JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARYWe study two natural classes of kernel density estimators for use with spherical data. Members of both classes have already been used in practice. The classes have an element in common, but for the most part they are disjoint. However, all members of the first class are asymptotically equivalent to one another, and to a single element of the second class. In this sense the second class 'contains' the first. It includes some estimators which out-perform all those in the first class, if loss is measured in either squared-error or Kullback-Leibler senses. Explicit formulae are given for bias, variance and loss, and large-sample properties of these quantities are described. Numerical illustrations are presented.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARYA bandwidth selection method is proposed for kernel density estimation. This is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth, but with two important modifications. The result is a bandwidth selector with the, by nonparametric standards, extremely fast asymptotic rate of convergence of n-i, where n -) o0 denotes sample size. Comparison is given to other bandwidth selection methods, and small sample impact is investigated.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Box-counting estimators are popular for estimating fractal dimension. However, very little is known of their stochastic properties, despite increasing statistical interest in their application. We show that, if the irregular curve to which the estimators are applied is modelled by a Gaussian process, concise formulae may be developed for asymptotic bias and variance of box-counting estimators. These formulae point to critical differences between a naive form of the box-counting estimator, based directly on the capacity definition of fractal dimension, and a regression-inspired version of that estimator.

The first members of a promising new family of hybrid amino acid-polyoxometalates have emerged from a search for modular functional molecules. Incorporation of glycine (Gly) or norleucine (Nle) ligands into an yttrium-tungstoarsenate structural backbone, followed by crystallization with p-methylbenzylammonium (p-MeBzNH3(+)) cations, affords (p-MeBzNH3)6K2(GlyH)[As(III)4(Y(III)W(VI)3)W(VI)44Y(III)4O159(Gly)8(H2O)14]⋅47 H2O (1) and enantiomorphs (p-MeBzNH3)15(NleH)3[As(III)4(Mo(V)2Mo(VI)2)W(VI)44Y(III)4O160(Nle)9(H2O)11][As(III)4(Mo(VI)2W(VI)2)W(VI)44Y(III)4O160(Nle)9(H2O)11] (generically designated 2: L-Nle, 2 a; D-Nle, 2 b). An intensive structural, spectroscopic, electrochemical, magnetochemical and theoretical investigation has allowed the elucidation of site-selective metal substitution and photoreduction of the tetranuclear core of the hybrid polyanions. In the solid state, markedly different crystal packing is evident for the compounds, which indicates the role of noncovalent interactions involving the amino acid ligands. In solution, mass spectrometric and small-angle X-ray scattering studies confirm maintenance of the structure of the polyanions of 2, while circular dichroism demonstrates that the chirality is also maintained. The combination of all of these features in a single modular family emphasizes the potential of such hybrid polyoxometalates to provide nanoscale molecular materials with tunable properties.

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SUMMARY To investigate the genetics of susceptibility to early onset pauciarticular juvenile chronic arthritis (JCA), 158 unrelated ethnic British patients with a mean disease onset of 3-2 years, together with controls, were tested for HLA-A, B, C, and DR antigens. Additionally, 117 patients were also investigated for complement Bf and C4 markers. New observations included an increased frequency of the C4B 2 allotype (p corrected(pc) <0.02) and C4A 4,B 2 phenotype (p

Self-exciting point processes (SEPPs), or Hawkes processes, have found applications in a wide range of fields, such as epidemiology, seismology, neuroscience, engineering, and more recently financial econometrics and social interactions. In the traditional SEPP models, the baseline intensity is assumed to be a constant. This has restricted the application of SEPPs to situations where there is clearly a self-exciting phenomenon, but a constant baseline intensity is inappropriate. In this paper, to model point processes with varying baseline intensity, we introduce SEPP models with time-varying background intensities (SEPPVB, for short). We show that SEPPVB models are competitive with autoregressive conditional SEPP models (Engle and Russell 1998) for modeling ultra-high frequency data. We also develop asymptotic theory for maximum likelihood estimation based inference of parametric SEPP models, including SEPPVB. We illustrate applications to ultra-high frequency financial data analysis, and we compare performance with the autoregressive conditional duration models.