A two-sample test for location

A new test that is based on U-statistics is oroposed for the two-sample local, on problem. This test is sensitive to heavy-tailed distributions.     

Some subset selection problems

Consider that we have a collection of k populations π₁, π₂…,π_k. The quality of the ith population is characterized by a real parameter θ_i and the population is to be designated as superior or inferior depending on how much the θ_i differs from θ_max = max{θ₁, θ₂,…,θ_k}. From the set {π₁, π₂,…,π_k}, we wish to select the subset of superior populations. In this paper we devise rules of selection which have the property that their selected set excludes all the inferior populations with probability at least 1?α, where a is a specified number.     

Bayesian optimal design for changepoint problems

We investigate Bayesian optimal designs for changepoint problems. We find robust optimal designs which allow for arbitrary distributions before and after the change, arbitrary prior densities on the parameters before and after the change, and any log‐concave prior density on the changepoint. We define a new design measure for Bayesian optimal design problems as a means of finding the optimal design. Our results apply to any design criterion function concave in the design measure. We illustrate our results by finding the optimal design in a problem motivated by a previous clinical trial. The Canadian Journal of Statistics 37: 495–513; 2009 © 2009 Statistical Society of Canada     

On nonparametric test for the residual probability order

Residual probability order is a partial ordering analog of the total order stochastic precedence which can be fruitfully applied in many reliability problems. We introduce a U-statistic test for the null hypothesis that two random variables are equally distributed versus the alternative that one strictly dominates the other in residual probability order. The critical value is obtained by normal approximation and bootstrap procedures. The performance of the test statistic is evaluated using a simulation study. Finally, a numerical example illustrating the theory is also given.     

Power of depth-based nonparametric tests for multivariate locations

Li and Liu [New nonparametric tests of multivariate locations and scales. Statist Sci. 2004;19(4):686–696] introduced two tests for a difference in locations of two multivariate distributions based on the concept of data depth. Using the simplicial depth [Liu RY. On a notion of data depth based on random simplices. Ann Stat. 1990;18(1):405–414], they studied the performance of these tests for symmetric distributions, namely, the normal and the Cauchy, in a simulation study. However, to the best of our knowledge, the performance of these tests for skewed distributions has not been studied in the current literature. This paper is a contribution in that direction and examines the performance of these depth-based tests in an extensive simulation study involving ten distributions belonging to five well-known families of multivariate skewed distributions. The study includes a comparison of the performance of these tests for four popular affine-invariant depth functions. Conclusions and recommendations are offered.     

Tables for computing shortest confidence intervals involving the f-distribution

Tables of multipliers necessary for computing the shortest confidence interval in situations involving the F-distribution are presented. Such situations are illustrated with several examples.     

Some asymptotics for U-statistics

We consider a one-sample U-statistic with kernel of dimension 2. We obtain its asymptotic bias and skewness and its Edgeworth and Cornish-Fisher type expansions. We also consider in less detail the one sample U-statistic with kernel of arbitrary dimension.     

A conservative nonparametric distribution-free confidence bound for the shift in the changepoint problem

The problem of testing whether there is a change in location in a sequence of random variables that are taken over time was discussed in several papers. In this paper we develop a conservative nonparametric distribution - free confidence bound for the amount of shift and give some Monte Carlo results to show how conservative the bound is.     

Exact and efficient Bayesian inference for multiple changepoint problems

We demonstrate how to perform direct simulation from the posterior distribution of a class of multiple changepoint models where the number of changepoints is unknown. The class of models assumes independence between the posterior distribution of the parameters associated with segments of data between successive changepoints. This approach is based on the use of recursions, and is related to work on product partition models. The computational complexity of the approach is quadratic in the number of observations, but an approximate version, which introduces negligible error, and whose computational cost is roughly linear in the number of observations, is also possible. Our approach can be useful, for example within an MCMC algorithm, even when the independence assumptions do not hold. We demonstrate our approach on coal-mining disaster data and on well-log data. Our method can cope with a range of models, and exact simulation from the posterior distribution is possible in a matter of minutes.     

A nonparametric procedure for testing partially ranked data

In consumer preference studies, it is common to seek a complete ranking of a variety of, say N, alternatives or treatments. Unfortunately, as N increases, it becomes progressively more confusing and undesirable for respondents to rank all N alternatives simultaneously. Moreover, the investigators may only be interested in consumers’ top few choices. Therefore, it is desirable to accommodate the setting where each survey respondent ranks only her/his most preferred k (k?N) alternatives. In this paper, we propose a simple procedure to test the independence of N alternatives and the top-k ranks, such that the value of k can be predetermined before securing a set of partially ranked data or be at the discretion of the investigator in the presence of complete ranking data. The asymptotic distribution of the proposed test under root-n local alternatives is established. We demonstrate our procedure with two real data sets.     

Some joint distributions for conditional random walks

Joint distributions concerning maxima, minima, and their indices are determined for certain conditional random walks called Bernoulli excursion and Bernoulli meander. The distribution of the local time of these processes is treated by generating function technique. Limiting distributions are also given, providing some partial results for Brownian excursion and meander.     

Some simple nonparametric methods to test for perfect ranking in ranked set sampling

A lot of research on ranked set sampling (RSS) is based on the assumption that the ranking is perfect. Hence, it is necessary to develop some tests that could be used to validate this assumption of perfect ranking. In this paper, we introduce some simple nonparametric methods for this purpose. We specifically define three test statistics, _Nk,_Sk$N_k,S_k$ and _Ak$A_k$, based on one-cycle RSS, which are all associated with the ordered ranked set sample (ORSS). We then derive the exact null distributions and exact power functions of all these tests. Next, by using the sum or the maximum of each statistic over all cycles, we propose six test statistics for the case of multi-cycle RSS. We compare the performance of all these tests with that of the Kolmogorov–Smirnov test statistic proposed earlier by Stokes and Sager [1988. Characterization of a ranked-set sample with application to estimating distribution functions. J. Amer. Statist. Assoc. 83, 35–42] and display that all proposed test statistics are more powerful. Finally, we present an example to illustrate the test procedures discussed here.     

Switching nonparametric regression models

We propose a methodology to analyse data arising from a curve that, over its domain, switches among J states. We consider a sequence of response variables, where each response y depends on a covariate x according to an unobserved state z. The states form a stochastic process and their possible values are j=1,?…?, J. If z equals j the expected response of y is one of J unknown smooth functions evaluated at x. We call this model a switching nonparametric regression model. We develop an Expectation–Maximisation algorithm to estimate the parameters of the latent state process and the functions corresponding to the J states. We also obtain standard errors for the parameter estimates of the state process. We conduct simulation studies to analyse the frequentist properties of our estimates. We also apply the proposed methodology to the well-known motorcycle dataset treating the data as coming from more than one simulated accident run with unobserved run labels.     

Parameterization and inference for nonparametric regression problems

We consider local likelihood or local estimating equations, in which a multivariate function () is estimated but a derived function () of () is of interest. In many applications, when most naturally formulated the derived function is a non-linear function of (). In trying to understand whether the derived non-linear function is constant or linear, a problem arises with this approach: when the function is actually constant or linear, the expectation of the function estimate need not be constant or linear, at least to second order. In such circumstances, the simplest standard methods in nonparametric regression for testing whether a function is constant or linear cannot be applied. We develop a simple general solution which is applicable to nonparametric regression, varying-coefficient models, nonparametric generalized linear models, etc. We show that, in local linear kernel regression, inference about the derived function () is facilitated without a loss of power by reparameterization so that () is itself a component of (). Our approach is in contrast with the standard practice of choosing () for convenience and allowing ()> to be a non-linear function of (). The methods are applied to an important data set in nutritional epidemiology.     

U-Statistics and Their Asymptotic Results for Some Inequality and Poverty Measures

U-statistics form a general class of statistics that have certain important features in common. This class arises as a generalization of the sample mean and the sample variance, and typically members of the class are asymptotically normal with good consistency properties. The class encompasses some widely used income inequality and poverty measures, in particular the variance, the Gini index, the poverty rate, the average poverty gap ratios, the Foster-Greer-Thorbecke index, and the Sen index and its modified form. This paper illustrates how these measures come together within the class of U-statistics, and thereby why U-statistics are useful in econometrics.     

A nonparametric test for a constant correlation matrix

We propose a nonparametric procedure to test for changes in correlation matrices at an unknown point in time. The new test requires constant expectations and variances, but only mild assumptions on the serial dependence structure, and has considerable power in finite samples. We derive the asymptotic distribution under the null hypothesis of no change as well as local power results and apply the test to stock returns.     

A note on the critical values used in stepwise tests for multiplicative components of interaction

A result is presented concerning the null distribution of a statistic used to determine the number of multiplicative components in a fixed two-way model. This result suggests critical values which are compared with previously suggested critical values.     

U-Statistics and Their Asymptotic Results for Some Inequality and Poverty Measures

U-statistics form a general class of statistics that have certain important features in common. This class arises as a generalization of the sample mean and the sample variance, and typically members of the class are asymptotically normal with good consistency properties. The class encompasses some widely used income inequality and poverty measures, in particular the variance, the Gini index, the poverty rate, the average poverty gap ratios, the Foster-Greer-Thorbecke index, and the Sen index and its modified form. This paper illustrates how these measures come together within the class of U-statistics, and thereby why U-statistics are useful in econometrics.     

Gaussian Hypergeometric Probability Distributions for Fitting Discrete Data

The distributions generated by the Gaussian hypergeometric function compose a tetraparametric family that includes many of the most common discrete distributions in the literature. In this article, probability aspects related to the whole family are reviewed and methods of estimation for fitting them to real data are developed. Several applied examples are also provided to illustrate the procedures and compare the methods of estimation.     

Some methods for constructing C-designs

Block designs having a property described by Caliński (1971) and Saha (1976), referred to as C-property, are further considered. A necessary and sufficient condition for a block design to have the C-property is given by Saha (1976) and another by Ceranka and Koz?owska (1983). In this paper some methods for constructing such block designs are presented.