Common Distributions and Generative Assumptions

Gaussian distributions are mathematically tractable, stable under linear combinations, and supported by central-limit behavior in many aggregate settings. They also lead to convenient least-squares objectives. Convenience is not evidence of adequacy.

A Gaussian observation model may assign probability to impossible negative values, underestimate heavy tails, or blur several subpopulations into one bell shape. Residual analysis should therefore inspect symmetry, scale changes, tails, dependence, and systematic structure, rather than rely on a histogram.

The price of the tail assumption has been quoted in public. In August 2007 Goldman Sachs' CFO, David Viniar, told the Financial Times that the firm was seeing “25-standard deviation moves, several days in a row”. Kevin Dowd and three co-authors, writing in the Journal of Portfolio Management in 2008, recorded the setting: “the CFO of Goldman Sachs, David Viniar, announced in August that Goldman's flagship GEO hedge fund had lost 27% of its value since the start of the year.”

The Bank of England then did the arithmetic the normal distribution implies. In a speech on 13 February 2009, Andrew Haldane, then its Executive Director for Financial Stability, spelled it out: “To provide some context, assuming a normal distribution, a 7.26-sigma daily loss would be expected to occur once every 13.7 billion or so years.” That is about the age of the universe. And it is the waiting time for a move far smaller than the one Viniar described. A 25-sigma event, under the same distribution, is expected once every 6 × 10^124 lives of the universe. Haldane did not conclude that the market had misbehaved several days running. “Fortunately, there is a simpler explanation – the model was wrong.” When a fitted family says an observed event is impossible, the family is the thing that has been falsified.

Choosing a distribution does more than choose a formula. It defines which values are possible, how variance relates to the mean, how tails behave, and which observations are treated as exchangeable. A Poisson model assumes a particular count mechanism and equates conditional mean with conditional variance. A Gaussian model allows all real values and gives exponentially light tails.

The right question is not which distribution is popular. It is which assumptions are defensible for the process, the measurement system, and the decision. Each of those assumptions is testable. The sections that follow test them against published records rather than against intuition: a nineteenth-century army casualty table, contact-tracing data from eight diseases, federal crash-count practice, and nearly a thousand networks whose assumed family did not survive the check.

The set of possible values rules out many families before parameter fitting begins. Binary outcomes, finite categories, nonnegative integers, positive continuous durations, and unrestricted real values each admit a different short list.

Crash counts are the case where that argument has been written into public practice instead of left to taste. A Federal Highway Administration report from February 2018 states the choice in its chapter on analysis: “A negative binomial error structure was specified, following the state of the art in modeling crash data.” The same report says why the reflexive default is wrong: “The negative binomial structure is now recognized as more appropriate for crash counts than the normal distribution assumed in conventional regression modeling.” Crashes at a site are nonnegative integers. A normal error structure puts probability on values the process cannot produce.

The specification is not decorative, because it carries a parameter that has to be estimated from data. Justice Appiah, calibrating Highway Safety Manual safety performance functions for Virginia freeway ramp terminals in May 2023, reported the method: “Dispersion parameters were estimated for each SPF and SPF grouping through maximum likelihood estimation assuming a negative binomial distribution of the errors.” The dispersion parameter k is fitted per function, not assumed once for the whole manual.

All three model counts, but they answer different generative questions. A binomial counts successes among a fixed number of trials, with variance tied to np(1−p). A Poisson counts events in an exposure interval under a rate model, takes exposure as an offset, and forces the conditional variance to equal the conditional mean. A negative binomial lets the variance exceed the mean, and it can arise as a Poisson–gamma mixture. It does not automatically solve zero inflation.

The gap between the second and the third is not a technicality. It is an entire epidemiological phenomenon. In Nature on 17 November 2005, Lloyd-Smith and three colleagues fitted a negative binomial offspring distribution — mean R0, dispersion parameter k, variance R(1+R/k) — to contact-tracing data from eight directly transmitted diseases. Their abstract states the result: “Using contact tracing data from eight directly transmitted diseases, we show that the distribution of individual infectiousness around R0 is often highly skewed.”

The fitted numbers show how far from Poisson the data sat. For SARS they estimated k = 0.16 (90% CI 0.11–0.64). Measles in the US from 1997 to 1999 gave 0.23 (0.16–0.39). Smallpox in Europe, Benin and Pakistan gave 0.32 to 0.72. A Poisson is the k → ∞ limit of that family, where mean and variance coincide. An estimate of 0.16 is nowhere near that limit, and the distance is precisely what superspreading means: most infected individuals transmit to nobody, while a few transmit to many, even when the average R0 is unchanged. A quantitative compendium of COVID-19 epidemiology states the modelling rule in one line: “The negative binomial distribution is generally used instead of Poisson to allow for unequal mean and variance.” Two models with the same mean can imply completely different probabilities for the outbreak that matters.

Resolution times are positive, skewed, and sometimes unfinished when the dataset closes. That last feature is not a nuisance to be dropped. It is the problem Edward Kaplan and Paul Meier solved in 1958. Their product-limit estimator uses durations known only to exceed the observation window, instead of discarding the rows that carry them.

The scale of the borrowing is measurable. Nature's 2014 analysis of Thomson Reuters' Web of Science ranked that paper number 11 on a list of the 100 most-cited papers of all time, a list that took 12,119 citations to enter: “the most frequently cited statistics paper (number 11) is a 1958 publication by US statisticians Edward Kaplan and Paul Meier that helps researchers to find survival patterns for a population, such as participants in clinical trials”. The article adds a detail that should reassure anyone whose own diagnostic work looks unglamorous: “The Kaplan–Meier paper was a sleeper hit, receiving almost no citations until computing power boomed in the 1970s, making the methods accessible to non-specialists.”

Challenge the model at the level of consequences, not only parameter estimates. List the impossible, bounded, discrete, censored, or truncated values. Write the mechanism in terms of trials, exposure, waiting, accumulation, or multiplicative growth. Compare mean–variance, skewness, and tail implications. Simulate replicated data and compare zeros, extremes, quantiles, and conditional patterns. Then inspect the summaries and tail regions that drive action.

The largest published run of that procedure overturned the family everyone expected. Anna Broido and Aaron Clauset tested nearly 1,000 real-world network data sets against five candidate degree distributions, in Nature Communications on 4 March 2019. Their abstract: “Across these networks, we find robust evidence that strongly scale-free structure is empirically rare, while for most networks, log-normal distributions fit the data as well or better than power laws.” The Results carry the counts: “only 10 and 4% of network data sets can be classified as belonging to the Strong or Strongest categories, respectively, in which the power-law distribution is not only statistically plausible, but the exponent falls within the special α ∈ (2,3) range”. Reporting on the same analysis, Quanta Magazine noted that “for 67 percent of the networks, including Facebook friendship networks, food webs and water distribution networks, the statistical tests rejected a power law as a plausible description of the network's structure.”

Fitting one family and reporting its parameters could not have produced any of those numbers. Only step four — comparing the candidate against rivals and against simulated replicates — can return the answer that the assumed family loses.