Experimental designs with dependent observations: New results and comments on previous results

We present a new sufficient condition on the covariance matrix of the normality distributed observations of an ANOVA model (with orthogonal decomposition of the total sum of squares) under which the F-statictics are distributed proportionally to Fisher F-random variables variables. A new proof of a previous result, a necessary and sufficient condition for applicability of Barlett’s test to the observations of a one-way ANOVA models, and comments on recent results are also given.     

On the existence of linear trend free block designs

We present a class of counerexamples for a conjecture on the existence or linear trend free block designs we will also prove a considerably weakened version of this conjecture which will determine all combinations of designs parmetres for which the class of linear trend free block designs is non empty.     

On the equivalence of two approaches to simplicity in the analysis of block designs and some related results

Two seemingly different approaches to simplicity in the analysis of connected block designs, and their relationship to the concepts of balance are discussed.     

Optimality and bias of some interpolation and extrapolation designs

We consider interpolation and extrapolation designs with controlled bias. A consistent estimate of an upper bound of the bias is presented. The estimate of some extrapolated value is obtained in a two-stage procedure. The first one provides an estimate of some interpolated value. The second one uses a Taylor expansion around this point. This procedure yields a new type of designs. We discuss their optimality properties with respect to the variance and the bias.     

On the enumeration of some D-optimal designs

Two matrices with elements taken from the set {-1,1}$\{-1,1\}$ are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.     

On some DS-optimal designs in spherical regions

For polynomial regression over spherical regions the d-th order D_s-optimal designs for the λ-th order models are derived for 1 ≤ λ ≤ d ≤ 4. Efficiencies of these designs with respect to the λ-th order D-optimal designs are obtained. The effects of estimating addtional parameters due to an m-th order model (d ≥ m >>λ) on the efficiencies are investigated.     

Robustness of some designs against missing data

This article studies the robustness of several types of designs against missing data. The robustness of orthogonal resolution III fractional factorial designs and second-order rotatable designs is studied when a single observation is missing. We also study the robustness of balanced incomplete block designs when a block is missing and of Youden square designs when a column is missing.     

Optimum mixture designs in some constrained experimental regions

In a mixture experiment, the response depends on the mixing proportions of the components present in the mixture. Optimum designs are available for the estimation of parameters of the models proposed in such situations. However, these designs are found to include the vertex points of the simplex Ξ defining the experimental region, which are not mixtures in the true sense. Recently, Mandal et al. (2015 Mandal, N.K., Pal, M., Sinha, B.K., and Das, P. (2015). Optimum mixture designs in a restricted region. Stat. Pap. 56(1):105–119.[Crossref], [Web of Science ®] , [Google Scholar]) derived optimum designs when the experiment is confined to an ellipsoidal region within Ξ, which does not include the vertices of Ξ. In this paper, an attempt has been made to find optimum designs when the experimental region is a simplex or is cuboidal inside Ξ and does not contain the extreme points.     

Effect of missing plots in some response surface designs

The effect of one or more missing observations for response surface designs arranged in blocks are examined in this paper. The results as applied to a central composite design with orthogonal blocking, and an equirdial design with orthogonal blocking, are reported.     

Generalised K-S confidence regions: some exact results

One-sided confidence regions for continuous cumulative distribution function are constructed using empirical cumulative distribution functions and the generalized Kolmogorov-Smirnov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. Significance levels necessary for implementation are given. Some other K-S type distances useful in constructing a confidence region with nonconstant width are also included.     

Generalized exponential distribution: Existing results and some recent developments

Mudholkar and Srivastava [1993. Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliability 42, 299–302] introduced three-parameter exponentiated Weibull distribution. Two-parameter exponentiated exponential or generalized exponential distribution is a particular member of the exponentiated Weibull distribution. Generalized exponential distribution has a right skewed unimodal density function and monotone hazard function similar to the density functions and hazard functions of the gamma and Weibull distributions. It is observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. The genesis of this model, several properties, different estimation procedures and their properties, estimation of the stress-strength parameter, closeness of this distribution to some of the well-known distribution functions are discussed in this article.     

Some results on two-level factorial designs with dependent observations

There are many situations in which observations in factorial experiments may be dependent. When this is so, run orders are needed that result in efficient estimates of contrasts. The Cheng and Steinberg reverse foldover algorithm, which gives a maximal number of level changes, is known to produce very efficient main-effects two-level designs using the D-criterion, but less is known about other designs, models and criteria. We present some further theoretical results, and give another statistic of importance in predicting efficiency under strong dependence. The theory is illustrated using some 16-run designs.     

Construction of some classes of balanced cross-over designs from terraces

5 and 6 have recently introduced power-sequence terraces. In this paper we have used these terraces for the construction of some new families of balanced cross-over designs of first and second order which are variance-balanced. We have also used them for the construction of some new families of balanced ternary cross-over designs.     

Construction of some mixed two- and four-level regular designs with GMC criterion

General minimum lower-order confounding (GMC) criterion is to choose optimal designs, which are based on the aliased effect-number pattern (AENP). The AENP and GMC criterion have been developed to form GMC theory. Zhang et al. (2015 Zhang, T.F., Yang, J.F., Li, Z.M., Zhang, R.C. (2015). Construction of regular 2ⁿ4¹ designs with general minimum lower-order confounding. Commun. Stat. - Theory Methods 46:2724–2735.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) introduced GMC 2ⁿ4^m criterion for choosing optimal designs and constructed all GMC 2ⁿ4¹ designs with N/4 + 1 ? n + 2 ? 5N/16. In this article, we analyze the properties of 2ⁿ4¹ designs and construct GMC 2ⁿ4¹ designs with 5N/16 + 1 ? n + 2 < N ? 1, where n and N are, respectively, the numbers of two-level factors and runs. Further, GMC 2ⁿ4¹ designs with 16-run, 32-run are tabulated.     

Bootstrapping a time series model: some empirical results

The bootstrap is a methodology for estimating standard errors. The idea is to use a Monte Carlo simulation experiment based on a nonparametric estimate of the error distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order autoregressive model fitted by least squares and maximum likelihood estimation. Additionally, a comparison of the bootstrap and the conventional methodology is made. As it turns out, the conventional asymptotic formulae (both the least squares and maximum likelihood estimates) for estimating standard errors appear to overestimate the true standard errors. But there are two problems:i. The first two observations y₁ and y₂ have been fixed, and ii. The residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional maximum likelihood and bootstrap estimates of the standard errors appear to be performing quite well.     

Constructing optimal router bit life sequential experimental designs: New results with a case study

Consider the experiment to improve router bit life as reported in Phadke (1986 Phadke, M. S. 1986. Design optimization case studies. AT &; T Technical Journal 65:51–68.[Crossref], [Web of Science ®] , [Google Scholar]). The goal of the experiment was to increase the life of the router bit before it gets dull, which causes excessive dust formation and consequent expensive cleaning operation to smooth the edges of the boards. A 32-run experimental design was used including seven two-level factors and two four-level factors (cf. Table 1). In this experiment and others, factorial designs with a mixture of two-level and μ( > 2)-level factors may be adopted. Sequential experiments composed of initial experiments and follow-up experiments are widely used to resolve ambiguities involving the aliasing of factorial effects. This article investigates the construction and theoretical properties of optimal designs for sequential experiments with a mixture of α two-level and β μ-level factors for the first time. Constructing optimal design for the router bit life sequential experiment will be discussed for a practical use. From the numerical results, it is found that using a uniform design as the initial experimental design for the router bit life experiment is highly recommended to get an efficient router bit life sequential experimental design. The novelty and significance of the work are evaluated by comparing our results to the existing literature.