This paper gives an account of the history of both small- and large-sample optimality for both estimation and testing. It covers this history from the beginnings by Laplace and Gauss to the ongoing research on optimal multiple comparison procedures, and also includes the optimality of designs.

This paper discusses some subtle, and largely overlooked, differences between conceptual and mathematical optimization goals in statistics, and illustrates them by examples.

Proportional hazards regression model assumes that the covariates affect the hazard function through a link function and an index which is a linear function of the covariates. Traditional approaches, such as the Cox proportional hazards model, focus on estimating the unknown index by assuming a known link function between the log-hazard function and covariates. A linear link function is often employed for convenience without any validation. This paper provides an approach to estimate the link function, which can then be used to guide the choice of a proper parametric link function. This is accomplished through a two-step algorithm to estimate the link function and the effects of the covariates iteratively without involving the baseline hazard estimate. The link function is estimated by a smoothing method based on a local version of partial likelihood, and the index function is then estimated using a full version of partial likelihood. Asymptotic properties of the non-parametric link function estimate are derived, which facilitates model checking of the adequacy of the Cox Proportional hazards model. The approach is illustrated through a survival data and simulations.

We consider classes of models related to those introduced by Lehmann [Ann. Math. Statist. 24 (1953) 23–43] and Sklar [L’Institut de Statistique de L’Universite de Paris 8 (1959) 229–231]. Recently developed algorithms for finding profile NP likelihood procedures are discussed, extended and implemented for such models by combining them with the MM algorithm. In particular we consider statistical procedures for a regression model with proportional expected hazard rates, and for transformation models including the normal copula. A variety of likelihoods introduced to deal with semiparametric models are considered. They all generate rank results, not only tests, but also estimates, confidence regions, and optimality theory, thereby, to paraphrase Lehmann [Ann. Math. Statist. 24 (1953) 23–43], demonstrating “the power of ranks”.

The size of the bootstrap test of hypotheses is studied for the normal and exponential one and two-sample problems. It is found that the size depends not only on the problem, but on the choice of test statistic and the nominal level. In some special cases, the bootstrap test is UMP, but in other cases, it can be totally useless, such as being completely randomized or rejecting the null hypothesis with probability one. More importantly, the size is usually greater than the nominal level, even in the limit as the sample size goes to infinity.

A Lévy model combines a Brownian motion with drift and a pure-jump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of the process, is studied under a discrete-sampling scheme. In that case, the jumps are latent variables whose statistical properties can in principle be assessed when the frequency of observations increase to infinity. We propose nonparametric estimators for the Lévy density following Grenander’s method of sieves. The associated problem of selecting a suitable approximating sieve is subsequently investigated using regular piece-wise polynomials as sieves and assuming standard smoothness conditions on the Lévy density. By sampling the process at a high enough frequency relative to the time horizon T, we show that it is feasible to choose the dimension of the sieve so that the rate of convergence of the risk of estimation off the origin is the best possible from a minimax point of view, and even if the estimation were based on the whole sample path of the process. The sampling frequency necessary to attain the optimal minimax rate is explicitly identified. The proposed method is illustrated by simulation experiments in the case of variance Gamma processes.

Consider distribution functions F and G and suppose that F is more peaked about a than G is about b. The problem of estimating F or G, or both, when F and G are symmetric, arises quite naturally in applications. The empirical distribution functions F_n and G_m will not necessarily satisfy the order constraint imposed by the experimental conditions. Rojo and Batun-Cutz [Series in Biostatistics vol. 3, Advances in Statistical Modeling and Inference, (2007) 649–670] proposed some estimators that are strongly uniformly consistent when both m and n tend to infinity. However the estimators fail to be consistent when only either m or n tend to infinity. Here estimators are proposed that circumvent these problems and the asymptotic distribution of the estimators is delineated. A simulation study compares these estimators in terms of Mean Squared Error and Bias behavior with their competitors.

A functional generalized linear model is applied to spectroscopic data to discriminate disease from non-disease in the diagnosis of cervical pre-cancer. For each observation, multiple functional covariates are available, and it is of interest to select a few of them for efficient classification. In addition to multiple functional covariates, some non-functional covariates are also used to account for systematic differences caused by these covariates. Functional principal components are used to reduce the model to multivariate logistic regression and a grouped Lasso penalty is applied to the reduced model to select useful functional covariates among multiple curves.

Hemodynamic response function (HRF) has played an important role in many recent functional magnetic resonance imaging (fMRI) based brain studies where the main focus is to investigate the relationship between stimuli and the neural activity. Standard statistical analysis of fMRI data usually calls for a “canonical” model of HRF, but it is uncertain how well this fits the actual HRF. The objective of this paper is to exploit the experimental designs by modeling the stimulus sequences using stochastic point processes. The identification of the stimulus-response relationship will be conducted in the frequency domain, which will be facilitated by fast Fourier transforms (FFT). The usefulness of this approach will be illustrated using both simulated and real human brain data. Under regularity conditions, it is shown that the estimated HRF possesses an asymptotic normal distribution.

This paper reviews the connections between estimators that derive from three different modeling methodologies: Mixed-effects models, Bayesian models and Penalized Least-squares. Extension of classical results on the equivalence for smoothing spline estimators and best linear unbiased prediction and/or posterior analysis of certain Gaussian signal-plus-noise models is examined in a more general setting. These connections allow for the application of an efficient, linear time algorithm, to estimate parameters, compute random effects predictions and evaluate likelihoods in a large class of model scenarios. We also show that the methods of generalized cross-validation, restricted maximum likelihood and unbiased risk prediction can be used to estimate the variance components or adaptively select the smoothing parameters in any of the three settings.

We prove that two seemingly-different models of random walk in random environment are generically quite close to one another. One model comes from statistical physics, and describes the behavior of a randomly-charged random polymer. The other model comes from probability theory, and was originally designed to describe a large family of asymptotically self-similar processes that have stationary increments.

The problem of recovering a cumulative distribution function (cdf) and corresponding density function from its moments is studied. This problem is a special case of the classical moment problem. The results obtained within the moment problem can be applied in many indirect models, e.g., those based on convolutions, mixtures, multiplicative censoring, and right-censoring, where the moments of unobserved distribution of actual interest can be easily estimated from the transformed moments of the observed distributions. Nonparametric estimation of a quantile function via moments of a target distribution represents another very interesting area where the moment problem arises. In all such models one can apply the present results to recover a function via its moments. In this article some properties of the proposed constructions are derived. The uniform rates of convergence of the approximation of cdf, its density function, quantile and quantile density function are obtained as well.

We consider the compound decision problem of estimating a vector of n parameters, known up to a permutation, corresponding to n independent observations, and discuss the difference between two symmetric classes of estimators. The first and larger class is restricted to the set of all permutation invariant estimators. The second class is restricted further to simple symmetric procedures. That is, estimators such that each parameter is estimated by a function of the corresponding observation alone. We show that under mild conditions, the minimal total squared error risks over these two classes are asymptotically equivalent up to essentially O(1) difference.

For a general family of one-dimensional skew-symmetric probability densities, the application of the maximum likelihood method to the estimation of the asymmetry parameter λ is studied. Under mild conditions, the existence and consistency of a sequence {λ̂_n} of maximum likelihood estimators is established, and the limit distributions of {λ̂_n} and the sequence of likelihood ratios are determined under the null hypothesis H₀: λ=0. These latter conclusions, which hold under differential singularity of the likelihood function at λ=0, extend to the present framework results recently obtained for general statistical models with null Fisher information.

Estimation of the proportion or the number of true null hypotheses is an important problem in multiple testing, especially when the number of hypotheses is large. Wu, Guan and Zhao [Biometrics 62 (2006) 735–744] found that nonparametric approaches are too conservative. We study two parametric mixture models (normal and beta) for the distributions of the test statistics or their p-values to address this problem. The components of the mixture are the null and alternative distributions with mixing proportions π₀ and 1−π₀, respectively, where π₀ is the unknown proportion to be estimated. The normal model assumes that the test statistics from the true null hypotheses are i.i.d. N(0, 1) while those from the alternative hypotheses are i.i.d. N(δ, 1) with δ≠0. The beta model assumes that the p-values from the null hypotheses are i.i.d. U[0, 1] and those from the alternative hypotheses are i.i.d. Beta(a, b) with a<1<b. All parameters are assumed to be unknown. Three methods of estimation of π₀ are developed for each model. The methods are compared via simulation with each other and with Storey’s [J. Roy. Statist. Soc. Ser. B 64 (2002) 297–304] nonparametric method in terms of the bias and mean square error of the estimators of π₀ and the achieved FDR. Robustness of the estimators to the model violations is also studied by generating data from other models. For the normal model, the parametric methods perform better compared to Storey’s method with the EM method (Dempster, Laird and Rubin [Roy. Statist. Soc. Ser. B 39 (1977) 1–38]) performing best overall when the assumed model holds; however, it is not very robust to significant model violations. For the beta model, the parametric methods do not perform as well because of the difficulties of estimation of parameters, and Storey’s nonparametric method turns out to be the winner in many cases. Therefore the beta model is not recommended for use in practice. An example is given to illustrate the methods.

Applying a decision theoretic approach to multiple comparisons very similar to that described by Lehmann [Ann. Math. Statist. 21 (1950) 1–26; Ann. Math. Statist. 28 (1975a) 1–25; Ann. Math. Statist. 28 (1975b) 547–572], we introduce a loss function based on the concept of the false discovery rate (FDR). We derive a Bayes rule for this loss function and show that it is very closely related to a Bayesian version of the original multiple comparisons procedure proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] to control the sampling theory FDR. We provide the results of a Monte Carlo simulation that illustrates the very similar sampling behavior of our Bayes rule and Benjamini and Hochberg’s procedure when applied to making all pair-wise comparisons in a one-way fixed effects analysis of variance setup with 10 and with 20 means.

We introduce a new methodology for determining the difficulty of selecting the modeling objective function and estimating the parameters for an ad-hoc network data set. The method utilizes formulation of the underlying optimization problem instance that consists of an objective function and a set of constraints. The method is illustrated on real distance measurement data used for estimating the locations of wireless nodes that is the most studied and a representative problem for ad-hoc networks estimation. The properties of the data set that could affect the quality of optimization are categorized. In large optimization problems with multiple properties (characteristics) that contribute to the solution quality, it is practically impossible to analytically study the effect of each property. A number of metrics for evaluating the effectiveness of the optimization on each data set are proposed. Using the well known Plackett and Burmann fast simulation methodology, for each metric, the impact of the categorized properties of the data are determined for the specified optimization. A new approach for combining the impacts resulting from different properties on various metrics is described. We emphasize that the method is generic and has the potential to be more broadly applicable to other parameter estimation problems.

It is shown that the capitalization-weighted portfolio is mathematically required to coincide with the minimum-variance portfolio, provided both portfolios are defined with respect to the same (arbitrary) collection of equities having linearly independent returns. This result is a logical consequence of the law of iterated expectations and has important implications for equity return covariance structure.