Estimation problems associated with the Weibull distribution

Series in descending powers of the sample size are developed for the moments of the coefficient of variation v* for the Weibull distribution F(t) = 1 -exp(-(t/b)/sup c/). A similar series for the moments of the estimator c* of the shape parameter c are derived from these. Comparisons are made with basic asymptotic assessments for the means and variances. From the first four moments, approximations are given to the distribution of v* and c*. In addition, an almost unbiased estimator of c is given when a sample is provided with the value of v*. Comments are given on the validity of the asymptotically normal assessments of the distributions

Plausibility functions and exact frequentist inference

In the frequentist program, inferential methods with exact control on error rates are a primary focus. The standard approach, however, is to rely on asymptotic approximations, which may not be suitable. This paper presents a general framework for the construction of exact frequentist procedures based on plausibility functions. It is shown that the plausibility function-based tests and confidence regions have the desired frequentist properties in finite samples---no large-sample justification needed. An extension of the proposed method is also given for problems involving nuisance parameters. Examples demonstrate that the plausibility function-based method is both exact and efficient in a wide variety of problems.Comment: 21 pages, 5 figures, 3 table

TRY plant trait database – enhanced coverage and open access

Plant traits—the morphological, anatomical, physiological, biochemical and phenological characteristics of plants—determine how plants respond to environmental factors, affect other trophic levels, and influence ecosystem properties and their benefits and detriments to people. Plant trait data thus represent the basis for a vast area of research spanning from evolutionary biology, community and functional ecology, to biodiversity conservation, ecosystem and landscape management, restoration, biogeography and earth system modelling. Since its foundation in 2007, the TRY database of plant traits has grown continuously. It now provides unprecedented data coverage under an open access data policy and is the main plant trait database used by the research community worldwide. Increasingly, the TRY database also supports new frontiers of trait‐based plant research, including the identification of data gaps and the subsequent mobilization or measurement of new data. To support this development, in this article we evaluate the extent of the trait data compiled in TRY and analyse emerging patterns of data coverage and representativeness. Best species coverage is achieved for categorical traits—almost complete coverage for ‘plant growth form’. However, most traits relevant for ecology and vegetation modelling are characterized by continuous intraspecific variation and trait–environmental relationships. These traits have to be measured on individual plants in their respective environment. Despite unprecedented data coverage, we observe a humbling lack of completeness and representativeness of these continuous traits in many aspects. We, therefore, conclude that reducing data gaps and biases in the TRY database remains a key challenge and requires a coordinated approach to data mobilization and trait measurements. This can only be achieved in collaboration with other initiatives

Global Retinoblastoma Presentation and Analysis by National Income Level

Importance: Early diagnosis of retinoblastoma, the most common intraocular cancer, can save both a child's life and vision. However, anecdotal evidence suggests that many children across the world are diagnosed late. To our knowledge, the clinical presentation of retinoblastoma has never been assessed on a global scale. Objectives: To report the retinoblastoma stage at diagnosis in patients across the world during a single year, to investigate associations between clinical variables and national income level, and to investigate risk factors for advanced disease at diagnosis. Design, Setting, and Participants: A total of 278 retinoblastoma treatment centers were recruited from June 2017 through December 2018 to participate in a cross-sectional analysis of treatment-naive patients with retinoblastoma who were diagnosed in 2017. Main Outcomes and Measures: Age at presentation, proportion of familial history of retinoblastoma, and tumor stage and metastasis. Results: The cohort included 4351 new patients from 153 countries; the median age at diagnosis was 30.5 (interquartile range, 18.3-45.9) months, and 1976 patients (45.4) were female. Most patients (n = 3685 84.7%) were from low-and middle-income countries (LMICs). Globally, the most common indication for referral was leukocoria (n = 2638 62.8%), followed by strabismus (n = 429 10.2%) and proptosis (n = 309 7.4%). Patients from high-income countries (HICs) were diagnosed at a median age of 14.1 months, with 656 of 666 (98.5%) patients having intraocular retinoblastoma and 2 (0.3%) having metastasis. Patients from low-income countries were diagnosed at a median age of 30.5 months, with 256 of 521 (49.1%) having extraocular retinoblastoma and 94 of 498 (18.9%) having metastasis. Lower national income level was associated with older presentation age, higher proportion of locally advanced disease and distant metastasis, and smaller proportion of familial history of retinoblastoma. Advanced disease at diagnosis was more common in LMICs even after adjusting for age (odds ratio for low-income countries vs upper-middle-income countries and HICs, 17.92 95% CI, 12.94-24.80, and for lower-middle-income countries vs upper-middle-income countries and HICs, 5.74 95% CI, 4.30-7.68). Conclusions and Relevance: This study is estimated to have included more than half of all new retinoblastoma cases worldwide in 2017. Children from LMICs, where the main global retinoblastoma burden lies, presented at an older age with more advanced disease and demonstrated a smaller proportion of familial history of retinoblastoma, likely because many do not reach a childbearing age. Given that retinoblastoma is curable, these data are concerning and mandate intervention at national and international levels. Further studies are needed to investigate factors, other than age at presentation, that may be associated with advanced disease in LMICs. © 2020 American Medical Association. All rights reserved

Continued fractions and the polygamma functions

AbstractIn a previous study we have shown that the polygamma functions (derivatives of the logarithm of the gamma function) relate to Stieltjes transforms in the square of the argument. These transforms in turn may be converted to Stieltjes continued fractions; in the background is a determined Stieltjes moment problem.In the present study we use the Hamburger form of the Stieltjes integral to produce a set of real monotonic increasing and monotonic decreasing approximants to each of the real and imaginary parts of a polygamma function when the argument is complex. The approximants involve rational fractions which appear to be new.Special attention is given to ln Γ(z) and the psi function

Asymptotic series and Stieltjes continued fractions for a gamma function ratio

AbstractAn analysis is given for the expansion (60 terms) of a gamma function ratio discussed by Stieltjes and others. A Stieltjes continued fraction is derived, affording lower and upper bound (but lacking a rigorous proof), along with continued fraction for the odd and even series

Replenishment-Depletion Urn in Equilibrium

Bernard's urn, beta integral transforms, finite difference calculus, generating functions, hypergeometric distributions, hypergeometric functions, moments, replenishment-depletion urn,

Moment series for moment estimators of the parameters of a Weibull density

Taylor series for the first four moments of the coefficients of variation in sampling from a 2-parameter Weibull density are given: they are taken as far as the coefficient of n/sup -24/. From these a four moment approximating distribution is set up using summatory techniques on the series. The shape parameter is treated in a similar way, but here the moment equations are no longer explicit estimators, and terms only as far as those in n/sup -12/ are given. The validity of assessed moments and percentiles of the approximating distributions is studied. Consideration is also given to properties of the moment estimator for 1/c