Papers by Brenton R Clarke
Nonsmooth analysis and Fr'echet differentiability of Mfunctionals. Research Report
primary 62E20
Fitting replicated multivariate time series models, extra supplementary material
Comment on "Estimating Mixtures of Normal Distributions and Switching Regressions
Journal of the American Statistical Association, 1979
We wish to commend Professors Quandt and Ramsey for presenting an article (Quandt and Ramsey 1978... more We wish to commend Professors Quandt and Ramsey for presenting an article (Quandt and Ramsey 1978) which is at once stimulating and certain to cause controversy. The problem of the estimation of the parameters of the mixture of normals is an old one which has resisted completely satisfactory resolution; the last word has not yet been heard on this problem. Presumably, more than a few more will be generated by the appearance of this article. The density in question is:
On a note on robustness of the ß-trimmed mean (Corrigendum)
Australian & New Zealand Journal of Statistics, 2017
Long-term historical daily temperatures are used in electricity forecasting to simulate the proba... more Long-term historical daily temperatures are used in electricity forecasting to simulate the probability distribution of future demand but can be affected by changes in recording site and climate. This paper presents a method of adjusting for the effect of these changes on daily maximum and minimum temperatures. The adjustment technique accommodates the autocorrelated and bivariate nature of the temperature data which has not previously been taken into account. The data are from Perth, Western Australia, the main electricity demand centre for the SouthWest of Western Australia. The statistical modelling involves a multivariate extension of the univariate time series 'interleaving method', which allows fully efficient simultaneous estimation of the parameters of replicated Vector Autoregressive Moving Average processes. Temperatures at the most recent weather recording location in Perth are shown to be significantly lower compared to previous sites. There is also evidence of long-term heating due to climate change especially for minimum temperatures.
Least Squares Theory
Wiley series in probability and statistics, Aug 8, 2008
This chapter contains sections titled: The Normal Equations The Gauss–Markov Theorem The Distribu... more This chapter contains sections titled: The Normal Equations The Gauss–Markov Theorem The Distribution of SΩ Some Simple Significance Tests Prediction Intervals Problems The Normal EquationsThe Gauss–Markov TheoremThe Distribution of SΩSome Simple Significance TestsPrediction IntervalsProblems
We provide a general approach for deriving the influence function for trimmed likelihood estimato... more We provide a general approach for deriving the influence function for trimmed likelihood estimators using the implicit function theorem. The approach is applied to lifetime models with exponential or lognormal distributions possessing a linear or nonlinear link function. A side result is that the functional form of the trimmed estimator for location and linear regression used by Bednarski and Clarke (1993, 2002) and Bednarski et al. (2010) is not generally always the correct functional form of a trimmed likelihood estimator. However, it is a version for which the influence function has a treatable form. A real data example shows the effect of trimming using a nonlinear link function for either the exponential or lognormal distribution.
The effects of changes in location and climate on weather records used for electricity demand simulation, extra supplementary material
Statistical Papers
This paper makes comparisons of automated procedures for robust multivariate outlier detection th... more This paper makes comparisons of automated procedures for robust multivariate outlier detection through discussion and simulation. In particular, automated procedures that use the forward search along with Mahalanobis distances to identify and classify multivariate outliers subject to predefined criteria are examined. Procedures utilizing a parametric model criterion based on a $$\chi ^2$$ χ 2 -distribution are among these, whereas the multivariate Adaptive Trimmed Likelihood Algorithm (ATLA) identifies outliers based on an objective function that is derived from the asymptotics of the location estimator assuming a multivariate normal distribution. Several criterion including size (false positive rate), sensitivity, and relative efficiency are canvassed. To illustrate relative efficiency in a multivariate setting in a new way, measures of variability of the multivariate location parameter when the underlying distribution is chosen from a multivariate generalization of the Tukey–Huber...
Asymptotic Theory for Multiple Solutions of Likelihood Equations with Application to a Single Ion Channel Model
Blackwell Publishing, 1993
This paper investigates local asymptotic theory when there are multiple solutions of the likeliho... more This paper investigates local asymptotic theory when there are multiple solutions of the likelihood equations due to non-identifiability arising as a consequence of incomplete information. Local asymptotic results extend in a stochastic way the notion of a Frechet expansion, which is then used to prove asymptotic normality of the maximum likelihood estimator under certain conditions. The local theory, which provides simultaneous consistency and asymptotic normality results, is applied to a bivariate exponential model exhibiting non-identifiability. This statistical model arises as an approximation to the distribution of observable sojourn-times in the states of a two-state Markov model of a single ion channel and the non-identifiability is a consequence of an inability to record sojourns less than some small detection limit. For each of two possible estimates, confidence intervals are constructed using the asymptotic theory. Although it is not possible to decide between the two estimates on the basis of a single realization, an appropriate solution of the likelihood equations may be found using the additional information contained in data based on two different detection limits; this is investigated numerically and by simulation.
TRIALS STUDY PROTOCOL Open Access Outcomes of Usual Chiropractic, Harm & efficacy
the OUCH study: study protocol for a randomized
The introduction of the Frechet derivative into statistics from the mathematical literature has s... more The introduction of the Frechet derivative into statistics from the mathematical literature has spomed a wealth of study of the relationships between robust statistics and mathematical• derivatives. The current view is that differentiability of the Frechet type with respect to the supremum norm is a valid approach and necessarily some well known statistics such as the arithmetic mean and the median do not have the attribute of differentiability in this sense. As an illustration of a Frechet differentiable estimating functional an L2 minimum distance estimator is applied to some data collected from seismic analysis and this provides estimates in an area where estimation previously was ad hoc and estimation was not always possible. The theoretical estimation techniques known from the study of mixtures of normal distributions are applied to this data and illustrate some of the features of using robust techniques as opposed to always resorting to the maximum likelihood estimation methods.
ASC2010 Statistics in the West: Understanding our world, including OZCOTS 2010 - The 7th Australian Conference on Teaching Statistics: program and abstracts
Biostatistics and Biometrics Open Access Journal, 2018
The trial assessed the accuracy of anaesthesia clinicians in estimating an anaesthetised patient'... more The trial assessed the accuracy of anaesthesia clinicians in estimating an anaesthetised patient's systolic blood pressure (SBP) by feeling the radial pulse. To credit their accuracy to luck, skill or circumstance, the volunteer medical participants were sequentially randomized to one of four groups: one group given no help (control), the second allowed to feel the pulse, the third given pre-and peri-operative clinical information, the fourth given both. We set out to collect 60 estimates for each group (240 estimates). The accuracy of their estimations was assessed to clinical and statistical significance. Specific objectives were to determine whether palpation statistically improved estimation of SBP and whether it could be clinically useful. Irrespective of the level of training or self-confidence, the doctors in the study performed better statistically against controls and to within pre-determined clinical relevance ranges when they were allowed to palpate the radial pulse. The degree of accuracy was enhanced by giving pre-and peri-operative information to the extent that the participant clinicians were able to estimate the systolic blood pressure to within 30mmHg accuracy 96.7% of the time.
Time series analysis of hourly ambient temperatures
This paper shows that the interleaving of replicated multiple time series allows the estimation m... more This paper shows that the interleaving of replicated multiple time series allows the estimation methods available in standard multiple time series packages to be applied simultaneously to each of the replicated series without loss of information. The methodology employs a non-trivial multivariate extension of an earlier univariate result involving interleaving. The interleaving approach is used to model more than sixty years of daily maximum and minimum temperatures for Perth, Western Australia.
The method of maximum likelihood using the EM-algorithm for fitting finite mixtures of normal dis... more The method of maximum likelihood using the EM-algorithm for fitting finite mixtures of normal distributions is the accepted method of estimation ever since it has been shown to be superior to the method of moments. Recent books testify to this. There has however been criticism of the method of maximum likelihood for this problem, the main criticism being when the variances of component distributions are unequal the likelihood is in fact unbounded and there can be multiple local maxima. Another major criticism is that the maximum likelihood estimator is not robust. Several alternative minimum distance estimators have since been proposed as a way of dealing with the first problem. This paper deals with one of these estimators which is not only superior due to its robustness, but in fact can have an advantage in numerical studies even at the model distribution. Importantly, robust alternatives of the EM-algorithm, ostensibly fitting t distributions when in fact the data are mixtures of...
Testing for outliers in gold assay data; unweighted and weighted samples
My involvement in this workshop has had its origins in a piece of consulting work with a gold min... more My involvement in this workshop has had its origins in a piece of consulting work with a gold mining company, Company A, in 1989. I received a fax from a concerned metallurgist who was responsible for overseeing the contracts and processing for ore crushed at the Company A Mine. The metallurgist was assessing the mean recoverable gold content from ore that was being purchased from Company B. Payment was on the basis of the assessed mean recoverable gold content less 2/t treatment charge. The problem is to determine the "best estimate" of the mean recoverable gold content.
In literature several estimates have been proposed for unimodal densities. They are typically der... more In literature several estimates have been proposed for unimodal densities. They are typically derived from the Grenander estimate (see Grenander (1956)) for decreasing densities which can be easily extended to the case of unimodal densities with a known mode at . The resulting estimator is the nonparametric maximum likelihood estimator (NPMLE). If the mode is unknown, the NPMLE does not exist anymore. One possible solution that has been proposed by Wegman (1969, 1970a,b) is to add the additional constraint of a modal interval of length , where has to be chosen by the statistician. More recently Bickel and Fan (1996) and Birg e (1997) proposed methods that are based on an initial mode estimate ̂ and an application of the NPMLE with mode ̂. The initial mode estimate requires the calculation of the MLE of f for O(n) candidate points for the mode, where n denotes the sample size. They showed their methods to provide good estimates both for nonsmooth densities and for the mode of nonsm...
Australian & New Zealand Journal of Statistics, 2017
For the lifetime (or negative) exponential distribution, the trimmed likelihood estimator has bee... more For the lifetime (or negative) exponential distribution, the trimmed likelihood estimator has been shown to be explicit in the form of a β-trimmed mean which is representable as an estimating functional that is both weakly continuous and Frechet differentiable and hence qualitatively robust at the parametric model. It also has high efficiency at the model. The robustness is in contrast to the maximum likelihood estimator (MLE) involving the usual mean which is not robust to contamination in the upper tail of the distribution. When there is known right censoring, it may be perceived that the MLE which is the most asymptotically efficient estimator may be protected from the effects of ‘outliers’ due to censoring. We demonstrate that this is not the case generally, and in fact, based on the functional form of the estimators, suggest a hybrid defined estimator that incorporates the best features of both the MLE and the β-trimmed mean. Additionally, we study the pure trimmed likelihood e...
Uploads
Papers by Brenton R Clarke