Likelihood methods are often difficult to use with large, irregularly sited spatial data sets, owing to the computational burden. Even for Gaussian models, exact calculations of the likelihood for "n" observations require "O"("n"-super-3) operations. Since any joint density can be written as a product of conditional densities based on some ordering of the observations, one way to lessen the computations is to condition on only some of the 'past' observations when computing the conditional densities. We show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation. Previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating. Through theoretical, numerical and practical examples, we show that there can often be considerable benefit in conditioning on some distant observations as well. Copyright 2004 Royal Statistical Society.

Log-linear models provide a statistically sound framework for Stochastic "Unification-Based" Grammars (SUBGs) and stochastic versions of other kinds of grammars. We describe two computationally-tractable ways of estimating the parameters of such grammars from a training corpus of syntactic analyses, and apply these to estimate a stochastic version of LexicalFunctional Grammar.

In the zebra finch forebrain nucleus robustus archistriatalis (RA), neurons burst during singing. We showed that the internal structure of spike bursts was regulated with a precision of circa 0.2 ms, and yielded alignment of acoustic features of song with a precision of circa 1 ms. In addition, interburst intervals and corresponding syllable durations displayed systematic variation within song (average elongation 0.3 ms/s song), and slower "drift" across songs. Systematic variation on even a coarser time scale might be difficult to detect in other systems, but could affect the analysis of temporal patterning. The close relationship between precise timing of individual spikes and stereotypic behavior suggests that song is represented in RA by a temporal code.

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The False Discovery Rate (FDR) paradigm aims to attain certain control on Type I errors with relatively high power for multiple hypothesis testing. The Benjamini--Hochberg (BH) procedure is a well-known FDR controlling procedure. Under a random effects model, we show that, in general, unlike the FDR, the positive FDR (pFDR) of the BH procedure cannot be controlled at an arbitrarily low level due to the limited evidence provided by the observations to separate false and true nulls. This results in a criticality phenomenon, which is characterized by a transition of the procedure's power from being positive to asymptotically 0 without any reduction in the pFDR, once the target FDR control level is below a positive critical value. To address the constraints on the power and pFDR control imposed by the criticality phenomenon, we propose a procedure which applies BH-type procedures at multiple locations in the domain of $p$-values. Both analysis and simulations show that the proposed procedure can attain substantially improved power and pFDR control.Comment: Published in at http://dx.doi.org/10.1214/009053607000000037 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multivariate statistics are often available as well as necessary in hypothesis tests. We study how to use such statistics to control not only false discovery rate (FDR) but also positive FDR (pFDR) with good power. We show that FDR can be controlled through nested regions of multivariate p-values of test statistics. If the distributions of the test statistics are known, then the regions can be constructed explicitly to achieve FDR control with maximum power among procedures satisfying certain conditions. On the other hand, our focus is where the distributions are only partially known. Under certain conditions, a type of nested regions are proposed and shown to attain (p)FDR control with asymptotically maximum power as the pFDR control level approaches its attainable limit. The procedure based on the nested regions is compared with those based on other nested regions that are easier to construct as well as those based on more straightforward combinations of the test statistics. by NSF DMS 0706048 and NIH MH 68028. The author is thankful to the comments of the reviewers. Y,iare the diagonal entries of Σ i . If the fraction of false nulls is 5%, then, by using t X,i alone, the minimum attainable pFDR is ≈ .289 and, by using t Y,i alone, the bound is even higher (≈ .447). The lower bounds are a consequence of Proposition 2.1. No procedure that only uses t X,i or t Y,i can get a pFDR lower than the bounds.One way to attain lower pFDR is to increase k, which may require significantly more resources. When resources are limited, a sensible solution is to exploit both t X,i and t Y,i or, equivalently, their marginal p-values. This then raises the question of pFDR control using multivariate p-values. FDR control using nested regions of p-values General descriptionLet {D t , 0 ≤ t ≤ 1} be a family of Borel sets in [0, 1] K such that D 1 = [0, 1] K , ℓ(D t ) = t, D s ⊂ D t , 0 ≤ s < t ≤ 1 {D t } is right-continuous, i.e., D t = ∩ s>t D s , t ∈ [0, 1).(3.1) Z. Chi/FDR with multivariate p-values 372

The functional organization giving rise to stimulus selectivity in higher-order auditory neurons remains under active study. We explored the selectivity for motifs, spectrotemporally distinct perceptual units in starling song, recording the responses of 96 caudomedial mesopallium (CMM) neurons in European starlings (Sturnus vulgaris) under awake-restrained and urethane-anesthetized conditions. A subset of neurons was highly selective between motifs. Selectivity was correlated with low spontaneous firing rates and high spike timing precision, and all but one of the selective neurons had similar spike waveforms. Neurons were further tested with stimuli in which the notes comprising the motifs were manipulated. Responses to most of the isolated notes were similar in amplitude, duration, and temporal pattern to the responses elicited by those notes in the context of the motif. For these neurons, we could accurately predict the responses to motifs from the sum of the responses to notes. Some notes were suppressed by the motif context, such that removing other notes from motifs unmasked additional excitation. Models of linear summation of note responses consistently outperformed spectrotemporal receptive field models in predicting responses to song stimuli. Tests with randomized sequences of notes confirmed the predictive power of these models. Whole notes gave better predictions than did note fragments. Thus in CMM, auditory objects (motifs) can be represented by a linear combination of excitation and suppression elicited by the note components of the object. We hypothesize that the receptive fields arise from selective convergence by inputs responding to specific spectrotemporal features of starling notes.