Double-blind randomized controlled trials are traditionally seen as the gold standard for causal inferences as the difference-in-means estimator is an unbiased estimator of the average treatment effect in the experiment. The fact that this estimator is unbiased over all possible randomizations does not, however, mean that any given estimate is close to the true treatment effect. Similarly, while predetermined covariates will be balanced between treatment and control groups on average, large imbalances may be observed in a given experiment and the researcher may therefore want to condition on such covariates using linear regression. This article studies the theoretical properties of both the difference-in-means and OLS estimators conditional on observed differences in covariates. By deriving the statistical properties of the conditional estimators, we can establish guidance for how to deal with covariate imbalances.

In a Data-Generating Experiment, the observed sample, x, has intractable or unavailable c.d.f., F_θ, and θ’s statistical nature is unknown; θ is element of metric space (Θ, dΘ). Matching estimates of θ are introduced, learning from the “best” x-matches with samples X^∗ from F_{θ^∗}, θ^∗ ∈ Θ. Under mild conditions, these nonparametric estimates are uniformly consistent and the upper bounds on their rates of convergence in probability have the same rate and depend on the Kolmogorov entropies of an increasing sequence of sets covering Θ. When Θ ⊆ Rm and the observations are i.i.d. the upper bounds can be, \sqrt{log n}/\sqrt{n} when m is known, and \sqrt{m_n·log n}/\sqrt{n} when m is unknown; m ≥ 1, m_n ↑ ∞ at a desired rate. Upper bounds can also be obtained for dependent observations. These rates hold for observations in R^d, complementing recent results obtained for real, i.i.d. observations, under stronger assumptions and using weak probability distances; d ≥ 1. In simulations, the Matching estimates are successful for the mixture of 2 normals and for Tukey’s (a, b, g, h) and the (a, b, g, k) models. Computers’ evolution will allow for more and faster comparisons, resulting in improved Matching estimates for universal use in Machine Learning.

When sample, X = x, is observed from intractable c.d.f. Fθ, or a Black-Box with input θ, an Approximate Bayesian Computation (ABC) method provides approximate posterior, π_ϵ. θ^∗ is included in the support of π_ϵ when the Matching distance ρ(S(x^∗), S(x)) ≤ ϵ; x^∗ is a sample drawn from F_{θ^∗}, θ^∗ is obtained from prior π, S is a summary statistic, ϵ > 0. ABC concerns include: the use of only one sample, x^∗, for each θ^∗; the choices of S, ρ and ϵ; π_ϵ(θ^∗), which is determined by arbitrary kernel, K(x, x^∗; ϵ), creating visual π_ϵ-artifacts. The concerns are accommodated with the introduced Fiducial(F)-ABC for all (θ^∗ drawn): M x^∗ are drawn from F_{θ^∗}; a universal S is used, the empirical measure indexed by sets which activate its sufficiency for exchangeable observations-vectors and have been neglected; a strong, probability distance ρ is used, inherently connected with S and Matching; light is thrown to ϵ’s nature and value; πϵ is obtained from the proportions of x∗ matching x. F-ABC for all posterior is closer to Bayesian philosophy, which does not use θ^∗-exclusions. Under few, mild assumptions, π_ϵ converges to the posterior, π(θ|x), when ϵ ↓ 0, and rates of concentration of T (F_{θ^∗}) to T (F_θ) are obtained when n ↑ ∞; T is a functional. Satisfactory F-ABC for all θ^∗ drawn posteriors are depicted for parametric and data generating models, including Tukey’s (a, b, g, h)-model, a 5-parameter normal mixture and a time series model. Various advantages of the F-ABC for all are presented over coarsened posteriors for observations in R^d, d ≥ 1.

In a Data-Generating Experiment (DGE), the data, X, is often obtained either from a Black-Box with inputs θ and Y, or from a Quantile function or a learning machine, f(Y, θ); θ is unknown, element of metric space (Θ, ρ), Y is random. If X has intractable or unknown c.d.f., Fθ, non-identifiability of θ cannot be confirmed and when present, among others, limits the predictive accuracy of the learned model, f(Y, \hat{θ}); \hat{θ} estimate of θ. In Machine Learning, non-identifiability of θ is ubiquitous and its extent is a criterion for selecting a learning machine. Empirical indices, EDI and PPVI, are introduced using P-values of Kolmogorov-Smirnov tests: i) to confirm almost surely, using generated data, the discrimination of θ from θ^∗, namely that the Kolmogorov distance, dK(Fθ, Fθ^∗), is positive, ii) to confirm identifiability of θ(∈ Θ) by repeating i) for θ^∗ in a sieve of Θ, since neighboring parameter values are in practice indistinguishable, and iii) most important, to compare EDI-graphs of DGEs, preferring more discrimination and less non-identifiability among parameters, and select one DGE to use. In applications, EDI-graphs confirm nonidentifiability in mixture models and in models parametrised with sums of parameters. EDI and PPVI explain why Tukey’s g-and-h model (DGE1) has better g-discrimination than the g-and-k model (DGE2), unless the sample size is extremely large; h_0 = k_0. EDIgraphs indicate that Normal learning machines have better parameter discrimination thanSigmoid learning machines and their parameters are non-identifiable.

We investigate meromorphic quasi-modular forms and their L-functions. We study the space of meromorphic quasi-modular forms and Rankin–Cohen brackets of meromorphic modular forms. Then we define their L-functions by using multiple techniques of regularized integral. Explicit formulas for the L-functions are given.