John Cowdery Kendrew helped establish how X-ray crystallography can reveal the three-dimensional arrangement of a protein. Nobel's primary nomination archive names John Cowdery Kendrew, also identifying J. C. Kendrew in its award-sharing comment.^[ref]

His team's1960 myoglobin paper presented a Fourier analysis at2-angstrom resolution. Its opening recalls how earlier phase determination compared native crystals with heavy-atom derivatives containing localized mercury or gold sites. Such comparisons supply information missing from a simple measurement of diffraction intensity.^[ref]

An original one-dimensional example explains that phase problem. Write a periodic electron density using Fourier series coefficients $F_k$. The two positive densitieshave $F_0=3$, $F_{\pm 1}=1/2$ and $F_{\pm 2}=\pm 1/4$, with the final sign distinguishing them. Their squared coefficient magnitudes are identical:9 for mode zero,1/4 for modes $\pm 1$ and1/16 for modes $\pm 2$. The relative phase of the second harmonic has changed while the first harmonic is fixed. Thus equal intensities need not identify one density.

The myoglobin experiment was more demanding than this toy calculation, but the distinction motivates its chemical strategy: modifying crystals at known heavy-atom sites changes their scattering in a way that constrains phase information. Reconstructing an electron density map therefore requires both measured diffraction and a justified phase-determination method, rather than treating the recorded spots as a direct photograph of the molecule.

Margaret M. Hilferty coauthored the 1931 paper The Distribution of Chi-Square by Edwin B. Wilson and Margaret M. Hilferty. The original byline identifies her with Harvard's Department of Vital Statistics; no fuller personal name is inferred from the initials.^[ref]

Their Wilson-Hilferty transformation reduces the skewness of a chi-squared distribution by taking a cube root. For $X\sim {\chi }_k^2$, the standardized quantity $((X/k{)}^{1/3}-(1-2/(9k)))/\sqrt{2/(9k)}$ is approximately standard normal. It provides a practical approximation to quantiles and tail probabilities, with accuracy depending on the statistical degrees of freedom and the probability region.^[ref]

Rajen D. Shah develops statistical inference for high-dimensional data. His author publication list verifies work on regression, model checking and testing conditional independence.^[ref]

With Jonas Peters, he introduced the generalized covariance measure statistic, based on the covariance of residuals after each response is regressed on the same conditioning variables. The resulting hypothesis test combines regression estimates with a calibrated covariance statistic. Their accompanying impossibility result explains why unrestricted distribution-free conditional independence testing needs additional structure; the regression-based method instead states approximation and moment assumptions explicitly.^[ref]

Jonas Peters studies causal inference and the foundations of conditional independence testing. With Rajen D. Shah, he proved that a test with uniform finite-sample validity over an unrestricted continuous-data null class cannot obtain nontrivial power against alternatives in their setting.^[ref]

Their constructive response is the generalized covariance measure statistic: regress each of two variables on the conditioning variables and examine the covariance of their residuals. Suitable regression-error rates and moment conditions yield asymptotic validity. Such a test targets conditional residual association; arbitrary forms of conditional dependence need not produce a nonzero residual covariance.^[ref]

John Carlin develops biostatistics for health and medical research, including methods for missing data and causal inference. His institutional profiles describe work on multiple imputation and the need to specify the scientific purpose of an analysis before choosing a regression model.^[ref][ref]

With multiple imputation, several plausible completed datasets represent uncertainty about unobserved values. Their analyses must account both for uncertainty within a completed dataset and for variation between completed datasets. This makes the assumptions about the missing values visible in the analysis. Carlin is a coauthor of Bayesian Data Analysis by Andrew Gelman and others.^[ref]

David Dunson develops Bayesian inference and statistical machine learning for complex data. His research seeks lower-dimensional structure in observations such as networks, functions and biological measurements, with applications in ecology, neuroscience and environmental health.^[ref]

A model may represent many dependent measurements through a smaller set of latent quantities, then carry uncertainty about those quantities into predictions. Dunson's work couples flexible modelling with algorithms for Bayesian inference and uncertainty quantification. He is also a coauthor of Bayesian Data Analysis by Andrew Gelman and others.^[ref]

Aki Vehtari develops computational Bayesian statistics, including probabilistic programming, model diagnostics, model assessment and Gaussian processes. His author research page describes these as connected parts of a Bayesian workflow: specify a model, compute its implications, check the computation and examine predictive adequacy.^[ref]

He coauthored Bayesian Data Analysis by Andrew Gelman and others. A posterior distribution summarizes uncertainty conditional on the model and data; checking whether that model reproduces important observations is a further task. This distinction connects Bayesian inference with posterior predictive checks, rather than treating successful computation as sufficient evidence that the model is suitable.^[ref]

Olivier Catoni develops methods for robust statistics. His work on the mean and variance constructs M-estimators with deviation guarantees for heavy-tailed distributions under moment assumptions.^[ref]

The sample mean lets a single very large observation exert arbitrarily large influence. Catoni's construction replaces a linear estimating equation by a controlled nonlinear function of each residual. With an appropriate tuning parameter, a mean estimator can achieve Gaussian-order deviation bounds at a chosen confidence level under bounded variance, even when the observations are not normally distributed. The tuning and assumptions are part of the guarantee; the result does not assert that one fixed setting automatically works at every confidence level.^[ref]

Vladimir Koltchinskii studies the mathematics of statistical learning. His joint work with Karim Lounici gives concentration inequalities and expectation bounds for sample covariance operators of centered Gaussian observations.^[ref]

Their bounds separate two questions: the typical size of the operator norm error, and its fluctuations around that typical size. In finite-dimensional Euclidean space, their effective dimension is comparable to $\mathrm{tr}(\Sigma )/\Vert \Sigma \Vert$. If most covariance eigenvalues are small, this quantity can be much smaller than the ambient dimension, explaining why statistical difficulty need not track the raw number of coordinates.^[ref]

Karim Lounici develops the theoretical foundations of statistical models and algorithms for structured data. His own research page identifies this focus.^[ref]

With Vladimir Koltchinskii, he bounded the error of sample covariance operators in the operator norm. For a centered Gaussian vector with covariance $\Sigma$, the expected error has scale comparable to $\Vert \Sigma \Vert \max \{\sqrt{r/n},r/n\}$, where $r=(\mathbb{E}\Vert X\Vert {)}^2/\Vert \Sigma \Vert$. This effective dimension reflects the distribution's geometry rather than merely the number of recorded coordinates.^[ref]

Tony Cai develops methods and theory for high-dimensional statistics, statistical decision theory and statistical machine learning. His research includes functional data analysis and applications to genomics.^[ref]

With Weidong Liu and Xi Luo, he introduced CLIME, estimating a sparse precision matrix through linear programming. Their results bound estimation error under sparsity and tail assumptions. The method makes a useful distinction between an optimization constraint and a theorem: positive definiteness of the final estimate is established under conditions, rather than imposed as a constraint in the columnwise programs.^[ref]

Weidong Liu works on high-dimensional statistics. Shanghai Jiao Tong University's institutional profile identifies this researcher and his focus on high-dimensional statistical inference.^[ref]

His CLIME paper with Tony Cai and Xi Luo estimates a sparse precision matrix even when direct inversion of the sample covariance matrix is unsuitable. It controls $\Vert \hat{\Sigma }\Omega -I{\Vert }_{\max }$ while minimizing an entrywise absolute-value objective, then uses a symmetrization rule. Statistical accuracy requires assumptions on sparsity and the distribution's tails.^[ref]

Xi Luo develops biostatistics and machine learning methods for biomedical data. His institutional profile identifies applications in neuroscience, genomics and disease research.^[ref]

With Tony Cai and Weidong Liu, he developed CLIME for sparse precision matrix estimation. The method constrains the error of an approximate inverse of the sample covariance matrix and minimizes an entrywise absolute-value objective. Its computation separates into columnwise linear programs.^[ref]

Jerome H. Friedman is the statistician who coauthored the Graphical Lasso with Trevor Hastie and Robert Tibshirani. Their paper estimates a sparse inverse covariance matrix using penalized likelihood and a computationally efficient iterative algorithm.^[ref]

The statistical benefit of a sparse precision matrix is an interpretable pattern of conditional relationships. In a multivariate normal distribution, an off-diagonal zero means that the corresponding variables are conditionally independent given the others. This conditional relationship can differ from their ordinary marginal correlation. The paper's repeated Lasso calculations connect that model with practical convex optimization.^[ref]

Trevor Hastie develops methods in statistical learning, including generalized additive models and regularized regression. His research page describes the interaction between interpretable statistical models, flexible prediction and efficient computation.^[ref]

His joint Graphical Lasso work with Jerome H. Friedman and Robert Tibshirani turns sparse precision matrix estimation into repeated Lasso subproblems. A blockwise algorithm updates parts of the estimate while solving the penalized likelihood problem, making large conditional-dependence models computationally feasible.^[ref]

Robert Tibshirani develops methods for statistical learning and biostatistics, with applications to genomics and medicine. His Stanford research profile connects methodological work with scientific data and computational tools.^[ref]

With Jerome H. Friedman and Trevor Hastie, he developed the Graphical Lasso for estimating a sparse precision matrix. An absolute-value penalty encourages zero entries, which encode conditional independence relationships in a multivariate normal distribution. The method combines a statistical model with an efficient optimization algorithm.^[ref]

Peter Gavin Hall, also cited as Peter Hall, developed theory for the bootstrap, nonparametric statistics and approximation of sampling distributions. His professional society's memorial explicitly verifies the fuller identity and these research areas.^[ref]

His The Bootstrap and Edgeworth Expansion by Peter Hall explains how Edgeworth expansions refine a normal approximation using higher cumulants. Such expansions help analyze the error of a bootstrap approximation, including whether a transformed or studentized statistic achieves a higher order of accuracy. Smoothness and sampling assumptions matter: the presence of a formal expansion alone does not establish a uniformly accurate approximation.^[ref]

Egon Pearson developed the theory of statistical hypothesis testing with Jerzy Neyman. Their framework distinguishes controlling type I error under a null hypothesis from maximizing statistical power against a specified alternative hypothesis. University College London's institutional history identifies this collaboration as a central contribution.^[ref]

The Neyman-Pearson lemma makes the tradeoff precise for two simple hypotheses with densities $f_0$ and $f_1$. At a fixed allowed type I error, the most powerful test rejects where the likelihood ratio $f_1/f_0$ is largest, with boundary randomization if needed. To see the principle, compare its test function ${\varphi }_{\ast }$ with another admissible $\varphi$: the product $(\varphi -{\varphi }_{\ast })(f_1-c{f}_0)$ is nonpositive pointwise. Integrating and using the size constraint shows that $\varphi$ cannot have greater statistical power.

D. G. Horvitz coauthored the 1952 paper that generalized finite-population sampling without replacement to unequal selection probabilities. The joint work with D. J. Thompson provides the foundation for the Horvitz-Thompson estimator.^[ref]

The central idea is to compensate for unequal chances of inclusion. For fixed population values $y_i$ and positive inclusion probabilities ${\pi }_i$, write a total estimate asUsing an inclusion indicator $I_i$ gives $\mathbb{E}(I_i/{\pi }_i)=1$, so $\mathbb{E}\hat{Y}={\sum }_i{y}_i$. Unbiasedness follows from the design probabilities rather than from assuming that the sampled values are an independent realization of a population model.^[ref]

For an illustrative population $(4,6)$, select exactly one unit with probabilities $(1/4,3/4)$. The total estimate is $16$ or $8$, respectively, with expected value $(1/4)16+(3/4)8=10$, the true total. The less frequently selected unit receives the larger multiplier. This example explains how a sampling design can deliberately favor some units without forcing an unweighted analysis to be correct.