Bruce M. Brown

UNSW Sydney | UNSW · School of Mathematics and Statistics

It is known that the Brownian bridge or L\'evy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the error. In particular, we show for geometric Brownian motion that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the expected...

It is known that the Brownian bridge or L\'evy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the uniform error. In particular, we show constructively that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error a...

A smoothed rank-based procedure is developed for the accelerated failure time model to overcome computational issues. The proposed estimator is based on an EM-type procedure coupled with the induced smoothing. The proposed iterative approach converges provided the initial value is based on a consistent estimator, and the limiting covariance matrix...

Large-sample Wilson-type confidence intervals (CIs) are derived for a parameter of interest in many clinical trials situations: the log-odds-ratio, in a two sample experiment comparing binomial success proportions, say between cases and controls. The methods cover several scenarios: (i) results embedded in a single 2 × 2 contingency table, (ii) a s...

The most general form of analysis in paired experiments investigates whether within-pair differences are stochastically positive, and the most natural measure of stochastic positivity is θ, the expected sign of Walsh averages, with values in [−1, + 1]. The estimate of this effect size measure is asymptotically equivalent to the generalized Wilcoxon...

In many forms of penalty function smoothing, the choice of smoothing parameter is critical. Existing procedures for this choice include cross-validation or variations on likelihood methods. An alternative is introduced here, based on taking an intuitive non-parametric approach. For the linear spline form of penalty function, a Cramér-von Mises test...

Let Z 0 = l, Z 1,Z 2, … denote a super-critical Galton-Watson branching process with 1 <EZ 1 =m <∞ and 0 <varZ 1 = σ2 < ∞It is well-known that there exists a non-degenerate random variable W such that \(\mathop {\lim }\limits_{n \to \infty } W_n = W\) almost surely, where W n = m −n Z n (e.g. Harris [3], p. 13). Also, a central limit analogue holds...

Interruptions and multitasking are implicated as a major cause of clinical inefficiency and error. The aim was to measure the association between emergency doctors' rates of interruption and task completion times and rates. The authors conducted a prospective observational time and motion study in the emergency department of a 400-bed teaching hosp...

The logit-stable distributions are distributions on the unit interval. They have unconventional shapes, are alternatives to beta distributions for random proportions, arise from an axiomatic derivation based upon latent utility scales, have closed-form probabilities, and greater tail-weight than beta distributions. Logit-stable-binomial distributio...

A simple method is introduced for finding large sample, boundary-respecting confidence intervals (CIs) for the two-sample Mann-Whitney measure, θ=Pr{X>Y}-Pr{X<Y}. This natural separation measure for two distributions occurs in stress-strength models, receiver operating characteristic curves, and nonparametrics generally. The usual estimate of θ is...

A modification to the least‐squares estimating equations in simple linear regression, leading to exact distribution‐free inference about slope, is outlined. The modification involves a general residual‐transformation iψ and the cases of ψ— sign, rank, and the identity are discussed in some detail. It is also shown how exact inference about individu...

Edwin Pitman was born on 29th October, 1897, in Melbourne, Victoria. He was Professor of Mathematics at the University of Tasmania from 1926 until his retirement in 1962. His publications on distribution‐free methods in Statistics established a sound basis for the development of these methods. His theoretical papers gave a sound mathematical treatm...

Various asymptotic-based criteria are used to assess the robustness against inhomogeneity of variance of various standard and/or distribution-free tests in the two sample problem. The extent of robustness is influenced greatly by the equality or not of the two sample sizes, conforming to an earlier principle of G. E. P. Box. With equal sample sizes...

For some locations both global and diffuse solar radiation are measured. However, for many locations, only global is measured, or inferred from satellite data. For modelling solar energy applications, the amount of radiation on a tilted surface is needed. Since only the direct component on a tilted surface can be calculated from trigonometry, we ne...

Estimation of scale and index parameters of positive stable laws is considered. Maximum likelihood estimation is known to be efficient, but very difficult to compute, while methods based on the sample characteristic function are computationally easy, but have uncertain efficiency properties. In this paper an estimation method is presented which is...

Estimates in simple linear regression which minimize the sum of absolute values of residuals are discussed. These estimates, called median-estimates, are an analogue of the sample median, and their properties reflect this. A simple graphical procedure, related to an early (1923) one of Edgeworth, is outlined for computing median estimates. This met...

SummaryA general way of testing in the presence of nuisance parameters is to choose from a family of tests the one to maximize evidence against null hypothesis; that is, to minimize the significance level. This method yields exact tests when applied to distribution-free testing in various statistical designs; arbitrary choice of score functions is...

Kendall's tau is a coefficient of concordance between two rankings of n objects. Its definition and large sample normal approximation are easily extended to the case where one of the rankings contains ties. In this paper, definition and normal approximation are extended further to the case where both rankings contain ties. The results are applied t...

Adaptions of weighted rank regression to the accelerated failure time model for censored survival data have been successful in yielding asymptotically normal estimates and flexible weighting schemes to increase statistical efficiencies. However, for only one simple weighting scheme, Gehan or Wilcoxon weights, are estimating equations guaranteed to...

This paper suggests a new type of mixture regression model, in which each mixture component is explained by its own regressors. Thus, the dependent variable can be driven by one of several unobservable explanatory mechanisms, each of which has its own distinct variables. An extension of the simulated annealing algorithm is introduced to fit this ge...

A circularity statistic, based upon pairwise Mann-Whitney statistics, and measuring the non-transitivity effect A > B > C > A, was introduced in Brown and Hettmansperger (2002). In the present paper, its large sample null distribution is shown to be logistic. To test for non-transitivity, possibly indicating the presence of mixture terms, one of th...

A 'pseudo-Bayesian' interpretation of standard errors yields a natural induced smoothing of statistical estimating functions. When applied to rank estimation, the lack of smoothness which prevents standard error estimation is remedied. Efficiency and robustness are preserved, while the smoothed estimation has excellent computational properties. In...

The Polarization Index, developed by Sabavala, Morrison and Kalwani (1977), is a well-known measure of behavioral loyalty. It is of considerable use in analyzing purchase data and repeated choice revealed preference data. Until now, it has only been possible to evaluate the accuracy of estimates or polarization through simulations. We present a clo...

The relationship between seagrass area and the commercial catch of some economically important South Australian fish species is not known in any quantitative sense. However, there is evidence in the literature to suggest that seagrass decline will detrimentally impact on species abundance and composition, and consequently that the presence or absen...

This paper makes the proposition that the only statistical analyses to achieve widespread popular use in statistical practice are those whose formulations are based on very smooth mathematical functions. The argument is made on an empirical basis, through examples. Given the truth of the proposition, the question ‘why should it be so?’ is intriguin...

The integral of the square of the error density function, called integral f-squared, is a factor in the expression for asymptotic standard error of parameter estimates derived from rank methods. Its estimation is considered by substituting a density estimate into the integral. The results of using two very different density estimation methods, base...

The Kruskal–Wallis test is a rank–based one way ANOVA. Its test statistic is shown here to be a quadratic form among the Mann–Whitney or Kendall tau concordance measures between pairs of treatments. But the full set of such concordance measures has more degrees of freedom than the Kruskal–Wallis test uses, and the independent surplus is attributabl...

Even in one dimension the sample median exhibits very poor performance when used in conjunction with the bootstrap. For example, both the percentile-t bootstrap and the calibrated percentile method fail to give second-order accuracy when applied to the median. The situation is generally similar for other rank-based methods, particularly in more tha...

Sullivan suggests that the alternative audit approaches adopted by accounting firms be expressed in terms of “structure” and “judgement”, with a division provided by the degree to which auditor judgement is replaced by structured quantitative algorithms. Cushing and Loebbecke attempt to operationalise this division by examining the guidance provide...

Sullivan (1984) suggests that the alternative audit approaches adopted by accounting firms be expressed in terms of 'structure' and 'judgement', with a division provided by the degree to which auditor judgement is replaced by structured quantitative algorithms. Cushing and Loebbecke (1986) attempt to operationalise this division by examining the gu...

Matching university places to students is not as clear cut or as straightforward as it ought to be. By investigating the matching algorithm used by the German central clearinghouse for university admissions in medicine and related subjects, we show that a procedure designed to give an advantage to students with excellent school grades actually harm...

In conventional empirical likelihood, there is exactly one structural constraint for every parameter. In some circumstances, additional constraints are imposed to reflect additional and sought-after features of statistical analysis. Such an augmented scheme uses the implicit power of empirical likelihood to produce very natural adaptive statistical...

We point out that inliers adversely affect performance of the spatial median and its generalization due to Gentleman. They are most deleterious in the case of the median itself, and in the important setting of two dimensions. There, the second term in a stochastic expansion of the median has a component with a Cauchy limiting distribution, and does...

The bivariate ranks and quantiles based on the bivariate affine equivariant median are considered. Correspondences between two different plots for bivariate data, the direct diagram and the Oja rank plot, are described. Several illustrative examples are given.

In this article we develop new plotting positions for normal plots. The use of the plots usually centers on detecting irregular tail behavior or outliers. Along with the normal plot, we develop tests for various departures from normality, especially for skewness and heavy tails. The tests can be considered as components of a Shapiro-Wilk type test...

A study is made of some unusual location estimates first proposed in 1977 by J. S. Maritz, M. Wu, and R. G. Staudte, Jr., who established some strong robustness properties of these estimates, such as redescending influence functions and, in some cases, full efficiency at the Cauchy model. It is further shown here that the estimates, called M-W-S es...

The first two authors [J. R. Stat. Soc., Ser. B 51, No. 1, 117-125 (1989; Zbl 0675.62036)] proposed an affine invariant bivariate analogue of the sign test, the OS test, based on the generalized median of Oja. On the other hand, the third and fourth author [J. Am. Stat. Assoc. 84, No. 405, 249-259 (1989; Zbl 0677.62043)] introduced a family of loca...

The generalized median of H. Oja [Stat. Probab. Lett. 1, 327-332 (1983; Zbl 0517.62051)] yields a notion of bivariate quantile and, in turn, an affine invariant bivariate analogue of the sign test. Its properties include a simple null covariance formula, facilitating a permutation or sign change test in the case of bivariate symmetry, normal effici...

The generalized multivariate median of Oja (Statist. and Prob. Letters 1(1983), 327–332) is used to define a multivariate notion of quantile, or rank, and to define a measure of scatter of multivariate linear models. The latter, when applied to the one‐ and two‐sample bivariate location models, yields affine invariant analogues of the Wilcoxon rank...

Oja (1983) introduced an affine invariant multivariate median. Brown and Hettsmansperger (1987) developed affine rank methods in the bivariate location based on Oja's measure of scatter. Analogs of the Wilcoxon signed rank and the Mann-Whitney-Wilcoxon rank sum tests were presented. This paper discusses scale, rotation and affine invariance. Tests...

In regression models having symmetric errors, exact distribution-free inference about individual parameters may be carried out by grouping observations, eliminating unwanted parameters within groups, and applying distribution free techniques for the symmetric location parameter problem. Models whose errors have identical but not symmetric distribut...

The spatial median, called the mediancentre in an earlier paper by J. C. Gower, is defined as the bivariate location measure to minimize the sum of absolute distances to observations. its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2/π = 0.637. its estimating equations have an an...

In the k-sample problem, or one-way analysis of variance, a multiple response permutation procedure based on within-group sums of absolute rank differences is shown to have a certain large sample limit distribution. This distribution is the convolution of k— 1 copies of the usual Cramér—von Mises distribution, and so is a Cramér-von Mises analogue...

A moment method based on the sample averages of X and X (with suitably chosen t) is suggested for estimating the index a and scale parameter c of the positive stable distribution with Laplace transform E exp(-ΛX) = exp(-cΛ). The corresponding Cramér-Rao lower bounds are computed by using a new and simple Fourier technique, which is based on an idea...

The average of two complementary order statistics is termed a symmetric quantile average. In addition to having quite good efficiency, robustness and computational properties, exact nonparametric confidence intervals for location can be derived from it. A related type of estimator called a combined quantile average is introduced; it is slightly mor...

The asymptotic structure of a vector of weighted sums of signs of residuals, in the general linear model, is studied. The vector can be used as a basis for outlier-detection tests, or alternatively, setting the vector to zero and solving for the parameter yields a class of robust estimators which are analogues of the sample median. Asymptotic resul...

A reduced $U$-statistic (of order 2) is defined as the sum of terms $f(X_i, X_j),$ where $f$ is a symmetric function, $(X_1, \cdots, X_N)$ are independent and identically distributed (i.i.d.) random variables (rv's), and $(i,j)$ are drawn from a restricted, though balanced, set of pairs. (A $U$-statistic corresponds to summation over all $(i, j)$ p...

A procedure is developed by which the variance of each fitted parameter of a watershed model can be determined. This information can be used to determine the degree to which the model parameters can be related to physical watershed characteristics. The method is illustrated by applying the Boughton model to several watersheds and examining the coef...

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit...

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit...

A statistical analysis is presented whereby the sensitivity of each parameter of a mathematical rainfall-runoff model can be estimated. Such an analysis is useful in assessing the potential of a mathematical model to predict the effects of land-use change on catchment response, and the possibility of relating a model's parameters to physical catchm...

SUMMARY The differences between the numbers of positive and negative residuals in each row and column of a two-way analysis of variance are used to define an approximate chi-squared statistic which can be used to test for the presence of outlying observations. The chi-squared approximation rests on a Central Limit Theorem for the number of positive...

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we a...

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we a...

The large-sample behaviour is investigated of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow e...

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow ea...

Let S = {Sn, n [greater-or-equal, slanted] 1} be a martingale. Expectations of mth order quantities associated with S are related by two forms of Wald-type identity, called Generalized Wald equations. The previously known sufficient conditions for the validity of Wald equations are shown to be of a set of three equivalent conditions, each of which...

One classical form of the central limit problem considers the asymptotic distribution of weighted sums of independent, identically distributed random variables (r.vs). In this paper we consider the asymptotic distribution of such sums where the constant weights are replaced by a sequence of non-negative r.vs, i.e. we consider the sum of weighted, i...

If T is a stopping time on a martingale {St}, the mth order moments of T and ST (for integers m ≥ 2) are related by an identity known as a Wald equation of order m. Wald equations hold under certain conditions, which are simplified in this paper by showing that among the set of previously known conditions, some are implied by the others.

If T is a stopping time on a martingale {St }, the mth order moments of T and ST (for integers m ≧ 2) are related by an identity known as a Wald equation of order m. Wald equations hold under certain conditions, which are simplified in this paper by showing that among the set of previously known conditions, some are implied by the others.

The purpose of this note is to derive an alternative expression to that given in Lemma 1 below (due to Hsu, and to von Bahr) for the absolute moments of a random variable, in terms of the characteristic function.

Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martingales within rows, and the results are obtained under conditions which parallel those of the classical case of convergence in distribution, to infinitely...

Let $\{\Omega, \mathscr{F}, P \}$ be a probability space. The subset of $\Omega$ on which an arbitrary sequence of random variables converges is shown to be equivalent to the intersection of three other sets, each specified by the almost sure convergence of a certain sequence of random variables. Kolmogorov's three-series theorem, which gives neces...

The classical Lindeberg-Feller CLT for sums of independent random variables (rv's) provides more than the convergence in distribution of the sum to a normal law. The independence of summands also guarantees the weak convergence of all finite dimensional distributions of an a.e. sample continuous stochastic process (suitably defined in terms of the...

Let {X n Fn ,n = 0, 1, 2, …} be a martingale with X 0 = 0 a.s., \(X_n = \sum {_{i = 1}^n Y_i,n\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 1} \), n ≧ 1, and Fn the σ-field generated by X 0, X 1, …, X n . Write $$\begin{array}{lr} \sigma^{2}_{n} = E(Y^2_n | \mathcal{F}_{n-1}), & S^{2}_{n} = \sum\nolimits^{n}_{i=1} E\sigma^...

In [2], Lindeberg conditions of order $\nu \geqq 2$ are defined and shown to be NSC for convergence of $\nu$th absolute moments in the Central Limit Theorem when $\nu = 2k, k = 2,3, \cdots$. Section 4 contains the extension of that result to the case of all $\nu > 2$, the proof depending on some of the theorems, given in Section 2, relating the exi...