Three mutually orthogonal idempotent Latin squares of orders 22 and 26

Three mutually orthogonal idempotent Latin squares of orders 22 and 26 are constructed, which can be used to obtain 3 HMOLS of type 5²² and type 23²² and to obtain a (110, 5, 1)-PMD and a (130 5, 1)-PMD.     

An optimal k of kth MA-ARIMA models under a class of ARIMA model

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated ARIMA(p, d, q) model. We run a simulation using the three above examining methods under specific conditions. The main finding is that 5th exponential weighted moving average (5th EWMA) ARIMA model is the best forecasting model among others, which means the optimal k = 5. For Turkish Telecommunications (TTKOM) stock market, real data reveal the similar results of simulation study.     

A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity

To solve the heteroscedastic problem in linear regression, many different heteroskedasticity-consistent covariance matrix estimators have been proposed, including HC0 estimator and its variants, such as HC1, HC2, HC3, HC4, HC5 and HC4m. Each variant of the HC0 estimator aims at correcting the tendency of underestimating the true variances. In this paper, a new variant of HC0 estimator, HC5m, which is a combination of HC5 and HC4m, is proposed. Both the numerical analysis and the empirical analysis show that the quasi-t inference based on HC5m is typically more reliable than inferences based on other covariance matrix estimators, regardless of the existence of high leverage points.     

Law of iterated logarithm and almost sure limit points for random sums

Let {X_n,n≥1} be a sequence of independent identically distributed (i.i.d) random variables with a common distribution function F. When F belongs to the domain of partial attraction of a Semi-Stable law with index ,0<<2, we give complete solution to the results of R. Vasudeva and G. Divanji [Law of iterated logarithm for random subsequences, Statist. Probab. Lett. 12 (1991) 189–194], where they obtained Chover’s form of the law of iterated logarithm for random subsequences. Further, we extended the situation in obtaining almost sure limit points for random subsequences.     

Firm size distributions and stochastic growth models: a comparison between ICT and Mechanical Italian Companies

In this paper we analyze the relationship between the distribution of firm size and stochastic processes of growth. Three main models have been suggested by Gibrat (1931), Kalecki (1945) and Champernowne (1973). The first two lead to lognormal distribution and the last to Pareto distribution. We fitted lognormal and Pareto distribution to two Italian sectors: ICT and mechanical. For ICT we found that lognormal distribution must be rejected and Pareto fits reasonably well to the last 30% of largest companies. For mechanical sector we can not reject lognormal distribution. Furthermore, we perform some experiments to corroborate the theoretical models. By means of transition matrices we found that ICT shows features very close to Gibrats and Champernownes models, while Kaleckis model strongly fits to mechanical.JEL Classification: L00, L25, D21Correspondence to: Luigi GrossiThis research was partially supported by grants from Ministero dellIstruzione, dellUniversitá e della Ricerca (MIUR). Despite being the results of a joint work, Sects. 1, 4, 8 and 10 should be attributed to Ganugi, Sects. 3, 6, and 7 to Grossi and Sects. 2, 5, and 9 to Crosato.     

A short table for computing normal orthant probabilities of dimensions 4 and 5

A short of the quadravariate normal orthant probability P ₄ is presented which, except for near singular correlation matrices, is generally accurate to 7+ significant digits. Since the quinvariate orthant probability P ₅is expressed in terms of five copies of P ₄, such a table is also useful for computing P ₅     

An optimal k of kth MA-ARIMA models under MA(q) models

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated MA(q) model. We run a simulation using the three above examining methods under specific conditions. The main finding is that, 5th Exponential Weighted Moving Average (5-th EWMA) Autoregressive Integrated Moving Average (ARIMA) model is the best forecasting model among others, which means the optimal k = 5. For Turkish Telecommunications (TTKOM), stock market real data reveals the similar results of the simulation study.     

Data depth and correlation

Summary: We describe depth–based graphical displays that show the interdependence of multivariate distributions. The plots involve one–dimensional curves or bivariate scatterplots, so they are easier to interpret than correlation matrices. The correlation curve, modelled on the scale curve of Liu et al. (1999), compares the volume of the observed central regions with the volume under independence. The correlation DD–plot is the scatterplot of depth values under a reference distribution against depth values under independence. The area of the plot gives a measure of distance from independence. Correlation curve and DD-plot require an independence model as a baseline: Besides classical parametric specifications, a nonparametric estimator, derived from the randomization principle, is used. Combining data depth and the notion of quadrant dependence, quadrant correlation trajectories are obtained which allow simultaneous representation of subsets of variables. The properties of the plots for the multivariate normal distribution are investigated. Some real data examples are illustrated. *This work was completed with the support of Ca Foscari University.     

Detection of outliers in high-dimensional data using nu-support vector regression

Support Vector Regression (SVR) is gaining in popularity in the detection of outliers and classification problems in high-dimensional data (HDD) as this technique does not require the data to be of full rank. In real application, most of the data are of high dimensional. Classification of high-dimensional data is needed in applied sciences, in particular, as it is important to discriminate cancerous cells from non-cancerous cells. It is also imperative that outliers are identified before constructing a model on the relationship between the dependent and independent variables to avoid misleading interpretations about the fitting of a model. The standard SVR and the μ-ε-SVR are able to detect outliers; however, they are computationally expensive. The fixed parameters support vector regression (FP-ε-SVR) was put forward to remedy this issue. However, the FP-ε-SVR using ε-SVR is not very successful in identifying outliers. In this article, we propose an alternative method to detect outliers i.e. by employing nu-SVR. The merit of our proposed method is confirmed by three real examples and the Monte Carlo simulation. The results show that our proposed nu-SVR method is very successful in identifying outliers under a variety of situations, and with less computational running time.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

An exact test for nbue class of survival functions

Let F(x) be a life distribution. An exact test is given for testing H₀ F is exponential, versusH₁Fε NBUE (NWUE); along with a table of critical values for n=5(l)80, and n=80(5)65. An asymptotic test is made available for large values of n, where the standardized normal table can be used for testing.     

On the orthogonal designs of order 40

Using a new method we construct all 17 remaining (unresolved for over 20 years) full orthogonal designs of order 40 in three variables. This implies that all full orthogonal designs OD(2^t5;x,y, 2^t5−x−y) exist for all t⩾3. The last two remaining orthogonal designs of order 40 in 2 variables are obtained as a special case of two of these designs.     

Grouping normal distributions with unknown parameters

Optimal group definitions, maximizing retained information, are well known for grouping normal distributions with known parameters. When these group definitions are used with sample estimates substituted for unknown parameters, there is a shrinkage of retained information. If there are k groups, and if the size of the sample yielding parameter estimates is n, then this shrinkage is less than 5% for n <= 30 and minimization of shrinkage yields less than 1 % improvement for n <= 5, when k >= 12.     

Computing location depth and regression depth in higher dimensions

The location depth (Tukey 1975) of a point relative to a p-dimensional data set Z of size n is defined as the smallest number of data points in a closed halfspace with boundary through . For bivariate data, it can be computed in O(nlogn) time (Rousseeuw and Ruts 1996). In this paper we construct an exact algorithm to compute the location depth in three dimensions in O(n2logn) time. We also give an approximate algorithm to compute the location depth in p dimensions in O(mp3+mpn) time, where m is the number of p-subsets used.Recently, Rousseeuw and Hubert (1996) defined the depth of a regression fit. The depth of a hyperplane with coefficients (1,...,p) is the smallest number of residuals that need to change sign to make (1,...,p) a nonfit. For bivariate data (p=2) this depth can be computed in O(nlogn) time as well. We construct an algorithm to compute the regression depth of a plane relative to a three-dimensional data set in O(n2logn) time, and another that deals with p=4 in O(n3logn) time. For data sets with large n and/or p we propose an approximate algorithm that computes the depth of a regression fit in O(mp3+mpn+mnlogn) time. For all of these algorithms, actual implementations are made available.     

Inferences on stress–strength parameter based on GLD5 distribution

Let X and Y be independent random variables distributed as generalized Lindley distribution type 5 (GLD5). This article deals with the estimation of the stress–strength parameter R = P(Y < X), which plays an important role in reliability analysis. For this purpose, the maximum likelihood and the uniformly minimum variance unbiased estimators are presented in the explicit form. Moreover, considering Arnold and Strauss’ bivariate Gamma distribution as an informative prior and Jeffreys’ as noninformative prior, the Bayes estimators are derived. Various bootstrap confidence intervals are also proposed and, finally, the presented methods are compared using a simulation study.     

Small-sample comparison of the exact and asymptotic upper tail probabilities of chi-squared goodness-of-fit statistics: the binomila and the mixture binomial

The exact and asymptotic upper tail probabilities (α = .10, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and powerdivergence statisticD ²(λ), with λ= 2/3 are compared by complete enumeration for the binomial and the mixture binomial. For the two-component mixture binomial, three cases have been distinguished. 1. Both success probabilities and the mixing weights are unknwon. 2. One of the two success probabilities is known. And 3., the mixing weights are known. The binomial was investigated for the number of cellsk, being between 3 and 6 with sample sizes between 5 and 100, for k = 7 with sample sizes between 5 and 45, and for k = 10 with sample sizes ranging from 5 to 20. For the mixture binomial, solely k = 5 cells were considered with sample sizes from 5 to 100 and k = 8 cells with sample sizes between 4 and 20. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions for the binomial: 1. Using G2 is not recommendable. 2. At the significance levels α=.10 and α=.05X ² should be preferred over D ²; D ² is the best choice at α = .01. 3. Cochran's (1954; Biometrics, 10, 417-451) rule for the minimum expectation when using X ² seems to generalize to the binomial for G ² and D ² ; as a compromise, it gives a rather strong lower limit for the expected cell frequencies in some circumstances, but a rather liberal in others. To draw similar conclusions concerning the mixture binomial was not possible, because in that case, the accuracy of the chi-squared approximation is not only a function of the chosen test statistic and of the significance level, but also heavily depends on the numerical value of theinvolved unknown parameters and on the hypothesis to be tested. Thereto, the present study may give rise only to warnings against the application of mixture models to small samples.     

Distributions of the ratio of two ranges of samples from nonnormal populations

The probability density function of the ratio of two ranges of samples from a population represented by an Edgeworth series is developed. The corrective functions arising due to nonnormality are tabulated for the upper 5 and 1 percent normal theory points and compared with those of F‐distribution. Indications are that both the variance ratio and the range ratio test are affected by nonnormality almost to the same extent. If there is any numerical difference for larger samples (say, n_:1 = n₂ >= 5), it is in the cases of symmetrical populations. The theoretical results are compared with those obtained from Monte Carlo sampling from x² and t‐distributions with various shape parameters.     

Classification of difference matrices over cyclic groups

The existence of difference matrices over small cyclic groups is investigated in this computer-aided work. The maximum values of the parameters for which difference matrices exist as well as the number of inequivalent difference matrices in each case is determined up to the computational limit. Several new difference matrices have been found in this manner. The maximum number of rows is 9 for an r ×15 difference matrix over Z₃, 8 for an r ×15 difference matrix over Z₅, and 6 for an r ×12 difference matrix over Z₆; the number of inequivalent matrices with these parameters is 5, 2, and 7, respectively.     

Equineighboured balanced nested row-column designs

This paper presents equineighboured balanced nested row-column designs for v treatments arranged in b blocks each comprising pq units further grouped into p rows and q columns. These are two-dimensional designs with the property that all pairs of treatments are neighbours equally frequently at all levels in both rows and columns. Methods of construction are given both for designs based on Latin squares and those where pq≤v. Cyclic equineighboured designs are defined and tabulated for 5≤v≤10, p≤3, q≤5, where r=bpq/v is the number of replications of each treatment.     

Model determination for categorical data with factor level merging

Summary.  We deal with contingency table data that are used to examine the relationships between a set of categorical variables or factors. We assume that such relationships can be adequately described by the cond`itional independence structure that is imposed by an undirected graphical model. If the contingency table is large, a desirable simplified interpretation can be achieved by combining some categories, or levels, of the factors. We introduce conditions under which such an operation does not alter the Markov properties of the graph. Implementation of these conditions leads to Bayesian model uncertainty procedures based on reversible jump Markov chain Monte Carlo methods. The methodology is illustrated on a 2×3×4 and up to a 4×5×5×2×2 contingency table.