We demonstrate that, under a theorem proposed by Vuong, the likelihood ratio statistic based on the Kullback-Leibler information criterion of the null hypothesis that a random sample is drawn from a k 0 -component normal mixture distribution against the alternative hypothesis that the sample is drawn from a k 1 -component normal mixture distribution is asymptotically distributed as a weighted sum of independent chi-squared random variables with one degree of freedom, under general regularity conditions. We report simulation studies of two cases where we are testing a single normal versus a two-component normal mixture and a two-component normal mixture versus a three-component normal mixture. An empirical adjustment to the likelihood ratio statistic is proposed that appears to improve the rate of convergence to the limiting distribution.

Testing the number of components in a normal mixture

This paper develops a framework for performing estimation and inference in econometric models with partial identification, focusing particularly on models characterized by moment inequalities and equalities. Applications of this framework include the analysis of game-theoretic models, revealed preference restrictions, regressions with missing and corrupted data, auction models, structural quantile regressions, and asset pricing models. Specifically, we provide estimators and confidence regions for the set of minimizers Θ I of an econometric criterion function Q(θ). In applications, the criterion function embodies testable restrictions on economic models. A parameter value θ that describes an economic model satisfies these restrictions if Q(θ) attains its minimum at this value. Interest therefore focuses on the set of minimizers, called the identified set. We use the inversion of the sample analog, Q n (θ), of the population criterion, Q(θ), to construct estimators and confidence regions for the identified set, and develop consistency, rates of convergence, and inference results for these estimators and regions. To derive these results, we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification.

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Estimation and Confidence Regions for Parameter Sets in Econometric Models

Current practice using probabilistic methods applied for designing hydraulic structures generally assume that extreme events are stationary. However, many studies in the past decades have shown that hydrological records exhibit some type of nonstationarity such as trends and shifts. Human intervention in river basins (e.g., urbanization), the effect of low-frequency climatic variability (e.g., Pacific Decadal Oscillation), and climate change due to increased greenhouse gasses in the atmosphere have been suggested to be the leading causes of changes in the hydrologic cycle of river basins in addition to changes in the magnitude and frequency of extreme floods and extreme sea levels. To tackle nonstationarity in hydrologic extremes, several approaches have been proposed in the literature such as frequency analysis, in which the parameters of a given model vary in accordance with time. The aim of this paper is to show that some basic concepts and methods used in designing flood-related hydraulic structures assuming a stationary world can be extended into a nonstationary frame- work. In particular, the concepts of return period and risk are formulated by extending the geometric distribution to allow for changing exceeding probabilities over time. Building on previous developments suggested in the statistical and climate change literature, the writers present a simple and unified framework to estimate the return period and risk for nonstationary hydrologic events along with examples and applications so that it can be accessible to a broad audience in the field. The applications demonstrate that the return period and risk estimates for nonstationary situations can be quite different than those corresponding to stationary conditions. They also suggest that the nonstationary analysis can be helpful in making an appropriate assessment of the risk of a hydraulic structure during the planned project-life. DOI: 10.1061/ (ASCE)HE.1943-5584.0000820. © 2014 American Society of Civil Engineers.

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Revisiting the Concepts of Return Period and Risk for Nonstationary Hydrologic Extreme Events

https://eprints.ncl.ac.uk/file_store/production/208845/37006749-05A2-482D-ADFB-1EB877CBF259.pdf

Stationarity is undead: Uncertainty dominates the distribution of extremes

The ability of a large ensemble of regional climate models to accurately simulate heat waves at the regional scale of Europe was evaluated. Within the EURO-CORDEX project, several state-of-the art models, including non-hydrostatic meso-scale models, were run for an extended time period (20 years) at high resolution (12 km), over a large domain allowing for the first time the simultaneous representation of atmospheric phenomena over a large range of spatial scales. Eight models were run in this configuration, and thirteen models were run at a classical resolution of 50 km. The models were driven with the same boundary conditions, the ERA-Interim re-analysis, and except for one simulation, no observations were assimilated in the inner domain. Results, which are compared with daily temperature and precipitation observations (ECA&D and E-OBS data sets) show that, even forced by the same re-analysis, the ensemble exhibits a large spread. A preliminary analysis of the sources of spread, using in particular simulations of the same model with different parameterizations, shows that the simulation of hot temperature is primarily sensitive to the convection and the microphysics schemes, which affect incoming energy and the Bowen ratio. Further, most models exhibit an overestimation of summertime temperature extremes in Mediterranean regions and an underestimation over Scandinavia. Even after bias removal, the simulated heat wave events were found to be too persistent, but a higher resolution reduced this deficiency. The amplitude of events as well as the variability beyond the 90th percentile threshold were found to be too strong in almost all simulations and increasing resolution did not generally improve this deficiency. Resolution increase was also shown to induce large-scale 90th percentile warming or cooling for some models, with beneficial or detrimental effects on the overall biases. Even though full causality cannot be established on the basis of this evaluation work, the drivers of such regional differences were shown to be linked to changes in precipitation due to resolution changes, affecting the energy partitioning. Finally, the inter-annual sequence of hot summers over central/southern Europe was found to be fairly well simulated in most experiments despite an overestimation of the number of hot days and of the variability. The accurate simulation of inter-annual variability for a few models is independent of the model bias. This indicates that internal variability of high summer temperatures should not play a major role in controlling inter-annual variability. Despite some improvements, especially along coastlines, the analyses conducted here did not allow us to generally conclude that a higher resolution is clearly beneficial for a correct representation of heat waves by regional climate models. Even though local-scale feedbacks should be better represented at high resolution, combinations of parameterizations have to be improved or adapted accordingly.

https://meetingorganizer.copernicus.org/EGU2013/EGU2013-2657-1.pdf

The simulation of European heat waves from an ensemble of regional climate models within the EURO-CORDEX project

In this paper, we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models, with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model. We give two major applications: the case where the number of populations has to be tested in a mixture and the case of stationary ARMA$(p, q)$ processes where the order $(p, q)$ has to be tested. We give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model. In the case of order selection for ARMAs, the asymptotic distribution is invariant with respect to the parameters generating the process. A locally conic parametrization is a key tool in deriving the limiting distributions; it allows one to discover the deep similarity between the two problems.

/pdf/testing-the-order-of-a-model-using-locally-conic-5f1oxqtnlx.pdf

Testing the order of a model using locally conic parametrization : population mixtures and stationary ARMA processes

The existence of an increasing trend in average temperatures during the last 50 years is widely acknowledged. Furthermore, there is compelling evidence of the variability of extremes, and rapid strides are made in studies of these events. Indeed, by extending the results of the “extreme value theory” (EVT) to the non-stationary case, analyses can examine the presence of trends in extreme values of stochastic processes. Definition of extreme events, their statistical significance as well as their interpretations have to be handled with great care when used for environmental concerns and public safety. Thus, we will discuss the validity of the hypothesis allowing the use of mathematical theories for these problems. To answer safety requirements, respect installation norms and reduce public risk, return levels are a major operational goal, obtained with the EVT. In this paper, we give quantitative results for observations of high temperatures over the 1950–2003 period in 47 stations in France. We examined the validity of the non-stationary EVT and introduced the notion of return levels (RL) in a time-varying context. Our analysis puts particular accent on the difference between methods used to describe extremes, to perform advanced fits and tests (climatic science), and those estimating the probability of rare future events (security problems in an evolving climate). After enouncing the method used for trend identification of extremes in term of easily interpretable parameters of distribution laws, we apply the procedure to long series of temperature measurements and check the influence of data length on trend estimation. We also address the problem of choosing the part of observations allowing appropriate extrapolation. In our analysis, we determined the influence of the 2003 heat wave on trend and return-level estimation comparing it to the RL in a stationary context. The application of the procedure to 47 stations spread over France is a first step for a refined spatial analysis. Working on the behavior of distribution parameters while assessing trend identification is a primary tool in order to classify climatic change with respect to the location of the station and open a systematic work using the same methodology for other variables and multivariate studies.

/pdf/trends-and-climate-evolution-statistical-approach-for-very-3jhytfg100.pdf

Trends and climate evolution : Statistical approach for very high temperatures in France

In this paper, we address the problem of testing hypotheses using maximum likelihood statistics in non identifiable models. We derive the asymptotic distribution under very general assumptions.
 The key idea is a local reparameterization, depending on the underlying distribution, which is called locally conic. This method enlights how the general model induces the structure of the limiting distribution in terms of dimensionality of some derivative space. We present various applications of the theory. The main application is to mixture models. Under very general assumptions, we solve completely the problem of testing the size of the mixture using maximum likelihood statistics. We derive the asymptotic distribution of the maximum likelihood statistic ratio which takes an unexpected form.

/pdf/testing-in-locally-conic-models-and-application-to-mixture-1wmgqholy2.pdf

Testing in locally conic models, and application to mixture models

We propose a new method to estimate the number of different populations when a large sample of a mixture of these populations is observed. It is possible to define the number of different populations as the number of points in the support of the mixing distribution. For discrete distributions having a finite support, the number of support points can be characterized by Hankel matrices of the first algebraic moments, or Toeplitz matrices of the trigonometric moments. Namely, for one-dimensional distributions, the cardinality of the support may be proved to be the least integer such that the Hankel matrix (or the Toeplitz matrix) degenerates. Our estimator is based on this property. We first prove the convergence of the estimator, and then its exponential convergence under wide assumptions. The number of populations is not a priori bounded. Our method applies to a large number of models such as translation mixtures with known or unknown variance, scale mixtures, exponential families and various multivariate models. The method has an obvious computational advantage since it avoids any computation of estimates of the mixing parameters. Finally we give some numerical examples to illustrate the effectiveness of the method in the most popular cases.

/pdf/the-estimation-of-the-order-of-a-mixture-model-4y5fjmsszo.pdf

The estimation of the order of a mixture model

[1] The important role of the evolution of mean temperature in the changes of extremes has been recently documented in the literature, and variability is known to play a role in the occurrence of extremes, too. This paper aims at further investigating the role of their evolutions in the observed changes of temperature extremes. Analyses are based on temperature time series for Eurasia and the United States and concern absolute minima in winter and absolute maxima in summer of daily minimum and maximum temperatures. A test is designed to check whether the extremes of the residuals after accounting for a time-varying mean and standard deviation can be considered stationary. This hypothesis is generally true for all extremes, seasons, and locations. Then, the comparison between the directly fitted parameters and the retrieved ones from those of the residuals compares favorably. Finally, a method is proposed to compute future return levels from the stationary return levels of the residuals and the projected mean and variance at the desired time horizon. Comparisons with return levels obtained through the extrapolation of significant linear trends identified in the parameters of the generalized extreme value (GEV) distribution show that the proposed method gives relevant results. It allows taking mean and/or variance trends into account in the estimation of extremes even though no significant trends in the GEV parameters can be identified. Moreover, the role of trends in variance cannot be neglected. Lastly, first results based on two CMIP5 climate models show that the identified link between mean and variance trends and trends in extremes is correctly reproduced by the models and is maintained in the future.

/pdf/the-importance-of-mean-and-variance-in-predicting-changes-in-10hlnl84q3.pdf

Didier Dacunha-Castelle

Papers

Testing the order of a model using locally conic parametrization : population mixtures and stationary ARMA processes

Trends and climate evolution : Statistical approach for very high temperatures in France

Testing in locally conic models, and application to mixture models

The estimation of the order of a mixture model

The importance of mean and variance in predicting changes in temperature extremes