On D-optimal robust designs for lifetime improvement experiments

Locating the optimal operating conditions of the process parameters is critical in a lifetime improvement experiment. For log-normal lifetime distribution with compound error structure (i.e., symmetry, inter-class and intra-class correlation error structures), we have developed methods for construction of D-optimal robust first order designs. It is shown that D-optimal robust first order designs are always robust first order rotatable but the converse is not always true.     

On D-optimal robust second order slope-rotatable designs

Das and Park (2006) introduced slope-rotatable designs overall directions for correlated observations which is known as A-optimal robust slope-rotatable designs. This article focuses D-optimal slope-rotatable designs for second-order response surface model with correlated observations. It has been established that robust second-order rotatable designs are also D-optimal robust slope-rotatable designs. A class of D-optimal robust second-order slope-rotatable designs has been derived for special correlation structures of errors.     

Modified Robust Second-Order Slope-Rotatable Designs

Generally it is very difficult to construct robust slope-rotatable designs along axial directions. Present paper focuses on modified second-order slope-rotatable designs (SOSRDs) with correlated errors. Modified robust second-order slope-rotatability conditions are derived for a general variance–covariance structure of errors. These conditions get simplified for intraclass correlation structure. A few robust second-order slope-rotatable designs (over all directions, or with equal maximum directional variance slope, or D-optimal slope) are examined with respect to modified robust slope-rotatability. It is observed that robust second-order slope-rotatable designs over all directions, or with equal maximum directional variance slope, or D-optimal slope are not generally modified robust second-order slope-rotatable designs.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.     

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.     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Optimal designs for additive mixture model with heteroscedastic errors

This paper presents a study of D- and A-optimality of direct sum designs for additive mixture models when the errors are heteroscedastic. Sufficient conditions are given so that D- and A-optimal designs for additive mixture models can be constructed from the D- and A-optimal designs for homogeneous models in sub-mixture systems.     

Bayesian D-optimal Accelerated Life Test plans for series systems with competing exponential causes of failure

This paper provides methods of obtaining Bayesian D-optimal Accelerated Life Test (ALT) plans for series systems with independent exponential component lives under the Type-I censoring scheme. Two different Bayesian D-optimality design criteria are considered. For both the criteria, first optimal designs for a given number of experimental points are found by solving a finite-dimensional constrained optimization problem. Next, the global optimality of such an ALT plan is ensured by applying the General Equivalence Theorem. A detailed sensitivity analysis is also carried out to investigate the effect of different planning inputs on the resulting optimal ALT plans. Furthermore, these Bayesian optimal plans are also compared with the corresponding (frequentist) locally D-optimal ALT plans.     

Locally optimal designs for some dose–response models with continuous endpoints

We consider the problem of constructing static (or non sequential), approximate optimal designs for a class of dose–response models with continuous outcomes. We obtain conditions for a design being D-optimal or c-optimal. The designs are locally optimal in that they depend on the model parameters. The efficiency studies show that these designs have high efficiency when the mis-specification of the initial values of model parameters is not severe. A case study indicates that using an optimal design may result in a significant saving of resources.     

The analytic construction of D-optimal designs for the two-variable binary logistic regression model without interaction

Candidate locally D-optimal designs for the binary two-variable logistic model with no interaction, which comprise 3 and 4 support points lying in the first quadrant of the two-dimensional Euclidean space, were introduced by Haines et al. (D-optimal designs for logistic regression in two variables. In: Lopez-Fidalgo J, Rodrigez-Diaz JM, Torsney B, editors. MODA8 – advances in model-oriented designs and analysis. Heidelberg: Physica-Verlag; 2007. p. 91–98). The authors proved algebraically the global D-optimality of the 3-point design for the special case in which the intercept parameter is equal to?1.5434. However for other selected values of the intercept parameter, the global D-optimality of the proposed 3- and 4-point designs was only demonstrated numerically. In this paper, we provide analytical proofs of the D-optimality of these 3- and 4-point designs for all negative and zero intercept parameters of the binary two-variable logistic model with no interaction. The results are extended to the construction of D-optimal designs on a rectangular design space and illustrated by means of two examples of which one is a real example taken from the literature.     

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

The design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T² control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T² control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.     

A Bayesian piecewise exponential phase II design for monitoring a time-to-event endpoint

A robust Bayesian design is presented for a single-arm phase II trial with an early stopping rule to monitor a time to event endpoint. The assumed model is a piecewise exponential distribution with non-informative gamma priors on the hazard parameters in subintervals of a fixed follow up interval. As an additional comparator, we also define and evaluate a version of the design based on an assumed Weibull distribution. Except for the assumed models, the piecewise exponential and Weibull model based designs are identical to an established design that assumes an exponential event time distribution with an inverse gamma prior on the mean event time. The three designs are compared by simulation under several log-logistic and Weibull distributions having different shape parameters, and for different monitoring schedules. The simulations show that, compared to the exponential inverse gamma model based design, the piecewise exponential design has substantially better performance, with much higher probabilities of correctly stopping the trial early, and shorter and less variable trial duration, when the assumed median event time is unacceptably low. Compared to the Weibull model based design, the piecewise exponential design does a much better job of maintaining small incorrect stopping probabilities in cases where the true median survival time is desirably large.     

An algorithm for calculating D-optimal designs for polynomial regression through a fixed point

Optimal designs are required to make efficient statistical experiments. By using canonical moments, in 1980, Studden found D_s-optimal designs for polynomial regression models. On the other hand, integrable systems are dynamical systems whose solutions can be written down concretely. In this paper, polynomial regression models through a fixed point are discussed. In order to calculate D-optimal designs for these models, a useful relationship between canonical moments and discrete integrable systems is introduced. By using canonical moments and discrete integrable systems, a new algorithm for calculating D-optimal designs for these models is proposed.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

D s -optimal designs for Kozak's tree taper model

In this work, we study D _s -optimal design for Kozak's tree taper model. The approximate D _s -optimal designs are found invariant to tree size and hence create a ground to construct a general replication-free D _s -optimal design. Even though the designs are found not to be dependent on the parameter value p of the Kozak's model, they are sensitive to the s×1 subset parameter vector values of the model. The 12 points replication-free design (with 91% efficiency) suggested in this study is believed to reduce cost and time for data collection and more importantly to precisely estimate the subset parameters of interest.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

Algorithmic and analytical construction of efficient designs in small blocks for comparing consecutive pairs of treatments

Optimal block designs in small blocks are explored under the A-, E- and D-criteria when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first formulate the problem via approximate theory which leads to a convenient multiplicative algorithm for obtaining A-optimal design measures. This, in turn, yields highly efficient exact designs, under the A-criterion, even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available in possibly several stages. Illustrative examples are given and tables of A-optimal design measures are provided. Approximate theory is also seen to yield analytical results on E- and D-optimal design measures.     

D-Optimal Two-Level Fractional Factorial Designs of Resolution III with Two Dispersion Factors

The problem of finding D-optimal designs, with two dispersion factors, for the estimation of all location main effects is investigated in the class of regular unreplicated two-level fractional factorial designs of resolution III. Designs having length three words involving both of the dispersion factors in the defining relation are shown to be inferior in terms of D-optimality. Tables of factors that are named as the two dispersion factors so that the resulting design is either D-optimal or has the largest determinant of the information matrix are provided. Rank-order of designs is studied when the number of length three words involving either one of the dispersion factors and the number of length four words involving both of the dispersion factors are fixed. Rank-order of designs when the numbers of aforementioned words are less than or equal to ten is given.     

Optimal Designs for Random Coefficient Regression Models with Heteroscedastic Errors

The purpose of this article is to present the optimal designs based on D-, G-, A-, I-, and D _β-optimality criteria for random coefficient regression (RCR) models with heteroscedastic errors. A sufficient condition for the heteroscedastic structure is given to make sure that the search of optimal designs can be confined at extreme settings of the design region when the criteria satisfy the assumption of the real valued monotone design criteria. Analytical solutions of D-, G-, A-, I-, and D _β-optimal designs for the RCR models are derived. Two examples are presented for random slope models with specific heteroscedastic errors.     

The beta log-normal distribution

For the first time, we introduce the beta log-normal (LN) distribution for which the LN distribution is a special case. Various properties of the new distribution are discussed. Expansions for the cumulative distribution and density functions that do not involve complicated functions are derived. We obtain expressions for its moments and for the moments of order statistics. The estimation of parameters is approached by the method of maximum likelihood, and the expected information matrix is derived. The new model is quite flexible in analysing positive data as an important alternative to the gamma, Weibull, generalized exponential, beta exponential, and Birnbaum–Saunders distributions. The flexibility of the new distribution is illustrated in an application to a real data set.