scholarly journals Comparison of Estimators for Parameters of Gamma Distributions with Left-Truncated Samples

2011 ◽  
Vol 50 (2) ◽  
pp. 296-310 ◽  
Author(s):  
Roger W. Johnson ◽  
Donna V. Kliche ◽  
Paul L. Smith
Keyword(s):  
Maximum Likelihood ◽  
Method Of Moments ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Minimum Size ◽  
Gamma Distributions ◽  
Raindrop Size Distribution ◽  
L Moments ◽  
Raindrop Size ◽  
Drop Size Distributions

Abstract When fitting a raindrop size distribution using a gamma model from data collected by a disdrometer, some consideration needs to be given to the small drops that fail to be recorded (typical disdrometer minimum size thresholds being in the 0.3–0.5-mm range). To this end, a gamma estimation procedure using maximum likelihood estimation has recently been published. The current work adds another procedure that accounts for the left-truncation problem in the data; in particular, an L-moments procedure is developed. These two estimation procedures, along with a traditional method-of-moments procedure that also accounts for data truncation, are then compared via simulation of volume samples from known gamma drop size distributions. For the range of gamma distributions considered, the maximum likelihood and L-moments procedures—which perform comparably—are found to outperform the procedure of method-of-moments. As these three procedures do not yield simple estimates in closed form, salient details of the R statistical code used in the simulations are included.

L-Moment Estimators as Applied to Gamma Drop Size Distributions

2008 ◽  
Vol 47 (12) ◽  
pp. 3117-3130 ◽  
Author(s):  
Donna V. Kliche ◽  
Paul L. Smith ◽  
Roger W. Johnson
Keyword(s):  
Maximum Likelihood ◽  
Method Of Moments ◽  
Moment Method ◽  
Traditional Approach ◽  
Full Range ◽  
Small Sample ◽  
Minimum Size ◽  
L Moments ◽  
Raindrop Size ◽  
The Moment

Abstract The traditional approach with experimental raindrop size data has been to use the method of moments in the fitting procedure to estimate the parameters for the raindrop size distribution function. However, the moment method is known to be biased and can have substantial errors. Therefore, the L-moment method, which is widely used by hydrologists, was investigated as an alternative. The L-moment method was applied, along with the moment and maximum likelihood methods, to samples taken from simulated gamma raindrop populations. A comparison of the bias and the errors involved in the L-moments, moments, and maximum likelihood procedures shows that, with samples covering the full range of drop sizes, L-moments and maximum likelihood outperform the method of moments. For small sample sizes the moment method gives a large bias and large error while the L-moment method gives results close to the true population values, outperforming even maximum likelihood results. Because the goal of this work is to understand the properties of the various fitting procedures, the investigation was expanded to include the effects of the absence of small drops in the samples (typical disdrometer minimum size thresholds are 0.3–0.5 mm). The results show that missing small drops (due to the instrumental constraint) can result in a large bias in the case of the L-moment and maximum likelihood fitting methods; this bias does not decrease much with increasing sample size. Because the very small drops have a negligible contribution to moments of order 2 or higher, the bias in the moment methods seems to be about the same as in the case of full samples. However, when moments of order less than 2 are needed (as in the case of modelers using moments 0 and 3), the moment method gives much larger bias. Therefore a modification of these methods is needed to handle the truncated-data situation.

scholarly journals Large Sample Comparison of Parameter Estimates in Gamma Raindrop Distributions

Atmosphere ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 333 ◽  
Author(s):  
Roger W. Johnson ◽  
Donna V. Kliche
Keyword(s):  
Maximum Likelihood ◽  
Method Of Moments ◽  
Estimation Procedure ◽  
Delta Method ◽  
Parameter Estimates ◽  
Large Sample Size ◽  
Large Sample ◽  
Estimation Procedures ◽  
Raindrop Size ◽  
Pseudo Maximum Likelihood

Raindrop size distributions have been characterized through the gamma family. Over the years, quite a few estimates of these gamma parameters have been proposed. The natural question for the practitioner, then, is what estimation procedure should be used. We provide guidance in answering this question when a large sample size (>2000 drops) of accurately measured drops is available. Seven estimation procedures from the literature: five method of moments procedures, maximum likelihood, and a pseudo maximum likelihood procedure, were examined. We show that the two maximum likelihood procedures provide the best precision (lowest variance) in estimating the gamma parameters. Method of moments procedures involving higher-order moments, on the other hand, give rise to poor precision (high variance) in estimating these parameters. A technique called the delta method assisted in our comparison of these various estimation procedures.

scholarly journals On the Shape–Slope Relation of Drop Size Distributions in Convective Rain

2005 ◽  
Vol 44 (7) ◽  
pp. 1146-1151 ◽  
Author(s):  
Axel Seifert
Keyword(s):  
Size Distribution ◽  
Gamma Distribution ◽  
Drop Size ◽  
Empirical Relation ◽  
Size Distributions ◽  
Shape Parameters ◽  
Raindrop Size Distribution ◽  
Raindrop Size ◽  
Drop Size Distributions ◽  
Good Agreement

Abstract The relation between the slope and shape parameters of the raindrop size distribution parameterized by a gamma distribution is examined. The comparison of results of a simple rain shaft model with an empirical relation based on disdrometer measurements at the surface shows very good agreement, but a more detailed discussion reveals some difficulties—for example, deviations from the gamma shape and the overestimation of collisional breakup.

scholarly journals Multivariate Gamma Regression: Parameter Estimation, Hypothesis Testing, and Its Application

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 813
Author(s):  
Anita Rahayu ◽  
Purhadi ◽  
Sutikno ◽  
Dedy Dwi Prastyo
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Hypothesis Testing ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Wald Test ◽  
Positive Outcome ◽  
Three Dimensions ◽  
Predictor Variables ◽  
Ratio Test

Gamma distribution is a general type of statistical distribution that can be applied in various fields, mainly when the distribution of data is not symmetrical. When predictor variables also affect positive outcome, then gamma regression plays a role. In many cases, the predictor variables give effect to several responses simultaneously. In this article, we develop a multivariate gamma regression (MGR), which is one type of non-linear regression with response variables that follow a multivariate gamma (MG) distribution. This work also provides the parameter estimation procedure, test statistics, and hypothesis testing for the significance of the parameter, partially and simultaneously. The parameter estimators are obtained using the maximum likelihood estimation (MLE) that is optimized by numerical iteration using the Berndt–Hall–Hall–Hausman (BHHH) algorithm. The simultaneous test for the model’s significance is derived using the maximum likelihood ratio test (MLRT), whereas the partial test uses the Wald test. The proposed MGR model is applied to model the three dimensions of the human development index (HDI) with five predictor variables. The unit of observation is regency/municipality in Java, Indonesia, in 2018. The empirical results show that modeling using multiple predictors makes more sense compared to the model when it only employs a single predictor.

scholarly journals DECONVOLUTING PREFERENCES AND ERRORS: A MODEL FOR BINOMIAL PANEL DATA

2010 ◽  
Vol 26 (6) ◽  
pp. 1846-1854 ◽  
Author(s):  
Mogens Fosgerau ◽  
Søren Feodor Nielsen
Keyword(s):  
Maximum Likelihood ◽  
Panel Data ◽  
Maximum Likelihood Estimation ◽  
Unknown Parameter ◽  
Random Variables ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Choice Experiments ◽  
Stated Choice ◽  
Stated Choice Experiments

In many stated choice experiments researchers observe the random variablesVt,Xt, andYt= 1{U+δ⊤Xt+ εt<Vt},t≤T, whereδis an unknown parameter andUand εtare unobservable random variables. We show that under weak assumptions the distributions ofUand εtand also the unknown parameterδcan be consistently estimated using a sieved maximum likelihood estimation procedure.

scholarly journals Maximum Likelihood Estimation for Three-Parameter Weibull Distribution Using Evolutionary Strategy

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Fan Yang ◽  
Hu Ren ◽  
Zhili Hu
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Weibull Distribution ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Evolutionary Strategy ◽  
Likelihood Method ◽  
Conventional Algorithm

The maximum likelihood estimation is a widely used approach to the parameter estimation. However, the conventional algorithm makes the estimation procedure of three-parameter Weibull distribution difficult. Therefore, this paper proposes an evolutionary strategy to explore the good solutions based on the maximum likelihood method. The maximizing process of likelihood function is converted to an optimization problem. The evolutionary algorithm is employed to obtain the optimal parameters for the likelihood function. Examples are presented to demonstrate the proposed method. The results show that the proposed method is suitable for the parameter estimation of the three-parameter Weibull distribution.

scholarly journals Statistical and Physical Descriptions of Raindrop Size Distributions in Equatorial Malaysia from Disdrometer Observations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Hong Yin Lam ◽  
Jafri Din ◽  
Siat Ling Jong
Keyword(s):  
Likelihood Estimation ◽  
Heavy Rain ◽  
Estimation Accuracy ◽  
Method Of Moment ◽  
Kuala Lumpur ◽  
Gamma Model ◽  
Essential Information ◽  
Raindrop Size Distribution ◽  
Raindrop Size

This work investigates the physical characteristics of raindrop size distribution (DSD) in an equatorial heavy rain region based on three years of disdrometer observations carried out at Universiti Teknologi Malaysia’s (UTM’s) campus in Kuala Lumpur, Malaysia. The natural characteristics of DSD are deduced, and the statistical results are found to be in accordance with the findings obtained from others disdrometer measurements. Moreover, the parameters of the Gamma distribution and the normalized Gamma model are also derived by means of method of moment (MoM) and maximum likelihood estimation (MLE). Their performances are subsequently validated using the rain rate estimation accuracy: the normalized Gamma model with the MLE-generated shape parameterµwas found to provide better accuracy in terms of long-term rainfall rate statistics, which reflects the peculiarities of the local climatology in this heavy rain region. These results not only offer a better understanding of the microphysical nature of precipitation in this heavy rain region but also provide essential information that may be useful for the scientific community regarding remote sensing and radio propagation.

scholarly journals The effect of reported high-velocity small raindrops on inferred drop size distributions and derived power laws

2010 ◽  
Vol 10 (4) ◽  
pp. 9121-9151 ◽  
Author(s):  
H. Leijnse ◽  
R. Uijlenhoet
Keyword(s):  
High Velocity ◽  
Drop Size ◽  
Doppler Radar ◽  
Probability Distributions ◽  
Power Laws ◽  
Size Distributions ◽  
Optical Links ◽  
Gamma Distributions ◽  
Raindrop Size Distribution ◽  
Drop Size Distributions

Abstract. It has recently been shown that at high rainfall intensities, small raindrops may fall with much larger velocities than would be expected from their diameters. These were argued to be fragments of recently broken-up larger drops. In this paper we quantify the effect of this phenomenon on raindrop size distribution measurements from a Joss-Waldvogel disdrometer, a 2-D Video Distrometer, and a vertically-pointing Doppler radar. Probability distributions of fall velocities have been parameterized, where the parameters are functions of both rainfall intensity and drop size. These parameterizations have been used to correct Joss-Waldvogel disdrometer measurements for this phenomenon. The effect of these corrections on fitted scaled drop size distributions are apparent but not major. Fitted gamma distributions for three different types of rainfall have been used to simulate drop size measurements. The effect of the high-velocity small drops is shown to be minor. Especially for the purpose of remote sensing of rainfall using radar, microwave links, or optical links, the errors caused by using the slightly different retrieval relations will be masked completely by other error sources.

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