Gamma Scale Parameters Research Articles

A numerical method to compute Fisher information for a special case of heterogeneous negative binomial regression

Negative binomial regression has been widely applied in various research settings to account for counts with overdispersion. Yet, when the gamma scale parameter, \begin{document}$ \nu $\end{document} , is parameterized, there is no direct algorithmic solution to the Fisher Information matrix of the associated heterogeneous negative binomial regression, which seriously limits its applications to a wide range of complex problems. In this research, we propose a numerical method to calculate the Fisher information of heterogeneous negative binomial regression and accordingly develop a preliminary framework for analyzing incomplete counts with overdispersion. This method is implemented in R and illustrated using an empirical example of teenage drug use in America.

On second-order improved estimation of a gamma scale parameter

ABSTRACTThis paper concludes our comprehensive study on point estimation of model parameters of a gamma distribution from a second-order decision theoretic point of view. It should be noted that efficient estimation of gamma model parameters for samples ‘not large’ is a challenging task since the exact sampling distributions of the maximum likelihood estimators and its variants are not known. Estimation of a gamma scale parameter has received less attention from the earlier researchers compared to shape parameter estimation. What we have observed here is that improved estimation of the shape parameter does not necessarily lead to improved scale estimation if a natural moment condition (which is also the maximum likelihood restriction) is satisfied. Therefore, this work deals with the gamma scale parameter estimation as a separate new problem, not as a by-product of the shape parameter estimation, and studies several estimators in terms of second-order risk.

A comparison of diameter distribution models for Khaya ivorensis A.Chev. plantations in Brazil

The purpose of this study was to compare Beta, Gamma, Johnson's SB and Weibull functions fitted by different methods for describing the horizontal structure of Khaya ivorensis (African mahogany) plantations in Brazil. The database comprised 128 plots from six plantations at varying ages. The function fits were compared using the Kolmogoroff–Smirnoff test, mean bias and mean absolute error for the number of trees and basal area per hectare per diameter class. Johnson's SB outperformed the other functions, although all functions provided an adequate fit. The best methods were method of moments and maximum likelihood fitted using 25% of the minimum observed diameter as the location parameter for the Johnson's SB function. The errors were greater in diameter classes with higher frequencies. Location and scale parameters were highly correlated with mean diameter and age for the Weibull and Johnson's SB functions, respectively, which is convenient for diameter prediction. Gamma's scale parameter had medium correlation with age. Beta's parameters had low correlation with stand attributes assessed.

On a threshold representation for complex load-sharing systems

Complex load-sharing systems are studied to incorporate dependencies among components through a load-sharing rule. As the load on the system increases, a series of cycles of Phase I/II failures occur where Phase I failure is a single component failure, which then causes a cascade of component failures (Phase II) due to the load transfer as these components fail. A threshold representation for the process of system failure is given. This representation is a gamma-type mixture representation when the component strengths are independent exponentials. In this case, for a given breaking pattern the mixture is over the gamma scale parameter and is based on a convolution of uniforms defined by the load-sharing parameters. Such convolutions can be approximated by normal densities which reduces the dimension of the parameter space. This representation can be generalized to independent component strengths with arbitrary distributions by transforming the strength and load-sharing to pseudo-strength and pseudo-load-sharing rules.

On the Admissibility of Estimators of Two Ordered Gamma Scale Parameters under Entropy Loss Function

Suppose that a random sample of size ni is drawn from a gamma distribution with known shape parameteri > 0 and unknown scale parameteri > 0, i=1,2, satis- fying 0 < �1 ≤ �2. In estimation of �1 and �2 under the entropy loss function, we consider the class of mixed estimators of �1 and �2. It is shown that a subclass of mixed estimators ofi beats the usual estimators Xi/�i, i=1,2, and the inadmissible estimators in the class of mixed estimators are derived. Also the asymptotic efficiency of mixed estimators relative to the usual estimators are obtained. Finally the results are extended to a subclass of the scale parameter exponential family and the family of transformed chi-square distributions.

Simulated changes due to global warming in daily precipitation means and extremes and their interpretation using the gamma distribution

The potential change in precipitation due to global warming is studied using five‐member ensembles of climate simulations by the CSIRO Mark 2 atmosphere‐ocean model for the period 1871–1990 and forward to 2100 under both the Special Report on Emission Scenarios (SRES) A2 (rapid CO2 increase) and B2 (moderate increase) forcing scenarios. The mean surface warming for the period 1961–1990 is 0.3 K. The warming from 1961–1990 to 2071–2100 is 3.5 K under A2, 29% more than for B2, and with a very similar spatial pattern. The daily precipitation (P) frequency distributions for January and July days in these periods are presented, focusing on the A2 case. The distributions for wet days at each point are approximated by the gamma distribution. The global mean P increase of around 6%, in both months, is related to a mean increase in the gamma's scale parameter of 18%, offset by small decreases in the shape parameter and wet day frequency. However, local changes of opposite signs also occur, especially in the tropics. Ensemble averages of 30‐year extreme daily precipitation for January and July, and other months, are generally greater for 2071–2100 than for 1961–1990, with an average increase of 14%. Extreme value theory based on the monthly gamma distributions provides a good match to these values. The theory is extended to the annual case. In general, the 1961–1990 extremes peak in the subtropical rainbands in the model, where increases of 10 to 30% are common. Larger relative increases occur in polar regions, and also over northern land in January.

Evidence for trends in heavy rainfall events over the UK.

Daily precipitation in the UK has changed over the period 1961-2000, becoming on average more intense in winter and less intense in summer. Recent increases in total winter precipitation are shown to be mainly due to an increase in the amount of precipitation on wet days, with a smaller contribution in the western UK from a trend towards more wet days. If the wet-day amounts are modelled using a gamma distribution, then positive trends in its scale parameter are found across almost all of the UK, consistent with an increased frequency of heavy winter precipitation. Non-parametric analyses confirm an increase in the contribution of heavy events to winter precipitation totals. Analysis of multi-day sequences of heavy rainfall indicate a corresponding increase in their frequency. Results for summer show almost opposite trends: decreased precipitation totals (driven more equally by fewer wet days and reduced wet-day amounts), decreases in gamma scale parameter (although accompanied by a trend towards a less positively skewed distribution) and decreases in the occurrence of heavy precipitation (whether defined parametrically or non-parametrically). A more sparse network of weather stations with data back to 1901 suggests that the recent winter changes are unusual, while the recent summer changes are not, though the poorer coverage reduces the confidence in these longer-period results.

The use of conditionally conjugate priors in the study of ratios of gamma scale parameters

Comparisons of gamma scale parameters (when shape parameters are known) are frequently of interest. Certain conditionally specified priors are shown to form convenient conjugate families for such situations. Hyperparameter elicitation techniques are discussed. Simulation from the resulting posterior distributions is readily accomplished using the Gibbs sampler or importance sampling. A re-analysis of a classical data set is provided to illustrate the technique.

Decision theory in maintenance strategy of standby system with gamma-distribution repair-time

This study develops an application for a previous decision-model analyzed via decision theory. The previous paper identifies a need for additional applications in that decision-model. This study introduces a gamma repair-time distribution with shape-parameter 2. The uncertainty of the gamma scale-parameter is accounted for in a Bayes approach, eliciting expert prior belief. Conflicting requirements of system availability and cost are combined through a multi-attribute utility function. A numerical application with sensitivity analysis, illustrates the use of the model.

All Admissible Linear Estimators of the Vector of Gamma Scale Parameters with Application to Random Effects Models

The paper is devoted to the problem of simultaneous estimation of scale and natural parameters of the multiparameter gamma distribution under a quadratic loss. The vector of the scale parameters is assumed to range over a certain subset of the Cartesian product $\mathscr{R}^n_+$ of $n$ positive half lines. We identify the class of all linear admissible estimators for the scale parameters and show that all linear estimators of the natural parameters are inadmissible. Since the problem of invariant quadratic estimation of variance components in balanced random effects normal models leads to a problem of linear estimation of parametric functions of gamma scale parameters restricted to subsets of $\mathscr{R}^n_+$ being considered in this paper, some results on admissibility of invariant quadratic estimators of variance components are also established.

Simple Test of Hypotheses on System Availability and Mean Time to Repair

A method is developed for testing hypotheses on system availability and mean time to repair, under the assumption that the failure times and repair times are, respectively, independent exponential and gamma random variables. System is assumed to be in a steady state; availability is defined in usual manner as the ratio of mean time to failure to the sum of mean time to failure and mean time to repair. Testing hypotheses on system availability is achieved by simple application of the F distribution, provided: a) the system mean time to repair is known or determinable from other assumptions, or b) the ratio of system mean time to failure to the gamma scale parameter is known. Similarly, hypotheses on system mean repair time can be tested using the chi-square distribution, if the gamma scale parameter is known.

Estimation of multiple gamma scale-parameters: bayes estimation subject to uniform domination

Simultaneous estimation of p gamma scale-parameters is considered under squared-error loss. The problem of minimizing, subject to uniform risk domination, the Bayes risk (or more generally the posterior expected loss) against certain conjugate or mixtures of conjugate priors is considered. Rather surprisingly, it is shown that the minimization can be done conditionally, thus avoiding variational arguments. Relative savings loss (and a posterior version thereof) are found, and it is found that in the most favorable situations, Bayesian robustness can be achieved without sacrificing substantial subjective Bayesian gains.

Some locally optimal subset selection rules for comparison with a control

Abstract Let π0,π1,…,πk be k+1 independent populations. For i=0,1,…,k,πi has the densit f(x,θi), where the (unknown) parameter θi belongs to an interval of the real line. Our goal is to select from π1,… πk (experimental treatments) those populations, if any, that are better (suitably defined) than π0 which is the control population. A locally optimal rule is derived in the class of rules for which Pr(πi is selected)γi, i=1,…,k, when θ0=θ1=⋯=θk. The criterion used for local optimality amounts to maximizing the efficiency in a certain sense of the rule in picking out the superior populations for specific configurations of θ=(θ0,…,θk) in a neighborhood of an equiparameter configuration. The general result is then applied to the following special cases: (a) normal means comparison — common known variance, (b) normal means comparison — common unknown variance, (c) gamma scale parameters comparison — known (unequal) shape parameters, and (d) comparison of regression slopes. In all these cases, the rule is obtained based on samples of unequal sizes.

A performance comparison of empirical Bayes and Bayes estimators of the Weibull and gamma scale parameters

AbstractA mean‐squared error comparison of smooth empirical Bayes and Bayes estimators for the Weibull and gamma scale parameters is studied based on a computer simulation. The smooth empirical Bayes estimators are determined as functions of up to 15 past estimates of the parameter of interest. Results indicate that at best the mean‐squared errors of the empirical Bayes estimators are about 20–40% larger than those of the corresponding squared‐error optimal Bayes estimators.

Comparison of some robust estimators of scale in gamma samples with known shape

In many practical situations a gamma model is used in which the shape parameter is known and the scale parameter is unknown. This paper is concerned with robust estimation of the gamma scale parameter in such circumstances. Three families of estimators are investigated: trimmed-mean type estimators, Huber-sense M-estimators and maximum likelihood estimators for censored samples. The robustness of these estimators is compared using influence curves and an asymptotic contamination model. Some small-sample results are also obtained for the special case of the exponential distribution

Improving on Inadmissible Estimators in Continuous Exponential Families with Applications to Simultaneous Estimation of Gamma Scale Parameters

A general technique is developed for improving upon inadmissible estimators of natural parameters (or integral powers thereof) from continuous exponential families. The technique is to reduce the problem to the study of a differential inequality. Typical differential inequalities are presented and solved. Explicit results are given for the simultaneous estimation of gamma scale parameters (and their inverses) for a variety of natural loss functions. Surprising behavior is observed for many of the estimators improving upon "standard" estimators. For squared error loss (and any continuous exponential family) it is shown explicitly how to establish inadmissibility of an estimator and construct improved estimators.