Maximum likelihood estimation in a class of nonregular cases

Smith, Richard L.

doi:10.1093/biomet/72.1.67

Cited by 719 publications

(321 citation statements)

References 29 publications

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“…23,[25][26][27] Because the support of G depends on the unknown parameter values, the usual regularity conditions underlying the asymptotic properties of maximum likelihood estimators are not satisfied. This problem is studied in depth by Stephens 28 . In the case  > -0.5, the usual properties of consistency, asymptotic efficiency and asymptotic normality hold.…”

Section: Maximum Likelihood Methodsmentioning

confidence: 99%

Modelling of extreme minimum rainfall using generalised extreme value distribution for Zimbabwe

Chikobvu¹,

Chifurira²

2015

S. Afr. J. Sci.

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We modelled the mean annual rainfall for data recorded in Zimbabwe from 1901 to 2009. Extreme value theory was used to estimate the probabilities of meteorological droughts. Droughts can be viewed as extreme events which go beyond and/or below normal rainfall occurrences, such as exceptionally low mean annual rainfall. The duality between the distribution of the minima and maxima was exploited and used to fit the generalised extreme value distribution (GEVD) to the data and hence find probabilities of extreme low levels of mean annual rainfall. The augmented Dickey Fuller test confirmed that rainfall data were stationary, while the normal quantile-quantile plot indicated that rainfall data deviated from the normality assumption at both ends of the tails of the distribution. The maximum likelihood estimation method and the Bayesian approach were used to find the parameters of the GEVD. The Kolmogorov-Smirnov and Anderson-Darling goodnessof-fit tests showed that the Weibull class of distributions was a good fit to the minima mean annual rainfall using the maximum likelihood estimation method. The mean return period estimate of a meteorological drought using the threshold value of mean annual rainfall of 473 mm was 8 years. This implies that if in the year there is a meteorological drought then another drought of the same intensity or greater is expected after 8 years. It is expected that the use of Bayesian inference may better quantify the level of uncertainty associated with the GEVD parameter estimates than with the maximum likelihood estimation method. The Markov chain Monte Carlo algorithm for the GEVD was applied to construct the model parameter estimates using the Bayesian approach. These findings are significant because results based on non-informative priors (Bayesian method) and the maximum likelihood method approach are expected to be similar.

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Section: Maximum Likelihood Methodsmentioning

confidence: 99%

Modelling of extreme minimum rainfall using generalised extreme value distribution for Zimbabwe

Chikobvu¹,

Chifurira²

2015

S. Afr. J. Sci.

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“…R functions fgev in package evd or fit.gev in package ismev for example perform this maximum likelihood estimation. When ξ i > −0.5, the maximum likelihood estimate has the usual asymptotic properties (Smith, 1985):μ i ,σ i andξ i are asymptotically unbiased and standard errors are approximately given by the square root of the diagonal of the inverse observed information matrix (Coles, 2001, chapter 3).…”

Section: Pointwise Estimation Of the Gev Distributionsmentioning

confidence: 99%

Mapping snow depth return levels: smooth spatial modeling versus station interpolation

Blanchet¹,

Lehning²

2010

Hydrol. Earth Syst. Sci.

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Abstract. For adequate risk management in mountainous countries, hazard maps for extreme snow events are needed. This requires the computation of spatial estimates of return levels. In this article we use recent developments in extreme value theory and compare two main approaches for mapping snow depth return levels from in situ measurements. The first one is based on the spatial interpolation of pointwise extremal distributions (the so-called Generalized Extreme Value distribution, GEV henceforth) computed at station locations. The second one is new and based on the direct estimation of a spatially smooth GEV distribution with the joint use of all stations. We compare and validate the different approaches for modeling annual maximum snow depth measured at 100 sites in Switzerland during winters 1965-1966 to 2007-2008. The results show a better performance of the smooth GEV distribution fitting, in particular where the station network is sparser. Smooth return level maps can be computed from the fitted model without any further interpolation. Their regional variability can be revealed by removing the altitudinal dependent covariates in the model. We show how return levels and their regional variability are linked to the main climatological patterns of Switzerland.

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“…Consistent with Smith (1985), it is only when j > À0.5 that the resulting maximum likelihood estimators are regular and the usual asymptotic properties of maximum likelihood estimators are valid. For the illustrative case study, the function fevd of the package evd from the R language was used to obtain the maximum partial-likelihood estimators, the standard errors, and the observed information matrix.…”

Section: Extreme Value Distributionsmentioning

confidence: 91%

Extreme value modeling for the analysis and prediction of time series of extreme tropospheric ozone levels: A case study

Escarela

2012

Journal of the Air & Waste Management Association

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The occurrence of high concentrations of tropospheric ozone is considered as one of the most important issues of air management programs. The prediction of dangerous ozone levels for the public health and the environment, along with the assessment of air quality control programs aimed at reducing their severity, is of considerable interest to the scientific community and to policy makers. The chemical mechanisms of tropospheric ozone formation are complex, and highly variable meteorological conditions contribute additionally to difficulties in accurate study and prediction of high levels of ozone. Statistical methods offer an effective approach to understand the problem and eventually improve the ability to predict maximum levels of ozone. In this paper an extreme value model is developed to study data sets that consist of periodically collected maxima of tropospheric ozone concentrations and meteorological variables. The methods are applied to daily tropospheric ozone maxima in Guadalajara City, Mexico, for the period January 1997 to December 2006. The model adjusts the daily rate of change in ozone for concurrent impacts of seasonality and present and past meteorological conditions, which include surface temperature, wind speed, wind direction, relative humidity, and ozone. The results indicate that trend, annual effects, and key meteorological variables along with some interactions explain the variation in daily ozone maxima. Prediction performance assessments yield reasonably good results.

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Maximum likelihood estimation in a class of nonregular cases

Cited by 719 publications

References 29 publications

Modelling of extreme minimum rainfall using generalised extreme value distribution for Zimbabwe

Modelling of extreme minimum rainfall using generalised extreme value distribution for Zimbabwe

Mapping snow depth return levels: smooth spatial modeling versus station interpolation

Extreme value modeling for the analysis and prediction of time series of extreme tropospheric ozone levels: A case study

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