Partitioning Anderson's statistic for tied data

Anderson (Biometrics 15 (1959) 582) proposed a χ²-type statistic for the nonparametric analysis of a randomized blocks design with no ties in the data. In this paper, we propose an Anderson statistic that allows for ties in the data. We show that the asymptotic distribution of the statistic under the null hypothesis of no treatment effect is a χ² distribution. Under weak assumptions on the tie structure it is shown that the degrees of freedom for the asymptotic distribution is unchanged compared to the untied case. An extended analysis based on a partition of the statistic into independent components is suggested. The first component is shown to equal the Friedman rank statistic corrected for ties. The subsequent components allow for the detection of dispersion effects, higher order effects and differences in distribution. A simulation study is given and the new analysis is applied to a sensory evaluation data set.     

The asymptotic distributions of maximum likelihood ratio test and maximally selected χ2-test in binomial observations

We considered binomial distributed random variables whose parameters are unknown and some of those parameters need to be estimated. We studied the maximum likelihood ratio test and the maximally selected χ²-test to detect if there is a change in the distributions among the random variables. Their limit distributions under the null hypothesis and their asymptotic distributions under the alternative hypothesis were obtained when the number of the observations is fixed. We discussed the properties of the limit distribution and found an efficient way to calculate the probability of multivariate normal random variables. Finally, those results for both tests have been applied to examples of Lindisfarne's data, the Talipes Data. Our conclusions are consistent with other researchers' findings.     

Random partition models with regression on covariates

Many recent applications of nonparametric Bayesian inference use random partition models, i.e. probability models for clustering a set of experimental units. We review the popular basic constructions. We then focus on an interesting extension of such models. In many applications covariates are available that could be used to a priori inform the clustering. This leads to random clustering models indexed by covariates, i.e., regression models with the outcome being a partition of the experimental units. We discuss some alternative approaches that have been used in the recent literature to implement such models, with an emphasis on a recently proposed extension of product partition models. Several of the reviewed approaches were not originally intended as covariate-based random partition models, but can be used for such inference.     

A weighted least-squares approach to clusterwise regression

Clusterwise regression aims to cluster data sets where the clusters are characterized by their specific regression coefficients in a linear regression model. In this paper, we propose a method for determining a partition which uses an idea of robust regression. We start with some random weighting to determine a start partition and continue in the spirit of M-estimators. The residuals for all regressions are used to assign the observations to the different groups. As target function we use the determination coefficient R²_wR^{2}_{w} for the overall model. This coefficient is suitably defined for weighted regression.     

On the distributions of multivariate sample skewness

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymptotic expectation up to the order N⁻¹ for the multivariate sample skewness and approximate χ²${\chi }^2$ test statistic, where N   is sample size. Under normality, we derive another expectation and variance for Srivastava's multivariate sample skewness in order to obtain a better test statistic. From this result, improved approximate χ²${\chi }^2$ test statistic using the multivariate sample skewness is also given for assessing multivariate normality. Finally, the numerical result by Monte Carlo simulation is shown in order to evaluate accuracy of the obtained expectation, variance and improved approximate χ²${\chi }^2$ test statistic. Furthermore, upper and lower percentiles of χ²${\chi }^2$ test statistic derived in this paper are compared with those of χ²${\chi }^2$ test statistic derived by Mardia (1974) which is used multivariate sample skewness defined by Mardia (1970).     

Addition of runs to an s-level supersaturated design

The purpose of the present work is to extend the work of Gupta et al. (2010) to s  -level column balanced supersaturated designs. Addition of runs to an existing E(χ²)-optimal$E({\chi }^2)\mathrm{-}\mathrm{optimal}$ supersaturated design and to study the optimality of the resulting design is an important issue. This paper considers the study of the optimality of the resulting design. A lower bound to E(χ²)$E({\chi }^2)$ has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(χ²)-optimal$E({\chi }^2)\mathrm{-}\mathrm{optimal}$ supersaturated designs are also presented.     

On empirical Bayes simultaneous selection procedures for comparing normal populations with a standard

In this paper, we derive statistical selection procedures to partition k normal populations into ‘good’ or ‘bad’ ones, respectively, using the nonparametric empirical Bayes approach. The relative regret risk of a selection procedure is used as a measure of its performance. We establish the asymptotic optimality of the proposed empirical Bayes selection procedures and investigate the associated rates of convergence. Under a very mild condition, the proposed empirical Bayes selection procedures are shown to have rates of convergence of order close to O(k^−1/2) where k is the number of populations involved in the selection problem. With further strong assumptions, the empirical Bayes selection procedures have rates of convergence of order O(k^{−α(r−1)/(2r+1)}), where 1<α<2 and r is an integer greater than 2.     

Empirical likelihood inference for probability density functions under association

In this paper, we consider to apply the empirical likelihood method to a probability density function under an associated sample. It is shown that the empirical likelihood ratio statistic is asymptotically χ²-type distributed under some mild conditions. The result is used to construct empirical likelihood-based confidence intervals on the probability density function.     

Nonlinear orthogonal series estimates for random design regression

Let (X,Y) be a pair of random variables with supp(X)⊆[0,1] and EY²<∞. Let m be the corresponding regression function. Estimation of m from i.i.d. data is considered. The L₂ error with integration with respect to the design measure μ (i.e., the distribution of X) is used as an error criterion.Estimates are constructed by estimating the coefficients of an orthonormal expansion of the regression function. This orthonormal expansion is done with respect to a family of piecewise polynomials, which are orthonormal in L₂(μ_n), where μ_n denotes the empirical design measure.It is shown that the estimates are weakly and strongly consistent for every distribution of (X,Y). Furthermore, the estimates behave nearly as well as an ideal (but not applicable) estimate constructed by fitting a piecewise polynomial to the data, where the partition of the piecewise polynomial is chosen optimally for the underlying distribution. This implies e.g., that the estimates achieve up to a logarithmic factor the rate n^−2p/(2p+1), if the underlying regression function is piecewise p-smooth, although their definition depends neither on the smoothness nor on the location of the discontinuities of the regression function.     

Flocks and partial flocks of quadrics: a survey

If O is an ovoid of PG(3,q), then a partition of all but two points of O into q−1 disjoint ovals is called a flock of O. A partition of a nonsingular hyperbolic quadric Q⁺(3,q) into q+1 disjoint irreducible conics is called a flock of Q⁺(3,q). Further, if O is either an oval or a hyperoval of PG(2,q) and if K is the cone with vertex a point x of PG(3,q)⧹PG(2,q) and base O, then a partition of K⧹{x} into q disjoint ovals or hyperovals in the respective cases is called a flock of K. The theory of flocks has applications to projective planes, generalized quadrangles, hyperovals, inversive planes; using flocks new translation planes, hyperovals and generalized quadrangles were discovered. Let Q be an elliptic quadric, a hyperbolic quadric or a quadratic cone of PG(3,q). A partial flock of Q is a set P consisting of β disjoint irreducible conics of Q. Partial flocks which are no flocks, have applications to k-arcs of PG(2,q), to translation planes and to partial line spreads of PG(3,q). Recently, the definition and many properties of flocks of quadratic cones in PG(3,q) were generalized to partial flocks of quadratic cones with vertex a point in PG(n,q), for n⩾3 odd.     

Weighted bootstraps for the variance

We consider the weighted bootstrap introduced by Mason and Newton [Ann. Statist. 20 (1992) 1611–1624] for estimators of the variance (in non-Studentized form). First, we give conditions on the moments of the weights to ensure that the weighted bootstrap leads to uniformly correct two-sided approximation up to the rate O(n^−3/2). Then, we discuss the practical choice of the random weights in order to construct one-sided confidence intervals accurate up to O(n^−3/2).     

On empirical likelihood for linear models with missing responses

Suppose that we have a linear regression model Y=X^′β+_ν0(X)ε$Y=X^{\prime }\beta +{\nu }_0(X)\epsilon$ with random error ε$\epsilon$, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random (MAR). In this paper, based on the ‘complete’ data set for Y after inverse probability weighted imputation, we construct empirical likelihood statistics on EY   and β$\beta$ which have the χ²${\chi }^2$-type limiting distributions under some new conditions compared with Xue (2009). Our results broaden the applicable scope of the approach combined with Xue (2009).     

Statistical inferences for partially linear single-index models with error-prone linear covariates

We consider statistical inference for partially linear single-index models (PLSIM) when some linear covariates are not observed, but ancillary variables are available. Based on the profile least-squared estimators of the unknowns, we study the testing problems for parametric components in the proposed models. It is to see whether the generalized likelihood ratio (GLR) tests proposed by Fan et al. (2001) are applicable to testing for the parametric components. We show that under the null hypothesis the proposed GLR statistics follow asymptotically the χ²-distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated experiments are conducted to illustrate our proposed methodology.     

A test statistic for weighted runs

A new test statistic based on runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic χ² test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute p-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.     

Empirical likelihood for probability density functions under negatively associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a negatively associated sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²‐type distributed. The result is used to obtain EL based confidence interval on the probability density function.     

Efficient estimates in linear and nonlinear regression with heteroscedastic errors

In this paper we consider the heteroscedastic regression model defined by the structural relation Y = r(V, β) + s(W)ε, where V is a p-dimensional random vector, W is a q-dimensional random vector, β is an unknown vector in some open subset B of R^m, r is a known function from R^p × B into R, s is an unknown function on R^q, and ε is an unobservable random variable that is independent of the pair (V, W). We construct asymptotically efficient estimates of the regression parameter β under mild assumptions on the functions r and s and on the distributions of ε and (V, W).     

Nonparametric product partition models for multiple change-points analysis

ABSTRACT

We propose an extension of parametric product partition models. We name our proposal nonparametric product partition models because we associate a random measure instead of a parametric kernel to each set within a random partition. Our methodology does not impose any specific form on the marginal distribution of the observations, allowing us to detect shifts of behaviour even when dealing with heavy-tailed or skewed distributions. We propose a suitable loss function and find the partition of the data having minimum expected loss. We then apply our nonparametric procedure to multiple change-point analysis and compare it with PPMs and with other methodologies that have recently appeared in the literature. Also, in the context of missing data, we exploit the product partition structure in order to estimate the distribution function of each missing value, allowing us to detect change points using the loss function mentioned above. Finally, we present applications to financial as well as genetic data.     

Construction of optimal supersaturated design with large number of levels

Inspired by the ideas of column and row juxtaposition in Liu and Lin (2009) and level transformation in Yamada and Lin (1999), this paper presents a new method for constructing optimal supersaturated designs (SSDs). This method provides a convenient way to construct mixed-level designs with relatively large numbers of levels, avoiding the blind search and numerous calculations by computers. The goodness of the resulting SSDs is judged by the χ² (Yamada and Lin, 1999 and Yamada and Matsui, 2002) and J₂ (Xu, 2002) criteria. Some nice properties of the new designs are also provided.     

On the strong convergence for weighted sums of ρ *-mixing random variables

A complete convergence result is obtained for weighted sums of identically distributed ρ ^*-mixing random variables with E|X ₁| ^α log(1 + |X ₁|) < ∞ for some 0 < α ≤ 2. This result partially extends the result of Sung (Stat Papers 52: 447–454, 2011) for negatively associated random variables to ρ ^*-mixing random variables. It also settles the open problem posed by Zhou et al. (J Inequal Appl, 2011, doi:10.1155/2011/157816).