Two-parameter Gamma Distribution Research Articles

Class I and II HLA Genes Are Associated with Susceptibility and Age at Onset in Finnish Families with Type 1 Diabetes

Objectives: We explored the properties of the long-term survivor model (LTS) in the genetic association studies and studied allelic and haplotypic associations between the age at onset and partially latent susceptibility of type 1 diabetes (T1DM) and Human Leucocyte Antigen (HLA) A, B and DR loci. Methods: The authors applied the long-term survivor model (LTS) for sibships collected in a population-based registry during a calendar time period. The method uses sibs that could not become probands and includes the proband’s age at onset during the recruitment period. Association between the candidate gene and the partially latent susceptibility is modeled with logistic regression and the age at onset with a two-parameter gamma distribution, where a scale parameter depends on the candidate genotypes. We also performed a simulation study of nuclear families to compare the power of the likelihood ratio tests of the genetic association based on the LTS model with those obtained using family-based association method (FBAT) and bias of the case-pseudo control design. In addition, we analysed allele and haplotype associations between HLA A, B and DR loci (IDDM1) with T1DM, using population-based ascertainment of 705 sibships with complete HLA information. Results: A simulation study showed that the estimates of the genetic association using an ascertainment-corrected LTS model are virtually unbiased and that the relative risk estimates obtained from case-pseudo control design (TDT) are negatively biased. In the analysis of the Finnish T1DM families we found that only B62 (p < 0.05) is positively significantly associated with susceptibility after adjusting for the haplotype effects. Five alleles were significantly associated with age at onset (B8 and DR3, p < 0.01; A2, B60 and DR6, p < 0.05). No significant three-locus haplotype associations with the susceptibility were found, but A3B18DR4 (p < 0.001) haplotype was associated with older age at onset than average. Conclusions: Estimates of genetic relative risk obtained from the case-pseudo control design are negatively biased and the prospective LTS model is an appropriate choice, when there are non-susceptible subjects in the population with variable age at onset. Based on the analysis of T1DM, we conclude that there are gene(s) in the HLA region that are associated with susceptibility and/or age at onset of T1DM, and this should be taken into account in future studies.

Simplified Two-Parameter Gamma Distribution for Derivation of Synthetic Unit Hydrograph

Several methods for synthetic unit hydrographs are available in the literature. Most of them involve manual, subjective fitting of a hydrograph through few data points. Because it is difficult, the...

Markov chain-based modeling techniques for stochastic generation of daily intermittent streamflows

This study is an attempt to generate synthetic daily streamflow data for intermittent streams. The method followed in the study is based on the Markov chain. Two two-state Markov chains or alternatively one three-state Markov chain is proposed for determination of the state of the stream. Ascension curve of the hydrograph is simulated by the two-parameter gamma distribution. Recession curve of the hydrograph is assumed to decay exponentially. Comparison of the statistics of the generated and observed streamflow series shows the applicability of the techniques. Not only long-term statistics such as mean, variance, skewness, and correlation but also short-term features such as the non-invertible shape of the daily streamflow hydrograph are preserved.

The feature of stratification in the blowing sand cloud

The profiles of particle concentration in saltation layer versus height are calculated, by the motion equations for a saltating grain in conjunction with different probability distribution functions of the vertical liftoff velocities of grains and an empirical expression of wind velocity. The numerical results demonstrated that the stratification phenomenon exists in the particle concentration profiles and showed increasing, saturating and decreasing features, respectively, when the probability distribution functions of liftoff velocities adopted in the calculation are similar to a normal distribution or a two-parameter gamma distribution. When the distribution function of liftoff velocities is taken as an exponential form, the profile of particle concentration decreases monotonically. A numerical simulation of mass flux of grains, performed by the model suggested in this paper, is in reasonable accordance with the measured data.

A linear model of the effects of residence time distribution on mixing pattern in a ventilated airspace

The ability of a simple linear response model is evaluated to explain the mixing efficiencies in an incomplete mixing ventilated airspace. Data interpretation and mean residence time calculation for a specified output concentration profile can also be evaluated. The residence time distribution (RTD) functions take the form of the two-parameter gamma distribution and account for different mixing types such as complete mixing, piston flow (no mixing), incomplete mixing, and various combinations of the above types. In these combinations, the different mixing types simulated by different RTDs conceptually represent airflow regions in series. The mixing efficiency was introduced to characterize the extent or degree of mixing in a ventilation system in that mixing efficiency equals zero for piston flow (no mixing), unity for complete mixing, and a value in between these two extremes for incomplete mixing. An environmental chamber experiment was conducted to generate several output profiles to evaluate the applicability of the model. Carbon dioxide was employed as the tracer gas. Our results show that an overall root-mean-squared error value of 8.64±5.25 ppm is low, indicating that the combination mixing patterns are generally found to be minimally biased and give better fitting than other simpler mixing patterns. Despite their neglect of molecular diffusion and possible temporal/spatial nonlinearities, these linear response models appear reasonably robust, making them at least as useful to building microenvironment designers in reconsidering the possibilities and consequences of various forms of incomplete mixing related to indoor air quality problems.

TESTING THE PERFORMANCE OF DIFFERENT PROCESSES TO GENERATE TEMPERATURE AND SOLAR RADIATION: A CASE STUDY AT LLEIDA (NORTHEAST SPAI N)

The type of weather data that is available largely conditions weather generation at any given location. Weathergenerators that provide flexibility in the selection of generation methods are advantageous. ClimGen, a multivariateautoregressive firstorder weather generation model, was used to compare different methods to generate weather series ofdaily maximum (Tx) and minimum (Tn) temperatures and bright sunshine hours (Bsh). Two of these methods (A1 and A2) canbe applied at stations where daily precipitation (Prec), Tx, Tn, and Bsh data are available. Another method (B) can be appliedat stations where Bsh data are lacking. Method A1 uses Tx, Tn, and Bsh in the multivariate generation process. Method A2uses Tx,
T = Tx Tn, and Bsh. Method B only uses Tx and Tn. Two temperaturebased methods to estimate solar radiationwere used to complement method B. In addition, the twoparameter Weibull and Gamma distributions were compared for thegeneration of precipitation amounts.<br><br>The different methods were evaluated for agricultural applications at the Lleida region, in northeast Spain, a continentalsemiarid climate. Different climatological variables and agricultural indices were calculated using actual and generateddata, and statistically compared at 5% and 10% levels of significance. It was concluded that: (1) The twoparameter Weibulldistribution performed better in this region than the twoparameter Gamma distribution; (2) method A1 was slightly superiorto method A2, mainly because the minimum temperature and the bright sunshine hours were better replicated; (3) methodB was found satisfactory, but the complementary methods to estimate solar radiation from temperature did not perform wellwhen they were calibrated at a given station and then used to generate for a station in a surrounding area; and (4) the methodstested well replicated chill units, growing degree days, the mean of heating degree days (but not its variance), and long hotor cold sequences, but they did not appear to reliably replicate short sequences as needed to assess problems such as cropfrost potential. The generation of bright sunshine hours, as introduced in this study, is an alternative for locations where solarradiation data are not available to parameterize the weather generator.

SW—Soil and Water: Two-component Transfer Function Modelling of Flow through Macroporous Soil

The macropore and micropore domains of a soil system were considered to be hydraulically distinct. Drainage and solute transport through the soil were then characterized by mixed probability distributions. The components of the mixed probability distributions were used to represent the distinct processes of drainage and solute transport in the macropore and micropore domains. A goodness-of-fit test showed that the lognormal distribution was the best theoretical distribution for solute transport in each of the two domains, while the two-parameter gamma distribution was best for drainage in each of the domains. A validation test showed that the mixed probability distribution representation of the transfer function adequately incorporated dual porosity in the transfer function model for drainage and solute transport through macroporous soil.

CHARACTERIZATION OF MIXING PATTERNS IN A VENTILATED AIRSPACE WITH A MULTIPLE AIRFLOW REGIONS GAMMA MODEL

A model called the multiple airflow regions gamma model (MARGM)was developed based on a continuous distribution of residence time for predictingthe mixing behavior in a ventilated airspace in that data interpretation andmean residence time calculation for a specified output concentration profilecan also be evaluated. The MARGM takes the form of the two-parameter gammadistribution and accounts for different mixing types: complete mixing, nomixing (piston flow), incomplete mixing, and various combinations of the abovetypes. In these combinations, the different mixing types simulated by theMARGM conceptually represent airflow regions in series. The mixing efficiencywas introduced to characterize the extent or degree of mixing in a ventilationsystem. Mixing efficiency equals zero for piston flow (no mixing), unity forcomplete mixing, and a value in between these two extremes for incompletemixing. The MARGM simulates the combinations of complete mixing, incompletemixing, and piston flow. Therefore, seven models are introduced in this effort:complete mixing model, piston flow model, complete-piston flow model, complete-incomplete-pistonflow model (the general model), complete-incomplete mixing model, incomplete-pistonflow model, and incomplete mixing model. The applicability of models was testedby several case studies. Results show that combination models give betterfitting than other simpler models. The MARGM enables building microenvironmentdesigners reconsider the possibilities and consequences of various forms ofincomplete mixing in investigating indoor air quality problems.

Bounds for the Maximum Likelihood Estimates in Two-Parameter Gamma Distribution

Bounds for the maximum likelihood estimator (MLE) of the shape parameter of the two-parameter gamma distribution are obtained for the first time. These bounds are then utilized to develop an efficient algorithm for obtaining the MLE of the shape parameter which is guaranteed to converge.

A stochastic weather generation model for Alaska

A specific problem encountered in ecosystem-level simulation of Arctic ecosystems is the depth and extent of the driving variable record. Often, climate records are of short duration, gathered at locations different from the area to be simulated, or do not contain all the variables required by a given model. This paper addresses this problem for ecosystem simulation in Alaska with the development of a weather generator. The generator, called WGENAL, is based on the WGEN climate generator developed and validated in the 48 conterminous states. Because of the extreme variability of weather in Alaska that is not accommodated by the statistical metrics in the earlier model, a new climate generator was developed. WGENAL generates daily values of precipitation, maximum temperature, minimum temperature, solar radiation, and wind run. Precipitation is generated using a Markov chain-gamma model. A two-parameter gamma distribution is used to generate wind run. Temperatures and solar radiation are generated using procedures developed in the earlier study. Validation of the generator shows it provides adequate diurnal and seasonal weather records for Fairbanks. Other comparisons of synthetic weather with observed weather for sites north of the Brooks Range in Alaska are also within the error of the original data.

The distribution of transaction intervals in common stock trading

This paper examines the characteristics of the distribution of time between successive transactions of common stocks. Using a large set of intraday stock trading data, we test two competing hypotheses on the functional form of the distribution of transaction time intervals. Although the exponential distribution is widely used for the study of time series properties of common stock prices, our result indicates that the two-parameter gamma distribution seems to provide a better description on the behavior of transaction time intervals. While our finding sheds light on the study of stochastic properties of stock return distribution, it also provides new insight into understanding stock investing behavior in today's ever-sensitive marketplace.

Simulated meteorological input for agricultural models

Many agricultural models need an input of daily weather, and a common way of accommodating this is by simulation. Long daily records are rare for agricultural sites, but it is usually possible to obtain climatic parameters to use as input for a s stochastic model. An additional advantage to simulated weather is that many likely sequences can be generated, whereas observation furnishes only a single realization. The objective of this study was to develop a model to simulate weather records, that places emphasis on realistic variability, sequences, extremes and cross-correlation, as well as mean conditions. The model was designated Simulated Meteorological Input for Agricultural Models (SIMIAM), and relies on selection of air mass as the first stochastic step. All required variables are then generated using air mass statistics. The day's wet or dry character is ascertained by first-order Markov chain and, if wet, precipitation amount simulated in a two-parameter gamma distribution. A weakly-stationary generating system, that incorporates matrices of cross and auto-correlation coefficients, is used to simulate all other variables, including winds. Amarillo, Texas records were used to demonstrate model feasibility. Significantly different air masses were shown to be identifiable by gradient level wind directions, and their sequence was adequately modeled with a first-order Markovian process. SIMIAM was tested for Amorillo, Oklahoma City, and New Orleans, using 15 years of record to compute input (1961–1975), and comparing simulations against an independent 15 years of observations (1976–1990). Results indicated that the study objectives were mostly satisfied for the two inland cities. As input was compiled from discontinuous, multiple, area sources, these tests were robust. They were also considered to demonstrate model applicability, not only to point locations, but to areas of the south central United States. Test results for New Orleans were less acceptable, and further study is clearly necessary to achieve model transportability to different climatic zones.

Stochastic models for daily rainfall in the Western Pacific

Abstract Several stochastic models of rainfall occurrence and daily rainfall amount have been applied to data sets, of 18–20 years duration, for 22 islands in the Western Pacific. The models for rainfall occurrence were first- and second-order Markov chains, the truncated negative binomial distribution, and the truncated geometric distribution; each was applied with parameters determined independently for each month, and with parameters assumed to vary smoothly through the year according to a Fourier series. The models tested for daily rainfall amounts were the Weibull distribution, a mixed exponential distribution, the two-parameter gamma distribution, a skewed normal distribution, the kappa distribution, and the Srikanthan-McMahon model. The rainfall data were classified according to the number of adjoining wet days, and the models tested with all data grouped together, solitary wet days fitted separately from other wet days, and each class of data fitted separately. The results of these tests showed that this classification improves model fitting, in spite of the additional number of parameters required. For several months of the year, there are significant differences between the distributions of rainfall amounts for the three classes. However, for 18 of the 22 stations, the Srikanthan-McMahon model outperformed all others, and successfully simulated the differences between the rainfall classes.

On an asymptotic goodness-of-fit test for a two-parameter gamma-distribution

In this paper, a formula for the test statistic is given for a two-parameter gamma-distribution which involves the first four moments.

Statistical properties of the failure time distribution for [formula omitted] steels

A generalised three-parameter gamma distribution was fitted to three creep failure data sets obtained from BSCC. The log normal, Weibull, gamma and exponential distributions are nested within this generalised distribution and emerge when the parameters of the generalised distribution take on particular values. Standard statistical tests for parameter restrictions were then used to identify which of the above-named distributions are most supported by the data. Confidence limits for 50%, 10% and 1% quantiles of log failure time were computed using distributions most and least supported by the data. The extent to which such uncertainty about distributional form effects the size of confidence limits was also analysed. At the three stress and temperature ranges studied, the log normal distribution was best supported by the data, the Weibull distribution was least supported, whilst the two-parameter gamma and exponential distributions were rejected by the data. In each case a wide variety of three-parameter gamma distributions were supported by the data and this significantly increased the confidence limits at the lower quantile ranges.

Monte-Carlo study of angular and edge length distributions in a three-dimensional Poisson-Voronoi tesselation

Several hundred thousand three-dimensional Poisson-Voronoi cells have been simulated to study accurately the distributions of the dihedral angles (angles between adjacent faces), bond angles (angles between adjacent edges), total edge length of a face and total edge length of a cell. The mean values of the dihedral angle and the bond angle are found to be very close to those required by the equilibrium surface tension criterion for the polycrystalline grains. A two-parameter gamma distribution can be used to describe the total edge length distribution of a cell. For a sample size of less than few thousand cells, a twoparameter lognormal distribution can also be used to describe the total edge length distribution of the cell.

A Monte Carlo study of size and angular properties of a three-dimensional Poisson-Delaunay cell

On the basis of simulation of 1.2×106 three-dimensional Poisson-Delaunay cells, the statistical properties of their size and angular parameters have been studied. The moments of the volume, face area, and edge length distributions are found to be equal to those obtained from the exact expressions of Miles and of Moller. The volume, surface area, and face area distributions can be described by the two-parameter gamma distribution. The normal distribution can be used to describe the distributions of the total edge length of a cell and the perimeter of a face. The edge length distribution has also been studied. The distribution of the angle in a face is found to be in accordance with its theoretical distribution.

Properties of a two-dimensional Poisson-Voronoi tesselation: A Monte-Carlo study

On the basis of simulation of 2 × 10 6 polygons, the distribution of the topological parameter (number of sides) as well as the distributions of area, perimeter, and edge length of the two-dimensional (2D) Poisson-Voronoi polygons have been accurately determined. A two-parameter gamma distribution has been found to be a good fit to both the topological and the area distributions. For sample sizes less than 5000 polygons, the two-parameter lognormal distribution can also be used to accurately describe these distributions. The area, perimeter, and edge length distributions for polygons having a fixed number of edges have also been obtained. The 2D cross section of a three-dimensional Voronoi tesselation has also been studied. The side and area distributions for these cross sections have been found to be significantly different from those in the case of pure 2D Voronoi tesselation. For the former case, our results for the mean number of sides of the neighboring cells of an n-sided cell are in agreement with Aboav's rule as well as his experimental results for the 2D cross section of magnesium oxide grains.

Likelihood-based confidence intervals for estimating floods with given return periods

This paper discusses aspects of the calculation of likelihood-based confidence intervals for T-year floods, with particular reference to (1) the two-parameter gamma distribution; (2) the Gumbel distribution; (3) the two-parameter log-normal distribution, and other distributions related to the normal by Box-Cox transformations. Calculation of the confidence limits is straightforward using the Nelder-Mead algorithm with a constraint incorporated, although care is necessary to ensure convergence either of the Nelder-Mead algorithm, or of the Newton-Raphson calculation of maximum-likelihood estimates. Methods are illustrated using records from 18 gauging stations in the basin of the River Itajai-Acu, State of Santa Catarina, southern Brazil. A small and restricted simulation compared likelihood-based confidence limits with those given by use of the central limit theorem; for the same confidence probability, the confidence limits of the simulation were wider than those of the central limit theorem, which failed more frequently to contain the true quantile being estimated. The paper discusses possible applications of likelihood-based confidence intervals in other areas of hydrological analysis.

Inferences on the Shape Parameter of a Gamma Distribution: A Conditional Approach

A conditional method for inferences on the shape parameter of a two-parameter gamma distribution is presented in this article. A simple numerical implementation of this conditional method is developed; this is a computer program that takes the observed data as input and produces the confidence distribution function for the shape parameter, which in turn provides approximate observed significance levels and confidence intervals for that parameter, as output. These approximations are extremely accurate even for very small sample size and are numerically simple and easy to obtain.