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Background: Environmental conditions early in life may imprint the circadian system and influence response to environmental signals later in life. We previously determined that a large springtime increase in solar insolation at the onset location was associated with a younger age of onset of bipolar disorder, especially with a family history of mood disorders. This study investigated whether the hours of daylight at the birth location affected this association. Methods: Data collected previously at 36 collection sites from 23 countries were available for 3896 patients with bipolar I disorder, born between latitudes of 1.4 N and 70.7 N, and 1.2 S and 413 S. Hours of daylight variables for the birth location were added to a base model to assess. the relation between the age of onset and solar insolation. Results: More hours of daylight at the birth location during early life was associated with an older age of onset, suggesting reduced vulnerability to the future circadian challenge of the springtime increase in solar insolation at the onset location. Addition of the minimum of the average monthly hours of daylight during the first 3 months of life improved the base model, with a significant positive relationship to age of onset. Coefficients for all other variables remained stable, significant and consistent with the base model. Conclusions: Light exposure during early life may have important consequences for those who are susceptible to bipolar disorder, especially at latitudes with little natural light in winter. This study indirectly supports the concept that early life exposure to light may affect the long term adaptability to respond to a circadian challenge later in life. (C) 2015 Elsevier Ltd. All rights reserved.

Using Maxwell-Boltzmann and Bose-Einstein models, we derive formulae for the exact probability that an adverse event (AE) report series will present with additional (ie, second, third, etc.) reports from a single geographical reporting region, under the assumption that each region has an equal probability of report generation. These formulae are used to calculate critical values for the number of regions beyond which the probability of additional same-region reporting falls below a predefined level. The results using either model suggest that for widely distributed drugs, the presence of even a single additional same-region AE report is extremely unlikely and provides reasonable evidence for differential reporting rates between regions with multiple versus single reports. Multiple same-region reports, therefore, provide evidence against the validity of report-based tests for a generalized increase in frequency, and evidence for the presence of a localized increase in frequency. These calculations imply th...

Background: The onset of bipolar disorder is influenced by the interaction of genetic and environmental factors. We previously found that a large increase in sunlight in springtime was associated with a lower age of onset. This study extends this analysis with more collection sites at diverse locations, and includes family history and polarity of first episode. Methods: Data from 4037 patients with bipolar I disorder were collected at 36 collection sites in 23 countries at latitudes spanning 3.2 north (N) to 63.4 N and 38.2 south (S) of the equator. The age of onset of the first episode, onset location, family history of mood disorders, and polarity of first episode were obtained retrospectively, from patient records and/or direct interview. Solar insolation data were obtained for the onset locations. Results: There was a large, significant inverse relationship between maximum monthly increase in solar insolation and age of onset, controlling for the country median age and the birth cohort. The effect was reduced by half if there was no family history. The maximum monthly increase in solar insolation occurred in springtime. The effect was one-third smaller for initial episodes of mania than depression. The largest maximum monthly increase in solar insolation occurred in northern latitudes such as Oslo, Norway, and warm and dry areas such as Los Angeles, California. Limitations: Recall bias for onset and family history data. Conclusions: A large springtime increase in sunlight may have an important influence on the onset of bipolar disorder, especially in those with a family history of mood disorders. (C) 2014 Elsevier B.V. All rights reserved.

Exact simultaneous one-sided confidence intervals for contrasts in m normal means are discussed. The set K of contrast vectors considered is of one of two forms: Either it is a cone whose coordinates are monotone for a partial ordering defined on the coordinate index set {1, …, m} or it is the polar of such a cone. It is shown that the intervals obtained are inverted from test statistics that may be used for testing the null hypothesis that the mean vector μ lies in the dual (or negative polar) cone of K, against the alternative that μ is not in the dual of K. Corresponding conservative two-sided intervals are also discussed. The structure of such cones is considered and results concerning dual norms for such cones are obtained. Illustrations for simple ordering, simple-tree, and umbrella orderings are discussed.

Two statistics are proposed for the simple goodness-of-fit problem. These are derived from a general principle for combining dependent test statistics that has been discussed elsewhere by the authors. It is shown that these statistics are relatively optimal in the sense of Bahadur efficiency and consequently, are more efficient than any weighted Kolmogorov statistic at every alternative. A curious pathology occurs: Under certain alternatives, the sequence of statistics has a Bahadur efficacy or exact slope only in the weak sense of convergence in law.

In this note we observe that for independent symmetric random variables X and Y, when the pdf of X is PF∞, the conditional distributions of ¦Y¦ given S = X + Y form a MLR family. We then show that for a function ϕ: R n→R that is symmetric in each coordinate and increasing on (0, ∞)n, E(ϕ(S1,...,Sn)¦Sn = s) is even and increasing in ¦s¦. Here S1,...,Sn are partial sums with independent symmetric PF∞ summands. Application is made to sequential tests that minimize the maximum expected sample size when the model is a one-parameter exponential family generated by a symmetric PF ∞ density.

For independent random variablesXandY,if the conditional distribution ofXgivenX+Ysatisfies certain conditions, then the joint distribution ofXandYis a member of a certain one-parameter exponential family. Extensions fornindependent random variables are given. A characterization for independent random variables involving order statistics is also given.

Asymptotic inequalities are obtained for the level attained by any test statistic in a given hypothesis testing problem. The bounds so obtained are themselves random. The inequalities subsume previous results by Bahadur and others about the asymptotic behavior of attained levels. Various remifications of the inequalities are discussed, including a notion of stochastic deficiency.

Motivated by Loynes' (1969) treatment of (weak) consistency of sequential estimators, we establish here some allied results on strong consistency. The strengthened conclusion is achieved by imposing further restrictions, so that our results are not as broadly applicable as Loynes'. The reader is referred to Loynes' paper for additional motivation and discussion. As in that paper, we are concerned with estimators that improve a given one by taking conditional expectations with respect to a sufficient statistic, in the case where the sample size may itself be a random variable. $x_1, x_2, \cdots$ will denote the data sequence, random variables defined on a measurable space $(\Omega, \mathscr{a})$. All probability measures considered on $(\Omega, \mathscr{a})$ will render the sequence iid. It is seen below (Theorem 3.4) that a given sequence $\{t_i\}$ of stopping times for the data sequence leads to a strongly consistent sequence of estimators if $\lim t_i = + \infty$ a.s., and the $t_i$ are $C$-ordered (Definition 2.3). This entails that the $t_i$ increase monotonically to $+ \infty$ a.s., but requires additional structure as well. The specific considerations are in Section 3; Section 2 presents some general notions.

Conditions are given under which a sequence of posterior distributions converges weakly to a degenerate distribution. It is not assumed that the model being used actually governs the data. Exponential models are studied in particular and are shown to be well behaved under mild assumptions. Large-deviation results for $U$-statistics and posterior odds are also given.

Conditions are given for the strong consistency and asymptotic normality of the MLE (maximum likelihood estimator) for multiparameter exponential models. Because of the special structure assumed, the conditions are less restrictive than required by general theorems in this area. The technique involves certain convex functions on Euclidean spaces that arise naturally in the present context. Some examples are considered; among them, the multinomial distribution. Some convexity and continuity properties of multivariate cumulant generating functions are also discussed.

In a standard linear model, we explore the optimality of the least squares estimator under assuptions stronger than those for the Gauss-Markov theorem. The conclusion is then much stronger than that of the Gauss-Markov theorem. Specifically, two results are cited below: Under the assumption that the unobserved error e has a spherically symmetric distribution, the least squares estimator for the regression coefficient β is shown to maximize the probability that β − β stays in any symmetric convex set among linear unbiased estimators β. With the additional assumption that e is unimodal, the conclusion holds among equivariant estimators. The import of these results for risk functions is also discussed.

A criterion for the asymptotic optimality of combined independent tests is given. Methods other than Fisher's are shown to satisfy the criterion. Certain problems call for a weighted combination of tests. It is shown that many weighted procedures are optimal. In particular, a weighted procedure based on gamma transforms is optimal. This weighted procedure has the additional advantage that critical values are easily determined. Good's weighting procedure is shown not to be asymptotically optimal.

In [1], P. J. Bickel and the author presented a theorem giving sufficient conditions for the equivalence of invariant and almost-invariant functions. In the subsequent discussion, it was intimated without proof that the results extend to certain other classes of distributions that do not satisfy the hypotheses of the main theorem. A small lacuna was glossed over in the discussion; however, the conclusions are correct for the examples given there. In this note we justify those conclusions. We adopt the notation of [1] without further comment.

Sequential tests for one-sided hypotheses are compared, asymptotically, with non-sequential counterparts. An analog of Pitman efficiency is obtained, as is another comparison that has no purely non-sequential analog. With these methods of comparison, the limiting relative efficiency of the sequential test is never less than one and for most parameter values, it is infinite. An asymptotic notion of minimal relative efficiency is also considered.

Consider the problem of sequentially testing a null hypothesis vs an alternative hypothesis when the risk function is a linear combination of probability of error in the terminal decision and expected sample size (i.e., constant cost per observation.) Assume that the parameter space is the union of null and alternative, the parameter space is convex, the intersection of null and alternative is empty, and the common boundary of the closures of null and alternative is nonempty and compact. Assume further that observations are drawn from a $p$-dimensional exponential family with an open $p$-dimensional parameter space. Sufficient conditions for Bayes tests to have bounded stopping times are given.

Let $\mathbf{x} = (x_1,\cdots, x_n)'$ be a random vector in $R^n$. Two characterizations of normality are given. One involves the existence of two linear combinations of the $\{x_j\}$ that are independent in every coordinate system. The other, which is actually a consequence of the first, assumes that $\mathbf{x}$ obeys a linear model with spherical errors and involves sufficiency of the least-squares estimator.

Consider the problem of sequentially testing composite, contiguous hypotheses where the risk function is a linear combination of the probability of error in the terminal decision and the expected sample size. Assume that the common boundary of the closures of the null and the alternative hypothesis is compact. Observations are independent and identically distributed. We study properties of Bayes tests. One property is the exponential boundedness of the stopping time. Another property is continuity of the risk functions. The continuity property is used to establish complete class theorems as opposed to the essentially complete class theorems in Brown, Cohen and Strawderman.

Sequential tests that are LMP for certain one-sided testing problems are discussed. In all cases considered, the stopping rule is the first time a certain random walk leaves a bounded interval. (Thus various inequalities and approximations due to Wald can be utilized in obtaining properties of these tests.) For models in a one-parameter exponential family, each LMP sequential test is shown to be a Wald SPRT for a family of paired (conjugate) simple hypotheses.

Some concepts of relative and absolute efficiency for sequential tests are considered. These are sequential analogs of Hodges-Lehmann and Chernoff efficiencies. Using these criteria, several sequential tests for the mean of a normal distribution are evaluated. Among them are Wald's SPRT, Anderson's triangular boundary, Bayes and APO tests and a repeated significance test. Truncated versions of these tests are also considered. Asymptotic expressions for the (expected) stopping times and error rates are given.

The notion of Bahadur efficiency for test statistics is extended to the sequential case and illustrated in the specific context of testing one-sided hypotheses about a normal mean. An analog of Bahadur's theorem on the asymptotic optimality of the likelihood ratio statistic is seen to hold in the normal case. Some possible definitions of attained level for a sequential experiment are considered.

Let $\mathbf{X}_1,\mathbf{X}_2,\cdots$ be a stochastic sequence and $\mathscr{P}$ and $\mathscr{L}$, two composite parametric hypotheses (models) under which the $\mathbf{X}_i$ are i.i.d. We consider SPRTs of $\mathscr{P}$ vs $\mathscr{L}$ that depend on a sequence of exchangeable densities. Included are SPRTs obtained by the method of weight-functions (Bayesian procedures) and many SPRTs obtained by invariance reduction. Conditions are established under which the stopping time of such a procedure is almost surely finite and has a nontrivial mgf. The ideas are illustrated using the sequential $t$-test.

The asymptotic behavior is given for the error rates and ASN of the Wald SPRT and of invariant sequential tests. An asymptotic justification of Bhate's conjecture is also provided for invariant sequential tests. Expressions are obtained for the asymptotic relative efficiency of the Wald SPRT as compared with the corresponding best non-sequential test.

A special group structure common to many statistical problems is presented. In the context of equivariant estimation, this leads to a characterization of equivariant estimators (Section 3). Under further conditions, measurable equivariant estimators are characterized (Section 4). Some examples are presented in Section 5 and Section 6 contains a brief discussion of related work in which the special group structure is useful.

Under certain conditions, it is shown that the invariant and almost-invariant $\sigma$-fields are equivalent if and only if the invariant $\sigma$-field is independent of an appropriate sufficient $\sigma$-field. This result helps unify work of Hall, Wijsman and Ghosh and of Pfanzagl, who dealt with the forward implication and work of Berk and Bickel, who treated the reverse implication. The conditions required are that the sufficient and invariant $\sigma$-fields be essentially disjoint and together generate the $\sigma$-field of the original data.

The large sample behavior of posterior distributions is examined without the assumption that the model is correct. Under certain conditions it is shown that asymptotically, the posterior distribution for a parameter $\theta$ is confined to a set (called the asymptotic carrier) which may, in general, contain more than one point. The asymptotic carrier depends on the model, the carrier of the prior distribution and the actual distribution of the observations. An example shows that, in general, there need be no convergence (in any sense) of the posterior distribution to a limiting distribution over the asymptotic carrier. This is in contrast to the (known) asymptotic behavior when the model is correct; see e.g. [7], p. 304: the asymptotic carrier then contains only one point, the "true value" of $\theta$ and the posterior distribution converges in distribution to the distribution degenerate at the "true value."

0. Summary. In this note we discuss the notions of biorthogonal and dual configurations and their relevance in certain statistical applications. The first application is to the distribution of a random matrix related to a multi-variatenormal sample matrix. As with the latter, the distribution is preserved by (certain) linear transformations. One consequence of this is the familiar result that if Q is a non-singular Wishart matrix, then for any non-zero vector a, 1/a'Q-'a is a multiple of a chi-square variable. Application is also made to the Gauss-Markov theorem and to certain estimates of mixing proportions due to Robbins. 1. Biorthogonal and dual configurations. Let S be a vector space with an inner-product, denoted by (. , .). The configurations (= ordered subsets) (xi, * - *, xp) and (xi*, ... *, x*) are said to be biorthogonal if (xi, xj*) = aij, the Kroneker delta. Clearly this relation is symmetric. Necessarily, the elements of {xi, * **, xp.} (respectively, {xi*, *. *, xp}) are linearly independent. For if (e.g.) xi E 'U {x2, * * *, xp,}, the subspace spanned by {x2, * * *, x}, then (xi, xi*) = 0, i = 2, ... , p, implies that (xi, xi*) = 0. In general, there are many configurations biorthogonal with a given configuration (xi, **.*, x One such is distinguished: There is a unique (y,, * **, y,) C 'U {xl, *. , xP} which is biorthogonal with (xi, * * *, xp). It is called the configuration dual to (xi, * - *, xp) and is constructed as follows: Let xi, be the projection of xi into 'L{xi, * , Xj_1, xi+1, ... , xp}. (q'L is the orthogonal complement of cl C SC in S.) (xi, xi.) = (xi., xi.) 0 0 by linear independence; of course (xi, xj.) = 0 if i 0 j. Then yi = xi.l(xi., xi.) gives the configuration dual to (xi, * * *, xp). It readily follows that any configuration (xi*, ..., xp*) biorthogonal with (xi, * , xp,) has the representation xi* = yi + ai, where t5 ? 'UL{Xi, ... Xp}. Since yi'U E x, **, xp}, we may write (y,, ***, yp) = (xi, xp)A,I where A is a non-singular p X p matrix. Letting Q denote the non-singular configuration matrix of (xi, *.* , xp) :qij = (xi, xj), the duality relations require that QA = I. Hence A = Q-1, (y,, * * , y,,) = (xi, * ,xp) Q' and the dual configuration has configuration matrix Q-1. Thus the dual configuration relation is also symmetric. If xl, * * *, xp are elements of RP and if X = ( (xl, * * *, xp) is the p X p matrix they generate by their representation as p-tuples, then Y' =

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