Inflated beta distributions

This paper considers the issue of modeling fractional data observed on [0,1), (0,1] or [0,1]. Mixed continuous-discrete distributions are proposed. The beta distribution is used to describe the continuous component of the model since its density can have quite different shapes depending on the values of the two parameters that index the distribution. Properties of the proposed distributions are examined. Also, estimation based on maximum likelihood and conditional moments is discussed. Finally, practical applications that employ real data are presented.

Information Matrix for Beta Distributions

2000 Mathematics Subject Classification: 33C90, 62E99.

On Finite Mixture of Pareto and Beta Distnbutions

Department of Statistics, Cochin University of Science and Technology

Exposure assessment of extended-spectrum beta-lactamases/AmpC beta-lactamases-producing Escherichia coli in meat in Denmark

INTRODUCTION Extended-spectrum beta-lactamases (ESBL) and AmpC beta-lactamases (AmpC) are of concern for veterinary and public health because of their ability to cause treatment failure due to antimicrobial resistance in Enterobacteriaceae. The main objective was to assess the relative contribution (RC) of different types of meat to the exposure of consumers to ESBL/AmpC and their potential importance for human infections in Denmark. MATERIAL AND METHODS The prevalence of each genotype of ESBL/AmpC-producing E. coli in imported and nationally produced broiler meat, pork and beef was weighted by the meat consumption patterns. Data originated from the Danish surveillance program for antibiotic use and antibiotic resistance (DANMAP) from 2009 to 2011. DANMAP also provided data about human ESBL/AmpC cases in 2011, which were used to assess a possible genotype overlap. Uncertainty about the occurrence of ESBL/AmpC-producing E. coli in meat was assessed by inspecting beta distributions given the available data of the genotypes in each type of meat. RESULTS AND DISCUSSION Broiler meat represented the largest part (83.8%) of the estimated ESBL/AmpC-contaminated pool of meat compared to pork (12.5%) and beef (3.7%). CMY-2 was the genotype with the highest RC to human exposure (58.3%). However, this genotype is rarely found in human infections in Denmark. CONCLUSION The overlap between ESBL/AmpC genotypes in meat and human E. coli infections was limited. This suggests that meat might constitute a less important source of ESBL/AmpC exposure to humans in Denmark than previously thought - maybe because the use of cephalosporins is restricted in cattle and banned in poultry and pigs. Nonetheless, more detailed surveillance data are required to determine the contribution of meat compared to other sources, such as travelling, pets, water resources, community and hospitals in the pursuit of a full source attribution model.

Reliability for Beta Models

In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability R = Pr(X2 < X1 ) when X1 and X2 are independent random variables belonging to the same univariate family of distributions. The algebraic form for R = Pr(X2 < X1 ) has been worked out for the majority of the well-known distributions including Normal, uniform, exponential, gamma, weibull and pareto. However, there are still many other distributions for which the form of R is not known. We have identified at least some 30 distributions with no known form for R. In this paper we consider some of these distributions and derive the corresponding forms for the reliability R. The calculations involve the use of various special functions.

A new approach to estimating trends in chlamydia incidence

Objectives Directly measuring disease incidence in a population is difficult and not feasible to do routinely. We describe the development and application of a new method of estimating at a population level the number of incident genital chlamydia infections, and the corresponding incidence rates, by age and sex using routine surveillance data. Methods A Bayesian statistical approach was developed to calibrate the parameters of a decision-pathway tree against national data on numbers of notifications and tests conducted (2001-2013). Independent beta probability density functions were adopted for priors on the time-independent parameters; the shape parameters of these beta distributions were chosen to match prior estimates sourced from peer-reviewed literature or expert opinion. To best facilitate the calibration, multivariate Gaussian priors on (the logistic transforms of) the time-dependent parameters were adopted, using the Matérn covariance function to favour changes over consecutive years and across adjacent age cohorts. The model outcomes were validated by comparing them with other independent empirical epidemiological measures i.e. prevalence and incidence as reported by other studies. Results Model-based estimates suggest that the total number of people acquiring chlamydia per year in Australia has increased by ~120% over 12 years. Nationally, an estimated 356,000 people acquired chlamydia in 2013, which is 4.3 times the number of reported diagnoses. This corresponded to a chlamydia annual incidence estimate of 1.54% in 2013, increased from 0.81% in 2001 (~90% increase). Conclusions We developed a statistical method which uses routine surveillance (notifications and testing) data to produce estimates of the extent and trends in chlamydia incidence.

Methods to determine resistance to anthelmintics when continuing larval development occurs

The in vivo faecal egg count reduction test (FECRT) is the most commonly used test to detect anthelmintic resistance (AR) in gastrointestinal nematodes (GIN) of ruminants in pasture based systems. However, there are several variations on the method, some more appropriate than others in specific circumstances. While in some cases labour and time can be saved by just collecting post-drench faecal worm egg counts (FEC) of treatment groups with controls, or pre- and post-drench FEC of a treatment group with no controls, there are circumstances when pre- and post-drench FEC of an untreated control group as well as from the treatment groups are necessary. Computer simulation techniques were used to determine the most appropriate of several methods for calculating AR when there is continuing larval development during the testing period, as often occurs when anthelmintic treatments against genera of GIN with high biotic potential or high re-infection rates, such as Haemonchus contortus of sheep and Cooperia punctata of cattle, are less than 100% efficacious. Three field FECRT experimental designs were investigated: (I) post-drench FEC of treatment and controls groups, (II) pre- and post-drench FEC of a treatment group only and (III) pre- and post-drench FEC of treatment and control groups. To investigate the performance of methods of indicating AR for each of these designs, simulated animal FEC were generated from negative binominal distributions with subsequent sampling from the binomial distributions to account for drench effect, with varying parameters for worm burden, larval development and drench resistance. Calculations of percent reductions and confidence limits were based on those of the Standing Committee for Agriculture (SCA) guidelines. For the two field methods with pre-drench FEC, confidence limits were also determined from cumulative inverse Beta distributions of FEC, for eggs per gram (epg) and the number of eggs counted at detection levels of 50 and 25. Two rules for determining AR: (1) %reduction (%R) < 95% and lower confidence limit <90%; and (2) upper confidence limit <95%, were also assessed. For each combination of worm burden, larval development and drench resistance parameters, 1000 simulations were run to determine the number of times the theoretical percent reduction fell within the estimated confidence limits and the number of times resistance would have been declared. When continuing larval development occurs during the testing period of the FECRT, the simulations showed AR should be calculated from pre- and post-drench worm egg counts of an untreated control group as well as from the treatment group. If the widely used resistance rule 1 is used to assess resistance, rule 2 should also be applied, especially when %R is in the range 90 to 95% and resistance is suspected.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

Identification of models using failure rate and mean residual life of doubly truncated random variables

In this paper, we study the relationship between the failure rate and the mean residual life of doubly truncated random variables. Accordingly, we develop characterizations for exponential, Pareto 11 and beta distributions. Further, we generalize the identities for fire Pearson and the exponential family of distributions given respectively in Nair and Sankaran (1991) and Consul (1995). Applications of these measures in file context of lengthbiased models are also explored

Exercise sheet 4 pdf

Exercises and solutions in PDF

Exercise sheet 5

Exercises and solutions in LaTex

Exercise sheet 4

Exercises and solutions in LaTex

Exercise sheet 5 pdf

Exercises and solutions in PDF

Cost models for engineering services

This thesis describes the procedure and results from four years research undertaken through the IHD (Interdisciplinary Higher Degrees) Scheme at Aston University in Birmingham, sponsored by the SERC (Science and Engineering Research Council) and Monk Dunstone Associates, Chartered Quantity Surveyors. A stochastic networking technique VERT (Venture Evaluation and Review Technique) was used to model the pre-tender costs of public health, heating ventilating, air-conditioning, fire protection, lifts and electrical installations within office developments. The model enabled the quantity surveyor to analyse, manipulate and explore complex scenarios which previously had defied ready mathematical analysis. The process involved the examination of historical material costs, labour factors and design performance data. Components and installation types were defined and formatted. Data was updated and adjusted using mechanical and electrical pre-tender cost indices and location, selection of contractor, contract sum, height and site condition factors. Ranges of cost, time and performance data were represented by probability density functions and defined by constant, uniform, normal and beta distributions. These variables and a network of the interrelationships between services components provided the framework for analysis. The VERT program, in this particular study, relied upon Monte Carlo simulation to model the uncertainties associated with pre-tender estimates of all possible installations. The computer generated output in the form of relative and cumulative frequency distributions of current element and total services costs, critical path analyses and details of statistical parameters. From this data alternative design solutions were compared, the degree of risk associated with estimates was determined, heuristics were tested and redeveloped, and cost significant items were isolated for closer examination. The resultant models successfully combined cost, time and performance factors and provided the quantity surveyor with an appreciation of the cost ranges associated with the various engineering services design options.