B.R. Clarke

Nonsmooth analysis and Fréchet differentiability of M-functionals

Abstract : A necessary requirement for existence of the Frechet derivative is that the defining psi function is uniformly bounded, and this naturally excludes those nonrobust estimators such as the maximum likelihood estimator in normal parametric models. In this paper the methods of nonsmooth analysis, described in the book by F.H. Clarke (1983), are introduced to the theory of statistical expansions, and are used here in the proofs of weak continuity and Frechet differentiability of M-functionals. Subsequently the conditions for Frechet differentiability given in Clarke (1983) can be relaxed to include most popular M-functionals. Additional keywords: distribution functions; M-estimators; robustness; gross error sensitivity; weak continuity; asymptotic expansions; asymptotic normality; selection functional; local uniqueness. (Author).

Outcomes of Usual Chiropractic. The OUCH Randomized Controlled Trial of Adverse Events

Study Design. Blinded parallel-group randomized controlled trial. Objective. Establish the frequency and severity of adverse effects from short-term usual chiropractic treatment of the spine when compared with a sham treatment group. Summary of Background Data. Previous studies have demonstrated that adverse events occur during chiropractic treatment. However, as a result of design limitations in previous studies, particularly the lack of sham-controlled randomized trials, understanding of these adverse events and their relation with chiropractic treatment is suboptimal. Methods. We conducted a trial to examine the occurrence of adverse events resulting from chiropractic treatment. It was conducted across 12 chiropractic clinics in Perth, Western Australia. The participants comprised 183 adults, aged 20 to 85 years, with spinal pain. Ninety-two participants received individualized care consistent with the chiropractors’ usual treatment approach; 91 participants received a sham intervention. Each participant received 2 treatments. Results. Completed adverse questionnaires were returned by 94.5% of the participants after appointment 1 and 91.3% after appointment 2. Thirty-three percent of the sham group and 42% of the usual care group reported at least 1 adverse event. Common adverse events were increased pain (sham 29%; usual care 36%), muscle stiffness (sham 29%; usual care 37%), and headache (sham 17%; usual care 9%). The relative risk (RR) was not significant for adverse event occurrence (RR = 1.24; 95% CI: 0.85–1.81), occurrence of severe adverse events (RR = 1.9; 95% CI: 0.98–3.99), adverse event onset (RR = 0.16; 95% CI: 0.02–1.34), or adverse event duration (RR = 1.13; 95% CI: 0.59–2.18). No serious adverse events were reported. Conclusion. A substantial proportion of adverse events after chiropractic treatment may result from natural history variation and nonspecific effects. Level of Evidence: 2

Statistical expansions and locally uniform Fréchet differentiability

Estimators which have locally uniform expansions are shown in this paper to be asymptotically equivalent to M-estimators. The M-functionals corresponding to these M-estimators are seen to be locally uniformly Frechet differentiable. Other conditions for M-functionals to be locally uniformly Frechet differentiable are given. An example of a commonly used estimator which is robust against outliers is given to illustrate that the locally uniform expansion need not be valid.

An unbiased minimum distance estimator of the proportion parameter in a mixture of two normal distributions

An estimator that minimizes an L2 distance used in studies of estimation of the location parameter is shown here to give an explicit formulation for the estimator of proportion in a mixture of two normal distributions when other parameters are known. This can prove to be an advantage over other minimum distance methods and the maximum likelihood estimator. Monte Carlo simulation demonstrates this and highlights good small sample behaviour of the estimator. It is shown that the estimator is also qualitatively robust both empirically and asymptotically, the latter being evidenced by the existence of a Frechet derivative.

A Single Series Representation of Multiple Independent ARMA Processes

This article shows that multiple independent time series from the same ARMA process can be represented by a single univariate ARMA time series through an interleaving of the original series. Using this result, existing univariate modelling software can be used to fit a single ARMA time series model simultaneously to multiple independent realizations of the same ARMA process. The interleaving approach and its properties will be presented and compared with alternative estimation options. It will be applied to the modelling of 66 years of daily maximum temperatures for Perth, Western Australia and to other time series models.

A remark on robustness and weak continuity of M-estimators

Global weak continuity of M-functionals in a neighbourhood of the parametric distribution is established. This has implications for robustness of M-estimators vis a vis definitions put forward by Hampel. For instance the Tukey bisquare location estimator is robust on neighbourhoods of the parametric model, but the median is not.

Post-emergent herbicidal activity of cineole derivatives

Essential oils are being investigated as potential herbicides or to provide leads to new environmentally and socially acceptable herbicides. Novel hydroxy and ester derivatives of 1,8-cineole and 1,4-cineole were synthesised, by chemical and biological methods, and have shown pre-emergence herbicidal activity against annual ryegrass and radish. Effects on post-emergence activity of these derivatives, as well as 1,8-cineole, eucalyptus oil and the carboxylic acids from which the esters were derived, against annual ryegrass and radish, are reported here. Results suggest that reduced root and shoot growth observed in pre-emergence herbicidal bioassays were due to post-emergence activity rather than delayed germination. All tested substances had a dose-dependent, post-emergence herbicidal activity against annual ryegrass and radish with many derivatives showing improved activity relative to 1,8-cineole and high-cineole eucalyptus oil. However, results do not support the postulate that cineole esters would be more active than their respective carboxylic acid and the hydroxy cineole. Phytotoxicity of ester derivatives may be due to metabolic cleavage of the esters to the hydroxy cineole and carboxylic acid within the plant.

Uncorrelated residuals and an exact test for two variance components in experimental design

The error contrasts from an experimental design can be constructed from uncorrelated residuals normally associated with the linear model. In this paper uncorrelated residuals are defined for the linear model that has a design matrix which is less than full rank, typical of many experimental design representations. It transpires in this setting, that for certain choices of uncorrelated residuals, corresponding to recursive type residuals, there is a natural partition of information when two variance components are known to be present. Under an assumtion of normality of errors this leads to construction of appropriate F-tests for testing heteroscedasticity. The test, which can be optimal, is applied to two well known data sets to illustrate its usefullness.

Asymptotic theory for description of regions in which Newton-Raphson iterations converge to location M-estimators

By examination of properties of the function of the sample and location parameter that describes the M-estimating equation for location a detailed probabilistic description of regions from which the Newton-Raphson algorithm converges to the location M-estimator is given. In finite samples an asymptotic distribution can be used to approximate the size of the region from which convergence occurs. This gives experimenters an idea of the tolerance acceptable for an initial estimate to begin an iteration. Empirical evidence is given which highlights the results.

The benefits and limitations of the sperm quality analyzer IIB in assessing porcine sperm motility

The Sperm Quality Analyzer (SQA) IIB, a member of the SQA-II family of machines which uses the scatter of light by sperm as an indicator of sperm motility, was systematically evaluated as a means of analyzing objectively the motility of porcine epididymal sperm. The sperm motility (%) and the Sperm Motility Index (SMI) are calculated by the machine using pre-programmed algorithms designed for human sperm. The machine performed well and was able to detect changes in sperm motility under experimental conditions. However, two major limitations of this machine were identified, (i) the readings obtained were influenced by the concentration of the sperm suspension despite the actual sperm motility remaining constant, and (ii) the machine was unable to differentiate between progressive and non-progressive motility. It should therefore be recognized that (a) the sperm concentration must be kept constant in studies in vitro if differences between treatment groups are to be identified, and (b) the inability to separate progressive motility from that of total motility will restrict the usefulness of this and similar machines to studies monitoring changes in total motility alone.