Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss

Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hat p$, its quality is measured by the risk associated with normalized loss functions of linear-linear or inverse-linear form. These functions are possibly asymmetric, with arbitrary slope parameters $a$ and $b$ for $\hat pp$ respectively. Interest in these functions is motivated by their significance and potential uses, which are briefly discussed. Estimators are given for which the risk has an asymptotic value as $p$ tends to $0$, and which guarantee that, for any $p$ in $(0,1)$, the risk is lower than its asymptotic value. This allows selecting the required number of successes, $r$, to meet a prescribed quality irrespective of the unknown $p$. In addition, the proposed estimators are shown to be approximately minimax when $a/b$ does not deviate too much from $1$, and asymptotically minimax as $r$ tends to infinity when $a=b$.Comment: 4 figure

Future ocean acidification will be amplified by hypoxia in coastal habitats

Ocean acidification is elicited by anthropogenic carbon dioxide emissions and resulting oceanic uptake of excess CO2 and might constitute an abiotic stressor powerful enough to alter marine ecosystem structures. For surface waters in gas-exchange equilibrium with the atmosphere, models suggest increases in CO2 partial pressure (pCO2) from current values of ca. 390 μatm to ca. 700–1,000 μatm by the end of the century. However, in typically unequilibrated coastal hypoxic regions, much higher pCO2 values can be expected, as heterotrophic degradation of organic material is necessarily related to the production of CO2 (i.e., dissolved inorganic carbon). Here, we provide data and estimates that, even under current conditions, maximum pCO2 values of 1,700–3,200 μatm can easily be reached when all oxygen is consumed at salinities between 35 and 20, respectively. Due to the nonlinear nature of the carbonate system, the approximate doubling of seawater pCO2 in surface waters due to ocean acidification will most strongly affect coastal hypoxic zones as pCO2 during hypoxia will increase proportionally: we calculate maximum pCO2 values of ca. 4,500 μatm at a salinity of 20 (T = 10 °C) and ca. 3,400 μatm at a salinity of 35 (T = 10 °C) when all oxygen is consumed. Upwelling processes can bring these CO2-enriched waters in contact with shallow water ecosystems and may then affect species performance there as well. We conclude that (1) combined stressor experiments (pCO2 and pO2) are largely missing at the moment and that (2) coastal ocean acidification experimental designs need to be closely adjusted to carbonate system variability within the specific habitat. In general, the worldwide spread of coastal hypoxic zones also simultaneously is a spread of CO2-enriched zones. The magnitude of expected changes in pCO2 in these regions indicates that coastal systems may be more endangered by future global climate change than previously thought