Set estimation and nonparametric detection

The authors consider the estimation of a set S ? R^d from a random sample of n points. They examine the properties of a detection method, proposed by Devroye & Wise (1980), which relies on the use of a “naive” estimator of S defined as a union of balls centered at the sample points with common radius ?_n. They obtain the convergence rate for the probability of false alarm and show that the smoothing parameter ?_n can be used to incorporate some prior information on the shape of S. They suggest two general methods for selecting ?_n and illustrate them with a simulation study and a real data example.     

Multivariate control chart based on the highest possibility region

The T ² control chart is widely adopted in multivariate statistical process control. However, when dealing with asymmetrical or multimodal distributions using the traditional T ² control chart, some points with relatively high occurrence possibility might be excluded, while some points with relatively low occurrence possibility might be accepted. Motived by the thought of the highest posterior density credible region, we develop a control chart based on the highest possibility region to solve this problem. It is shown that the proposed multivariate control chart will not only meet the false alarm requirement, but also ensure that all the in-control points are with relatively high occurrence possibility. The advantages and effectiveness of the proposed control chart are demonstrated by some numerical examples in the end.     

Identifying change-point in polynomial profiles based on data-segmentation

Profile monitoring is the use of control charts for cases in which the quality of a process or product can be characterized by a functional relationship between a response variable and one or more explanatory variables. Unlike the linear profile's simple structure, the nonlinear profile has relatively less attainment because of high complexity. Regression model is the initial method to analyze the phase I of nonlinear profiles, but it lacks sensitivity for local characteristic changes. This article presents a strategy comprising two major components: data-segmentation, to concisely detect the location of local change by overlaying grid points onto horizontal axis, and change-point detection via the maximum likelihood estimate. Simulated data set of a polynomial profile is used to illustrate the effectiveness of the proposed strategy, and is compared with Williams' T² multi-variable statistics.     

SOME PROPERTIES OF A FOLDOVER DESIGN

The foldover design of the 12-run Plackett-Burman design is shown to give a 2⁴and 2^4-1design in every set of four factors, and to be resolution V in every set of five factors. In addition, the design allows the search and estimation of up to two non-zero interactions.     

On Tests of Uniformity for Randomly Distributed Arcs on a Circle

Let n points be independently distributed on a circle. Moving in a counter-clockwise direction, place arcs of length a on the circle with the i^th are starting at the i^th point. We describe three simple tests of the hypothesis of uniformity based on vacancy, on number of spacings and on the length of the maximal spacing. The tests do not require knowledge of the random points. The asymptotic power of these tests is investigated, and it is shown that vacancy-based tests perform best of the three.     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Estimating and testing zones of abrupt change for spatial data

We propose a method for detecting the zones where a variable irregularly sampled in the plane changes abruptly. Our general model is that under the null hypothesis the variable is the realisation of a stationary Gaussian process with constant expectation. The alternative is that the mean function presents abrupt changes. We define potential Zones of Abrupt Change (ZACs) by the points where the gradient, estimated under the null hypothesis, exceeds a determined threshold. We then design a global test to assess the global significance of the potential ZACs, an issue missing in all existing methods. The theory that links the threshold and the global level is based on asymptotic distributions of excursion sets of non-stationary χ ² fields for which we provide new results. The method is evaluated by a simulation study and applied to a soil data set in the context of precision agriculture.     

An algorithm for computing principal points with respect to a loss function in the unidimensional case

In Flury (1990) the k principal points of a random vector X are defned as the points p(1),..., p(k) minimizing EX–p(i)²; i=1,..., k. We extend this concept to that of k principal points with respect to a loss function L, and present an algorithm for their computation in the univariate case.     

The asymptotic behavior of monotone percentile regression estimates

The least-absolute-deviation estimate of a monotone regression function on an interval has been studied in the literature. If the observation points become dense in the interval, the almost sure rate of convergence has been shown to be O(n^1/4). Applying the techniques used by Brunk (1970, Nonparametric, Techniques in Statistical Inference. Cambridge Univ. Press), the asymptotic distribution of the l₁ estimator at a point is obtained. If the underlying regression function has positive slope at the point, the rate of convergence is seen to be O(n^1/3). Monotone percentile regression estimates are also considered.     

Cautionary Note about R 2

The coefficient of determination (R ²) is perhaps the single most extensively used measure of goodness of fit for regression models. It is also widely misused. The primary source of the problem is that except for linear models with an intercept term, the several alternative R ² statistics are not generally equivalent. This article discusses various considerations and potential pitfalls in using the R ²'s. Specific points are exemplified by means of empirical data. A new resistant statistic is also introduced.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

Random r-content of an r-simplex from beta-type-2 random points

We study the r-content Δ of the r -simplex generated by r+ 1 independent random points in R”. Each random point Z_j is isotropic and distributed according to λ||Z_j||² ～ beta-type-2(n/2, v), λ, v > 0. We provide an asymptotic normality result which is analogous to the conjecture made by Miles (1971). A method is introduced to work out the exact density of W = (rλ)^r(r!Δ)²/(r + |)^r+l and hence that of Δ. The distribution of W is also related to some hypothesis-testing problems in multivariate analysis. Furthermore, by using this method, the distribution of W or Δ can easily be simulated.     

Performance characteristics of the adjusted r2 algorithm for determining the start of the terminal disposition phase and comparison with a simple r2 algorithm and a visual inspection method

The adjusted r² algorithm is a popular automated method for selecting the start time of the terminal disposition phase (t_z) when conducting a noncompartmental pharmacokinetic data analysis. Using simulated data, the performance of the algorithm was assessed in relation to the ratio of the slopes of the preterminal and terminal disposition phases, the point of intercept of the terminal disposition phase with the preterminal disposition phase, the length of the terminal disposition phase captured in the concentration‐time profile, the number of data points present in the terminal disposition phase, and the level of variability in concentration measurement. The adjusted r² algorithm was unable to identify t_z accurately when there were more than three data points present in a profile's terminal disposition phase. The terminal disposition phase rate constant (λ_z) calculated based on the value of t_z selected by the algorithm had a positive bias in all simulation data conditions. Tolerable levels of bias (median bias less than 5%) were achieved under conditions of low measurement variability. When measurement variability was high, tolerable levels of bias were attained only when the terminal phase time span was 4 multiples of t_1/2 or longer. A comparison of the performance of the adjusted r² algorithm, a simple r² algorithm, and t_z selection by visual inspection was conducted using a subset of the simulation data. In the comparison, the simple r² algorithm performed as well as the adjusted r² algorithm and the visual inspection method outperformed both algorithms. Recommendations concerning the use of the various t_z selection methods are presented.     

Estimating the number of clusters

Hartigan (1975) defines the number q of clusters in a d ‐variate statistical population as the number of connected components of the set {f > c}, where f denotes the underlying density function on R^d and c is a given constant. Some usual cluster algorithms treat q as an input which must be given in advance. The authors propose a method for estimating this parameter which is based on the computation of the number of connected components of an estimate of {f > c}. This set estimator is constructed as a union of balls with centres at an appropriate subsample which is selected via a nonparametric density estimator of f. The asymptotic behaviour of the proposed method is analyzed. A simulation study and an example with real data are also included.     

Clustering of Variables Based on Watson Distribution on Hypersphere: A Comparison of Algorithms

We consider n individuals described by p variables, represented by points of the surface of unit hypersphere. We suppose that the individuals are fixed and the set of variables comes from a mixture of bipolar Watson distributions. For the mixture identification, we use EM and dynamic clusters algorithms, which enable us to obtain a partition of the set of variables into clusters of variables.

Our aim is to evaluate the clusters obtained in these algorithms, using measures of within-groups variability and between-groups variability and compare these clusters with those obtained in other clustering approaches, by analyzing simulated and real data.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

New difference-based estimator of parameters in semiparametric regression models

The paper introduces a new difference-based Liu estimator β?_Ldiff=([Xtilde]′[Xtilde]+I)^?1([Xtilde]′[ytilde]+η β?_diff) of the regression parameters β in the semiparametric regression model, y=Xβ+f+?. Difference-based estimator, β?_diff=([Xtilde]′[Xtilde])^?1[Xtilde]′[ytilde] and difference-based Liu estimator are analysed and compared with respect to mean-squared error (mse) criterion. Finally, the performance of the new estimator is evaluated for a real data set. Monte Carlo simulation is given to show the improvement in the scalar mse of the estimator.     

On uniformly distributed random points in an n-Ball

Consider a set of r+1 independently and identically and uniformly distributed random points X₀, X₁,…,X_r in RⁿThese points determine almost surely via their convex hull a unique r-simplex in R^e This article deals with the exact density of the r-content of this random r-simplex when the points are such that p of them are in the interior and r+l?p of them are on the surface of a unit n-ball. This problem is transformed into a distribution problem connected with multivariate test statistics. Various possible representations of the exact density in the general case, are also pointed out.     

Another Cautionary Note about R 2: Its Use in Weighted Least-Squares Regression Analysis

A recent article in this journal presented a variety of expressions for the coefficient of determination (R ²) and demonstrated that these expressions were generally not equivalent. The article discussed potential pitfalls in interpreting the R ² statistic in ordinary least-squares regression analysis. The current article extends this discussion to the case in which regression models are fit by weighted least squares and points out an additional pitfall that awaits the unwary data analyst. We show that unthinking reliance on the R ² statistic can lead to an overly optimistic interpretation of the proportion of variance accounted for in the regression. We propose a modification of the estimator and demonstrate its utility by example.     

Representative points for location-biased datasets

Representative points (RPs) are a set of points that optimally represents a distribution in terms of mean square error. When the prior data is location biased, the direct methods such as the k-means algorithm may be inefficient to obtain the RPs. In this article, a new indirect algorithm is proposed to search the RPs based on location-biased datasets. Such an algorithm does not constrain the parameter model of the true distribution. The empirical study shows that such algorithm can obtain better RPs than the k-means algorithm.