2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>4<Superscript>1</Superscript> designs with minimum aberration or weak minimum aberration

For measuring the goodness of 2 ^m 4¹ designs, Wu and Zhang (1993) proposed the minimum aberration (MA) criterion. MA 2 ^m 4¹ designs have been constructed using the idea of complementary designs when the number of two-level factors, m, exceeds n/2, where n is the total number of runs. In this paper, the structures of MA 2 ^m 4¹ designs are obtained when m>5n/16. Based on these structures, some methods are developed for constructing MA 2 ^m 4¹ designs for 5n/16<m<n/2 as well as for n/2≤m<n. When m≤5n/16, there is no general method for constructing MA 2 ^m 4¹ designs. In this case, we obtain lower bounds for A ₃₀ and A ₃₁, where A ₃₀ and A ₃₁ are the numbers of type 0 and type 1 words with length three respectively. And a method for constructing weak minimum aberration (WMA) 2 ^m 4¹ designs (A ₃₀ and A ₃₁ achieving the lower bounds) is demonstrated. Some MA or WMA 2 ^m 4¹ designs with 32 or 64 runs are tabulated for practical use, which supplement the tables in Wu and Zhang (1993), Zhang and Shao (2001) and Mukerjee and Wu (2001).     

Accurate ARL computation for EWMA-S<Superscript>2</Superscript> control charts

Originally, the exponentially weighted moving average (EWMA) control chart was developed for detecting changes in the process mean. The average run length (ARL) became the most popular performance measure for schemes with this objective. When monitoring the mean of independent and normally distributed observations the ARL can be determined with high precision. Nowadays, EWMA control charts are also used for monitoring the variance. Charts based on the sample variance S² are an appropriate choice. The usage of ARL evaluation techniques known from mean monitoring charts, however, is difficult. The most accurate method—solving a Fredholm integral equation with the Nyström method—fails due to an improper kernel in the case of chi-squared distributions. Here, we exploit the collocation method and the product Nyström method. These methods are compared to Markov chain based approaches. We see that collocation leads to higher accuracy than currently established methods.     

Probability generating function of GPED<Subscript>2</Subscript>

The Probability generating function of a random variable which has Generalized Polya Eggenberger Distribution of the second kind (GPED ₂) is obtained. The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable from GPED ₂. The results of Bazargan-Lari (2004) follow as special cases.     

Paired permutation testing in 2<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> unreplicated factorials

Exact permutation testing of effects in unreplicated two-level multifactorial designs is developed based on the notion of realigning observations and on paired permutations. This approach preserves the exchangeability of error components for testing up tok effects. Advantages and limitations of exact permutation procedures for unreplicated factorials are discussed and a simulation study on paired permutation testing is presented.     

Distribution of the range when sample size has a GPED<Subscript>1</Subscript>

The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable having the Generalized Polya Eggenberger Distribution of the first kind (GPED ₁). The results of Raghunandanan and Patil (1972) and Bazargan-lari (1999) follow as special cases. The p.d.f of rangeR is obtained, when the distribution of the sample sizeN belongs to Katz family of distributions, as a special case. An erratum to this article is available at .     

A uniform saddlepoint expansion for the null‐distribution of the wilcoxon‐mann‐whitney statistic

The authors give the exact coefficient of 1/N in a saddlepoint approximation to the Wilcoxon‐Mann‐Whitney null‐distribution. This saddlepoint approximation is obtained from an Edgeworth approximation to the exponentially tilted distribution. Moreover, the rate of convergence of the relative error is uniformly of order O (1/N) in a large deviation interval as defined in Feller (1971). The proposed method for computing the coefficient of 1/N can be used to obtain the exact coefficients of 1/Nⁱ, for any i. The exact formulas for the cumulant generating function and the cumulants, needed for these results, are those of van Dantzig (1947‐1950).     

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.     

Robust mixture modeling using the skew <Emphasis Type="Italic">t</Emphasis> distribution

A finite mixture model using the Student's t distribution has been recognized as a robust extension of normal mixtures. Recently, a mixture of skew normal distributions has been found to be effective in the treatment of heterogeneous data involving asymmetric behaviors across subclasses. In this article, we propose a robust mixture framework based on the skew t distribution to efficiently deal with heavy-tailedness, extra skewness and multimodality in a wide range of settings. Statistical mixture modeling based on normal, Student's t and skew normal distributions can be viewed as special cases of the skew t mixture model. We present analytically simple EM-type algorithms for iteratively computing maximum likelihood estimates. The proposed methodology is illustrated by analyzing a real data example.     

<Emphasis Type="Italic">R</Emphasis>-charts for the exponential,Laplace and logistic processes

This paper (i) discusses theR-chart with asymmetric probability control limits under the assumption that the distribution of the quality characteristic under study is either exponential, Laplace, or logistic, (ii) examines the effect of the estimated probability limits on the performance of theR-chart, and (iii) obtains the desired probability limits of theR-chart that has a specified false alarm rate when probability limits must be estimated from preliminary samples taken from either the exponential, Laplace, or logistic processes.     

Testing multivariate uniformity: The distance‐to‐boundary method

Given a random sample taken on a compact domain S ? ^d, the authors propose a new method for testing the hypothesis of uniformity of the underlying distribution. The test statistic is based on the distance of every observation to the boundary of S. The proposed test has a number of interesting properties. In particular, it is feasible and particularly suitable for high dimensional data; it is distribution free for a wide range of choices of 5; it can be adapted to the case that the support of S is unknown; and it also allows for one‐sided versions. Moreover, the results suggest that, in some cases, this procedure does not suffer from the well‐known curse of dimensionality. The authors study the properties of this test from both a theoretical and practical point of view. In particular, an extensive Monte Carlo simulation study allows them to compare their methods with some alternative procedures. They conclude that the proposed test provides quite a satisfactory balance between power, computational simplicity, and adaptability to different dimensions and supports.     

Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures

Let Z ₁, Z ₂, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z _t  = 1) and q = Pr(Z _t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E₁{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E₂{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E₃{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by N_n,k⁽ⁱ⁾{ N_{n,k}^{(i)}} (respectively M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}) the number of occurrences of the pattern E_i{\mathcal{E}_{i}} , i = 1, 2, 3, in Z ₁, Z ₂, . . . , Z _n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} (resp. W_r,k⁽ⁱ⁾){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern E_i{\mathcal{E}_{i}}, i = 1, 2, 3, in Z ₁, Z ₂, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N_n,k⁽ⁱ⁾{N_{n,k}^{(i)}}, M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}, T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} and W_r,k⁽ⁱ⁾{W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.     

Improved estimators of common variance of <Emphasis Type="Italic">p</Emphasis>-populations when Kurtosis is known

The pooled variance of p samples presumed to have been obtained from p populations having common variance σ², has invariably been adopted as the default estimator for σ². In this paper, alternative estimators of the common population variance are developed. These estimators are biased and have lower mean-squared error values than . The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (a ratio of mean-squared error values).     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.     

Errors of misclassification in discrimination with data from truncated <Emphasis Type="Italic">t</Emphasis> populations

The distribution of the probabilities of misclassification is derived in this paper, which are reproduced by the use of the linear discriminant function. The statistical background is two independent doubly truncated t populations with distinct location parameters and common scale parameter and degrees of freedom. The behavior of the linear discriminant function is studied by comparing the distribution function of the errors of misclassification under the truncated t and truncated normal models.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Goodness‐of‐fit tests for the hyperbolic distribution

The authors give tests of fit for the hyperbolic distribution, based on the Cramér‐von Mises statistic W². They consider the general case with four parameters unknown, and some specific cases where one or two parameters are fixed. They give two examples using stock price data.     

Cramér-von Mises tests of fit for the Poisson distribution

Goodness-of-fit tests based on the Cramér-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A², give good overall tests of fit. The statistic A² will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.     

Modeling stock markets’ volatility using GARCH models with Normal,Student’s <Emphasis Type="Italic">t</Emphasis> and stable Paretian distributions

As GARCH models and stable Paretian distributions have been revisited in the recent past with the papers of Hansen and Lunde (J Appl Econom 20: 873–889, 2005) and Bidarkota and McCulloch (Quant Finance 4: 256–265, 2004), respectively, in this paper we discuss alternative conditional distributional models for the daily returns of the US, German and Portuguese main stock market indexes, considering ARMA-GARCH models driven by Normal, Student’s t and stable Paretian distributed innovations. We find that a GARCH model with stable Paretian innovations fits returns clearly better than the more popular Normal distribution and slightly better than the Student’s t distribution. However, the Student’s t outperforms the Normal and stable Paretian distributions when the out-of-sample density forecasts are considered.     

Estimation and testing of P(Y > X) in two-parameter exponential distributions

In this article, we obtain the UMVUE of the reliability function ξ=P(Y>X) and the UMVUE of ξ ^k =[P(Y>X)] ^k in the two-parameter exponential distributions with known scale parameters. We also derive the distribution of the UMVUE of ξ and further considering the tests of hypotheses regarding the reliability function ξ.     

On the Breakdown Properties of Some Multivariate M‐Functionals*

Abstract. For probability distributions on ? ^q, a detailed study of the breakdown properties of some multivariate M‐functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution‐free’ M‐functional of scatter is given. These include a symmetrized version of Tyler's M‐functional of scatter, and the multivariate t M‐functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M‐functional of scatter and of its symmetrized version are 1/q and , respectively. For the multivariate t M‐functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/( q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low‐dimensional subspaces.