

During an adaptive walk toward a fixed optimum, the remaining distance shrinks. Substantial changes can be favorable early, whereas fine adjustments dominate late. In a classical treatment of this process, the distribution of magnitudes of the phenotypic factors fixed over the walk is approximately an exponential distribution, with many small changes and a diminishing tail of large ones. This can reconcile a few prominent changes with numerous minor contributors; it is not the assertion that every step must be tiny.^[ref] The shape concerns the stated model and effect measure. A moving optimum, structured developmental effects, standing genetic variation or strong epistasis can change it.

A related but distinct argument uses extreme value theory. If the current genotype lies high in a distribution of possible genotype fitnesses, beneficial neighbors sample its upper tail. For a broad class of rapidly declining tails, excesses above a high threshold approximately follow an exponential distribution. But bounded tails and heavy tails need not give the same answer; they correspond to other cases of a generalized Pareto distribution. Neither an exponential tail nor the assumption of a high initial threshold is a law of nature. Also, changing the genetic background changes the set of available beneficial effects.

Even if the distribution of beneficial fitness effects among newly arising mutations is exponential at a particular stage, successful substitutions at that stage are biased toward larger advantages. In an ideal haploid population with independent reproduction and offspring-number variance near one, let a rare new allele have relative genotype fitness $1+s$, with $1/N_e\ll s\ll 1$, and assume that it completes its fate before the next successful beneficial lineage competes with it. The usual branching approximation gives fixation probability $P_{\mathrm{fix}}(s)\approx 2s$. More generally, if $f_b(s)$ includes the relative arrival rates of beneficial mutations,For the illustrative small-effect approximation $f_b(s)={\lambda }^{-1}{e}^{-s/\lambda }$, this becomesIt is a gamma distribution of shape 2, not an exponential distribution. The mathematical tail extends beyond the small-$s$ approximation, so this is biologically appropriate when $\lambda$ is small enough that that tail carries negligible mass. With $\lambda =0.01$, the fraction exceeding $s=0.02$ is $e^{-2}\approx 0.135$ among arrivals but $3e^{-2}\approx 0.406$ among substitutions in this approximation. This calculation and the pooled phenotypic-walk result answer different questions and are not contradictory.

Tests can begin with experimental evolution. Start replicate biological populations from a known ancestor, archive samples, sequence through time, and reconstruct individual mutations in the ancestral and contemporaneous genetic backgrounds. Competitive assays against a marked reference estimate selection coefficients; swap reference labels and include unmodified reconstruction controls to detect marker and manipulation costs. A short assay that directly samples one-step mutants tests available effects, while time-series substitutions test the filtering caused by genetic drift and competing lineages. In particular, experiments on two viruses rejected an exponential model for the beneficial effects they sampled, showing why tail shape must be tested rather than assumed.^[ref]

Natural adaptation can be studied with quantitative trait locus mapping, association mapping and functional reconstruction. However, a mapped interval can contain several causal changes, and detection favors large effects and common alleles. Effect estimates from selected discoveries suffer winner's curse; independent samples and confidence intervals are essential. The appropriate comparison fits the predicted distribution after accounting for detection thresholds, mutation rate differences and uncertainty, rather than drawing a histogram of significant hits and comparing its outline with an exponential curve. The biggest known genetic locus need not be the biggest actual causal substitution.

Further obstacles include distinguishing adaptive changes from neutral passengers, measuring lifetime genotype fitness in the historical environment, and recovering substitutions whose effects changed through epistasis. clonal interference favors large advantages when several beneficial lineages compete; genetic recombination changes that competition. diminishing-returns epistasis can make the same mutation less beneficial in fitter descendants. Sampling only surviving, detectable lineages misses small effects; observing a rise in allele frequency does not prove eventual allele fixation. Theory offers conditional distributional predictions that are testable only when the effect measure, sampling stage, population regime and observation process are matched.

For a regular two-parameter model, write $\psi$ for the parameter of interest and $\lambda$ for the nuisance parameter. If $u_{\psi },u_{\lambda }$ are its score functions, the parameters are orthogonal statistical parameters when the cross entry of the expected Fisher information matrix vanishes:This condition may hold at a particular parameter value or throughout the parameter domain. It concerns the information geometry, not exact independence of finite-sample estimators. Under regular asymptotic efficiency, a diagonal information matrix gives zero first-order asymptotic covariance.

An interest-respecting reparametrization leaves $\psi$ unchanged, or replaces it by a one-to-one function of $\psi$ alone, while changing the nuisance coordinate by a smooth invertible transformation which may depend on both old parameters. Thus it preserves the level sets of the inferential target. Without loss of generality, keep $\psi$ itself and write the old nuisance as $\lambda =g(\psi ,\eta )$, where $\eta$ is the new nuisance coordinate and $g_{\eta }\ne 0$.

The chain rule for the log-likelihood gives new score functionsTheir expected cross product isSetting this to zero, with $I_{\lambda \lambda }>0$, gives the orthogonalization equation for two statistical parametersSolve this ordinary differential equation along curves of constant $\eta$, using $\eta$ to label distinct initial conditions. Smooth local solutions with nonzero transverse derivative produce an invertible interest-respecting reparametrization. This informal construction identifies precisely which derivative is held at fixed new nuisance coordinate.

For the given positive-shape generalized Pareto distribution, its one-observation log-likelihood isDefine $L=\log (1+\psi Y/\sigma )$ and $D=e^{-L}$. The density transformation $y=\sigma (e^L-1)/\psi$, $dy/dL=\sigma e^L/\psi$, givesThus $L$ has an exponential distribution of mean $\psi$. Direct differentiation of the original log-likelihood, holding the other original parameter fixed, givesThe following exponential integrals determine their expected products:For example, $\mathbb{E}{e}^{-kL}=(1+k\psi {)}^{-1}$ follows by integrating the exponential density, and differentiating this integral in $k$ gives $\mathbb{E}(L{e}^{-kL})=\psi /(1+k\psi {)}^2$. These formulas also give $\mathbb{E}{u}_{\psi }=\mathbb{E}{u}_{\sigma }=0$.

For clarity, the nuisance information and cross information can be calculated explicitly. Put $a=1+\psi$. ThenUsing $\mathbb{E}[L-a+aD]=0$, the cross product isThe same moments give $I_{\psi \psi }={\psi }^{-4}\mathbb{E}[L-a+aD{]}^2=2/[(1+\psi )(1+2\psi )]$. Therefore the Fisher information matrix for one observation isIn particular, $I_{\psi \sigma }>0$ for every admissible pair, proving the original shape and scale parameters are not orthogonal. No finite-mean assumption on $Y$ was needed: the transformed logarithm $L$ and the bounded variable $D$ have the required integrable score products even when the Pareto mean is infinite.

The orthogonalization equation for two statistical parameters now becomesIntegration along a constant-$\eta$ curve gives $\sigma (1+\psi )=\eta$. Hence a global interest-respecting reparametrization isIt is smoothly invertible on the entire positive parameter domain, with inverse $\sigma =\eta /(1+\psi )$. The Fisher information matrix transforms as $J^{T}IJ$, whereSubstitution givesThis verifies the orthogonal shape and scale for a generalized Pareto distribution directly, as well as by the differential equation. For an iid sample, every information entry is multiplied by the sample size, so the same parameter orthogonality remains valid.

This is the generalized Pareto distribution for the excess $X-u$, with scale $\sigma >0$ and shape $\xi$. Its support requires $1+\xi (x-u)/\sigma >0$. The Pickands-Balkema-de Haan theorem says that, for distributions in an appropriate extreme-value domain of attraction, the conditional distribution of excesses above a sufficiently high threshold approaches a generalized Pareto form. This makes it the natural peaks-over-threshold method model; it is an asymptotic justification, not an assertion that every threshold is sufficiently high.

For $\xi >0$, the tail decays as a power and has no finite upper endpoint, giving a heavy tail. For $\xi =0$, the limiting distribution is exponential, with survival probability $\exp \{-(x-u)/\sigma \}$. For $\xi <0$, there is a finite upper endpoint $u-\sigma /\xi$, giving a bounded tail. Thus the sign of the shape parameter distinguishes heavy, exponential-type and bounded tails.

In the original PDF's probability plot, most points lie close to the diagonal. The quantile plot likewise agrees well over most of the range, but the largest empirical flow, around $68.5$, is well above its fitted quantile, around $66$. The density plot captures the strongly decreasing right-skewed bulk of the excesses. The return level plot shows a broadly reasonable fit to most observations, with one conspicuously high extreme and increasingly wide uncertainty at long return periods. Consequently the generalized Pareto distribution is a reasonable first approximation, but its most extreme tail is less convincingly represented.

Reading the return-level curve at the 100-year tick givesThese are approximate graphical readings, not parameter-based calculations; rounding to whole units would give about $67$ with an interval roughly $65$ to $69$. The two blue curves are uncertainty bounds for the fitted return level, not prediction bounds for individual observations. The daily-data independence assumption and the stability of the fit as the threshold changes should be checked, for example by examining clusters of high flows and using declustering of extremes where necessary. A 100-year level also extrapolates beyond the 25-year record.

The peaks-over-threshold method fits a generalized Pareto distribution to exceedances of a high threshold and also estimates their frequency. Together these determine tail probabilities and return levels. For dependent observations, threshold exceedances may occur in clusters; declustering of extremes can separate extreme events before fitting a model for independent events.

For a distribution in an extreme-value domain of attraction, the conditional distribution of the excess above a high threshold is asymptotically approximated by a generalized Pareto distribution with a threshold-dependent scale and a limiting shape parameter. This theorem justifies the peaks-over-threshold method. Threshold selection must balance approximation error against the reduced number of excess observations.