Non-negative demand in newsvendor models:The case of singly truncated normal samples

This paper considers the classical newsvendor model when demand is normally distributed but with a large coefficient of variation. This leads to observe with a non-negligible probability negative values that do not make sense. To avoid the occurrence of such negative values, first, we derive generalized forms for the optimal order quantity and the maximum expected profit using properties of singly truncated normal distributions. Since truncating at zero produces non-symmetric distributions for the positive values, three alternative models are used to develop confidence intervals for the true optimal order quantity and the true maximum expected profit under truncation. The first model assumes traditional normality without truncation, while the other two models assume that demand follows (a) the log-normal distribution and (b) the exponential distribution. The validity of confidence intervals is tested through Monte-Carlo simulations, for low and high profit products under different sample sizes and alternative values for coefficient of variation. For each case, three statistical measures are computed: the coverage, namely the estimated actual confidence level, the relative average half length, and the relative standard deviation of half lengths. Only for very few cases the normal and the log-normal model produce confidence intervals with acceptable coverage but these intervals are characterized by low precision and stability.Inventory Management; Newsvendor model; Truncated normal; Demand estimation; Confidence intervals; Monte-Carlo simulations

Confidence Intervals for the Coefficient of Quartile Variation of a Zero-inflated Lognormal Distribution

There are many types of skewed distribution, one of which is the lognormal distribution that is positively skewed and may contain true zero values. The coefficient of quartile variation is a statistical tool used to measure the dispersion of skewed and kurtosis data. The purpose of this study is to establish confidence and credible intervals for the coefficient of quartile variation of a zero-inflated lognormal distribution. The proposed approaches are based on the concepts of the fiducial generalized confidence interval, and the Bayesian method. Coverage probabilities and expected lengths were used to evaluate the performance of the proposed approaches via Monte Carlo simulation. The results of the simulation studies show that the fiducial generalized confidence interval and the Bayesian based on uniform and normal inverse Chi-squared priors were appropriate in terms of the coverage probability and expected length, while the Bayesian approach based on Jeffreys' rule prior can be used as alternatives. In addition, real data based on the red cod density from a trawl survey in New Zealand is used to illustrate the performances of the proposed approaches. Doi: 10.28991/esj-2021-01289 Full Text: PD

Confidence Interval Estimation for Continuous Outcomes in Cluster Randomization Trials

Cluster randomization trials are experiments where intact social units (e.g. hospitals, schools, communities, and families) are randomized to the arms of the trial rather than individuals. The popularity of this design among health researchers is partially due to reduced contamination of treatment effects and convenience. However, the advantages of cluster randomization trials come with a price. Due to the dependence of individuals within a cluster, cluster randomization trials suffer reduced statistical efficiency and often require a complex analysis of study outcomes. The primary purpose of this thesis is to propose new confidence intervals for effect measures commonly of interest for continuous outcomes arising from cluster randomization trials. Specifically, we construct new confidence intervals for the difference between two normal means, the difference between two lognormal means, and the exceedance probability. The proposed confidence intervals, which use the method of variance estimates recovery (MOVER), do not make certain assumptions that existing procedures make on the data. For instance, symmetry is not forced when the sampling distribution of the parameter estimate is skewed and the assumption of homoscedasticity is not made. Furthermore, the MOVER results in simple confidence interval procedures rather than complex simulation-based methods which currently exist. Simulation studies are used to investigate the small sample properties of the MOVER as compared with existing procedures. Unbalanced cluster sizes are simulated, with an average range of 50 to 200 individuals per cluster and 6 to 24 clusters per arm. The effects of various degrees of dependence between individuals within the same cluster are also investigated. When comparing the empirical coverage, tail errors, and median widths of confidence interval procedures, the MOVER has coverage close to the nominal, relatively balanced tail errors, and narrow widths as compared to existing procedure for the majority of the parameter combinations investigated. Existing data from cluster randomization trials are then used to illustrate each of the methods

Estimation of greenhouse gas emissions from a tracer gas study

This thesis models the emission of three greenhouse gases that exist in nature, CH4, CO2, and N2O, from a hoop structure, using the artificial introduction of a tracer gas, SF6, which exists in nature at levels below detection limit. Hoop structures are facilities used to house pigs before their slaughter. Many other studies of hoop structure emissions measure only one sample of the tracer gas and greenhouse gas at a time. However, the data sets in our study consist of 25 to 45 samples of each gas, taken on fixed grids of points. We will construct models to account for the relation between the greenhouse gases and SF6 as well as spatial relations in the data set. We will use these models along with the known emission rate of SF6 to estimate the relative rate of emission of the greenhouse gases;We fit a Bayesian hierarchical model to the data sets. In this model, we relate the pointwise concentrations of one greenhouse gas and SF6 and then analyze the posterior distribution of a parameter representing the relative rates of emission of the greenhouse gas and SF6. We assume lognormal measurement errors of the greenhouse gas and SF6 around the true concentration of each gas;We also fit geostatistical models to estimate the rates of emission of these gases. We consider block kriging, block co-kriging, and lognormal block kriging to estimate the concentration of each gas. An advantage of geostatistical models over the Bayesian hierarchical model is that we do not assume strict proportionality of the concentrations of the gases. These estimates can be related to the relative rate of emission of the gases. Due to the small size of these data sets, we take into consideration the uncertainty of the variogram parameters and how this uncertainty affects block kriging averages and variances;We use simulations from both geostatistical models and Bayesian hierarchical models to determine superiority of one set of models in terms of coverage probabilities, bias, and length of coverage sets or confidence intervals. We also address the concern of spatial design for the geostatistical models

Bayesian evaluation of groundwater age distribution using radioactive tracers and anthropogenic chemicals

pre-printThe development of a Bayesian modeling approach for estimation of the age distribution of groundwater using radioactive isotopes and anthropogenic chemicals is described. The model considers the uncertainties associated with the measured tracer concentrations as well as the parameters affecting the concentration of tracers in the groundwater, and it provides the posterior probability densities of the parameters defining the groundwater age distribution using a Markov chain Monte Carlo method. The model also incorporates the effect of dissolution of aquifer minerals on diluting the 14C signature and the uncertainties associated with this process on the inferred age distribution parameters. Two demonstration modeling cases have been performed. First, the method was applied to simulated tracer concentrations at a discharge point of a hypothetical 2-D vertical aquifer with two recharge zones, leading to a mixed groundwater age distribution under different presumed uncertainties. When the error variance of the observed tracer concentrations is considered unknown, the method can estimate the parameters of the fitted exponential-lognormal distribution with a relatively narrow credible interval when five hypothetical samples are assumed to be collected at the discharge point. However, when a single sample is assumed, the credible intervals become wider, and credible estimations of the parameters are not obtained. Second, the method was applied to the data collected at La Selva Biological Station in Costa Rica. In this demonstration application, nine different forms of presumed groundwater age distributions have been considered, including four single forms and five mixed forms, assuming the groundwater consists of distinct young and old fractions. For the medium geometrical standard deviation dc,i = 1.41, the model estimates a young groundwater age of between 0 and 350 years, with the largest odds being given to a mean age of approximately 100 years, and a fraction of young groundwater of between 15% to roughly 60%, with the largest odds for 30%. However, the method cannot definitively rule out larger fractions of young groundwater. The model provides a much more uncertain estimation of the age of old groundwater, with a credible interval of between 20,000 to 200,000 years

The gamma distribution as an alternative to the lognormal distribution in environmental applications

In environmental applications dealing with data from contaminated sites the positively skewed lognormal distribution has been the most commonly used model. The upper confidence limit (UCL) of the arithmetic mean of a lognormal population is computed by using the H-statistics. Recent concerns have arisen to the effectiveness of the H-Statistic based UCL for the mean of the lognormal distribution in instances of moderately to highly skewed data sets. In this paper the positively skewed Gamma distribution is considered as an alternative to the lognormal distribution and is shown to produce more reasonable UCL\u27s for the mean

Random Sampling of Skewed Distributions Implies Taylor’s Power Law of Fluctuation Scaling

Taylor’s law (TL), a widely verified quantitative pattern in ecology and other sciences, describes the variance in a species’ population density (or other nonnegative quantity) as a power-law function of the mean density (or other nonnegative quantity): Approximately, variance = a(mean)b, a \u3e 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analytically that observations randomly sampled in blocks from any skewed frequency distribution with four finite moments give rise to TL. We do not claim this is the only way TL arises. We give approximate formulae for the TL parameters and their uncertainty. In computer simulations and an empirical example using basal area densities of red oak trees from Black Rock Forest, our formulae agree with the estimates obtained by least-squares regression. Our results show that the correlated sampling variation of the mean and variance of skewed distributions is statistically sufficient to explain TL under random sampling, without the intervention of any biological or behavioral mechanisms. This finding connects TL with the underlying distribution of population density (or other nonnegative quantity) and provides a baseline against which more complex mechanisms of TL can be compared

Computationally Efficient and Robust BIC-Based Speaker Segmentation

An algorithm for automatic speaker segmentation based on the Bayesian information criterion (BIC) is presented. BIC tests are not performed for every window shift, as previously, but when a speaker change is most probable to occur. This is done by estimating the next probable change point thanks to a model of utterance durations. It is found that the inverse Gaussian fits best the distribution of utterance durations. As a result, less BIC tests are needed, making the proposed system less computationally demanding in time and memory, and considerably more efficient with respect to missed speaker change points. A feature selection algorithm based on branch and bound search strategy is applied in order to identify the most efficient features for speaker segmentation. Furthermore, a new theoretical formulation of BIC is derived by applying centering and simultaneous diagonalization. This formulation is considerably more computationally efficient than the standard BIC, when the covariance matrices are estimated by other estimators than the usual maximum-likelihood ones. Two commonly used pairs of figures of merit are employed and their relationship is established. Computational efficiency is achieved through the speaker utterance modeling, whereas robustness is achieved by feature selection and application of BIC tests at appropriately selected time instants. Experimental results indicate that the proposed modifications yield a superior performance compared to existing approaches