Kai-Tai Fang

• Professor Emeritus at Hong Kong Baptist University

Time series models (TSMs) are used to forecast events based on verified historical data that are widely applied to describe natural and social phenomena. The spectral density function (SDF) of the TSM has the benefit to determine trends, periodic patterns and behavior of observable events and is used in degree of memory (DOM) which is used to ident...

The mixture of two 2-parameter Weibull distributions (MixW), as a specialized variant of the mixture of Weibull distributions, serves as an ideal model for heterogeneous data sets within the realms of reliability studies and survival analysis. A principal challenge in dealing with MixW lies in the estimation of parameters. Inspired by the exemplary...

Good lattice point (GLP) sets are a type of number-theoretic method widely utilized across various fields. Their space-filling property can be further improved, especially with large numbers of runs and factors. In this paper, Kullback-Leibler (KL) divergence is used to measure GLP sets. The generalized good lattice point (GGLP) sets obtained from...

Assuming the underlying statistical distribution of data is critical in information theory, as it impacts the accuracy and efficiency of communication and the definition of entropy. The real-world data are widely assumed to follow the normal distribution. To better comprehend the skewness of the data, many models more flexible than the normal distr...

Many obstacles are causing Chinese shadow puppeteering, an endangered art form, to disappear. This state of affairs has been exacerbated by a number of factors, including the impact of varied entertainment options, aging puppeteers, declining audiences, and a dearth of youthful successors. This study suggests using real-time finger-tracking technol...

The mixture of two 2-parameter Weibull distributions (MixW), as a specialized variant of the mixtureof Weibull distributions, serves as an ideal model for heterogeneous data sets within the realms ofreliability studies and survival analysis. A principal challenge in dealing with MixW lies in the estima-tion of parameters. Inspired by the exemplary...

The bootstrap method relies on resampling from the empirical distribution to provide inferences about the population with a distribution F. The empirical distribution serves as an approximation to the population. It is possible, however, to resample from another approximating distribution of F to conduct simulation-based inferences. In this paper,...

In clinical trials studying paired parts of a subject with binary outcomes, it is expected to collect measurements bilaterally. However, there are cases where subjects contribute measurements for only one part. By utilizing combined data, it is possible to gain additional information compared to using bilateral or unilateral data alone. With the co...

Generating minimum energy points (MEPs) is an optimal solution of many real- world problems, such as the selection of best locations for hospitals inside a city that reduce the overcrowding and competition and avoid the less-populated regions. The key idea is considering these locations as charged particles with the same sign (i.e., repel each othe...

Statistical modeling relies on a diverse range of statistical distributions, encompassing both univariate and multivariate distributions and/or discrete and continuous distributions. In the literature, numerous statistical methods have been proposed to approximate continuous distributions. The most commonly used approach is the use of the empirical...

Gaussian process (GP) is a stochastic process that has been successfully applied in finance, black-box modeling of biosystems, machine learning, geostatistics, multitask learning or robotics and reinforcement learning. Effectively estimating the spectral density function (SDF) and degree of memory (DOM) of a long-memory stationary GP (LMSGP) is nee...

A uniform experimental design (UED) is an extremely used powerful and efficient method- ology for designing experiments with high-dimensional inputs, limited resources and unknown under- lying models. A UED enjoys the following two significant advantages: (i) It is a robust design, since it does not require to specify a model before experimenters c...

The construction of uniform designs (UDs) has received much attention in computer experiments over the past decades, but most of the previous works obtain uniform designs over a U-type by lattice domain. Due to increasing demands for continuous factors, UDs over the continuous domain is in lack. Moreover, the uniformity can be further improved over...

Two fractional factorial designs are called isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs and switching the levels of factors. Given a set of all orthogonal designs (ODs) with n runs, q levels and s factors, it may have several non-isomorphic sub-classes. Once a new OD with this design size is gener...

In recent years, the mixture of two-component normal distributions (MixN) has attracted considerable interest due to its flexibility in capturing a variety of density shapes. In this paper, we investigate the problem of discretizing a MixN by a fixed number of points under the minimum mean squared error (MSE-RPs). Motivated by the Fang-He algorithm...

This paper proposes a bias-corrected expression for maximum likelihood estimators using the sequential number-theoretic method for optimization (SNTO) to improve the efficiency and accuracy of the estimators in three extreme value distributions (EVDs). It is well known that the widely used maximum likelihood estimation (MLE) could be often biased f...

Mixtures of two-component normal distributions (MixN) have various applications in statistical inference with flexibility in density fitting. The best estimation of the five model parameters still represents a challenge. This article proposes more accurate density fittings given a random sample in both the moment-based and likelihood-based estimati...

We propose a new criterion “Entropy based on projection Hamming distance pattern”, denoted as S(Dk), to evaluate and detect combinatorially non-isomorphic orthogonal designs. It is defined by the entropy based on the distribution of all Hamming distance patterns of projections, that provides a new perspective to evaluate orthogonal designs. Hamming...

The fundamental problem in the orthogonal design theory is the design isomorphism, which involves two classes of methods in the statistical literature. One is to identify the isomorphic designs by costly computation, another is only to detect the non-isomorphic designs as a feasible alternative. In this paper we explore the design structure to prop...

It is known that the exponential distribution has many nice properties. Graf and Luschgy (2000) pointed out that the mean squared error of the set of representative points of the exponential distribution is fully determined by the smallest representative point. In this paper we concern with the representative points of the exponential distribution...

Construction of orthogonal designs (ODs) has received much attention over the past decades, where previous work were originated from either mathematical theory or algorithmic search. A new algorithm is proposed to construct symmetric ODs. It is established on a well-designed framework of sequential construction, combining the deep Q-network (DQN) a...

Limited by practical operation, most experimental designs were constructed on discrete space, even if the influencing factors themselves are continuous. While in computer experiments, one can explore more possibilities to improve the modeling performance without increasing the test cost. As a space-filling design, the construction of uniform design...

There are many difficulties that may be encountered in experiments of chemometrics as well as most computer experiments, that ought to explore an approximate model instead of the true one that is complicated. Space-filling designs including uniform designs are robust for such situation. However, when the underlying regression model is known, the D-...

Good lattice point (GLP) sets are sets of points that are uniformly distributed over the domain of interest and thus have good space-filling property. GLP sets are an important kind of points for many applications, such as multidimensional quadrature, simulation, computer experiments, quasi-Monte Carlo techniques and design of experiments. It is a...

This paper explores the properties of the gap of representative points (RPs) in the sense of minimum mean square error for various univariate statistical distributions. We illustrate the relationship between RPs and doubly truncated mean residual life (DMRL) as well as mean residual life (MRL), which are widely used in survival analysis. The limiti...

Discrete approximations of statistical continuous distributions have been widely requested in various fields. Using random samples generated by Monte Carlo (MC) method to infer the population has been dominant in statistics. The empirical distribution of a random sample can be regarded as a discrete approximation of the population distribution in a...

Several techniques are proposed for designing experiments in scientific and industrial areas in order to gain much effective information using a relatively small number of trials. Uniform design (UD) plays a significant role due to its flexibility, cost- efficiency and robustness when the underlying models are unknown. UD seeks its design points to...

For clarity and due to the limitation of the space, we relegate all the proofs and many tables (Tables B1–B10) to the supplementary material of this paper. The proofs of the main lemmas and theorems are given. A catalogue of smallest m(2) forgiven 0 ≤ m(4) ≤ 15, 0 ≤ m(3) ≤ 15 and 8 ≤ n ≤ 52 are given in Tables B1– B6. From Tables B1-B6, we can show...

For clarity and due to the limitation of the space, we relegate the R codes, algorithms and many numerical results in Tables A1-A9 to the following online supplementary material. --------------------------- Appendix A: Tables A1 and A2 give the TEPE for 3 ≤ n ≤ 100 and 101 ≤ n ≤ 200 MEPs from ICPs in a 1D unit hypercube, respectively. Tables A3 an...

By the affine resolvable design theory, there are 68 non-isomorphic classes of symmetric orthogonal designs involving 13 factors with 3 levels and 27 runs. This paper gives a comprehensive study of all these 68 non-isomorphic classes from the viewpoint of the uniformity criteria, generalized word-length pattern and Hamming distance pattern, which p...

Tables A1, A2 and A3 give the minimum values of MD, CD and WD, respectively for the uniform k-factor projected level permuted designs from all the ${13 \choose k} (68)(3!)^k$ values from the 68 non-isomorphic classes and the corresponding classes for these minimum values. The PGWLPs for all the five-factor projected designs of $L^{61}_{27}(3^{13}...

Due to the limited resources and budgets in many real-life projects, it is unaffordable to use full factorial experimental designs and thus fractional factorial (FF) designs are used instead. The aliasing of factorial effects is the price we pay for using FF designs and thus some significant effects cannot be estimated. Therefore, some additional o...

For clarity and due to the limitation of the space, we relegate all the Matlab codes of the new algorithms of the paper to this supplementary material.

The paper is motivated by the structural alloy steel standard that has been used in China for a long period. This standard indicates the scope of several chemical elements in the steel and requests several mechanical properties for qualification. Fang and Wu (Acta Math Appl Sin 2:132–148, 1979) established the relationships between the percents of...

Follow-up experiments are used extensively to provide precious information about the relationships between inputs and outputs to gain a better understanding of a process or system under study. This article gives a new look at designing optimal follow-up experiments that involve any number of factors with any number of different levels in light of t...

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the exper- imenters do not know which factors turn out to be important before conducting the experiment. Th...

In this paper, we propose a criterion “the main effect confounding pattern (MECP)” for comparing projection designs based on saturated symmetric orthogonal designs. Some studies for \(L_9(3^4)\), \(L_{27}(3^{13})\) and \(L_{16}(4^5)\) are given. They show that the new criterion MECP is mostly consistent with the criteria: the generalized word-lengt...

The foldover plan is a transformation map for adding a foldover design to the initial design, thus resulting in a combined design which can be used for breaking the links between aliased effects. This paper discusses the optimality of foldover plans for four-level designs via the most common criteria: the generalized word-length pattern (GWLP), Lee...

Computer experiments have become ubiquitous across the engineering, physical and chemical sciences. Computer experiments are constructed to emulate the behavior of a physical system. Assume that we perform an experiment using a two-level uniform design. If, after obtaining data, we decide additional runs of the computer simulator are needed, how to...

Representative points (RPs) are a set of points that optimally represents a distribution in terms of mean square error. When the prior data is location biased, the direct methods such as the k-means algorithm may be inefficient to obtain the RPs. In this paper, a new indirect algorithm is proposed to search the RPs based on location-biased data set...

The purpose of this book is to introduce theory, methodology and applications of the Uniform experimental design. The uniform experimental design can be regarded as a fractional factorial design with model uncertainty, a space filling design for computer experiments, a robust design against the model specification, a supersaturated design and can b...

This chapter introduces uniform design and modeling for experiments with mixtures and for experiments with restricted mixtures. Firstly, some designs for experiments with mixtures including the Scheffé simplex-lattice, simplex-centroid designs, and axial designs are introduced. Secondly, the uniform design of experiments with mixtures and the corre...

In this chapter and the following chapters, the different construction methods for uniform design tables are given. Typically, there are three approaches of constructing uniform design tables: (i) quasi-Monte Carlo methods; (ii) combinatorial methods; (iii) numerical search.

From the previous chapter, we see that there exist close relationships between uniformity and several other design criteria. In this chapter, we will show that the uniformity is also a useful criterion in some classical designs, such as block design, supersaturated design, Latin hypercube design and so on.

Let \(y=f(\varvec{x})=f(x_1,\ldots ,x_s)\) be the true model of a system where \(X_1,\ldots ,X_s\) are factors that take values \((x_1,\ldots ,x_s)\) on a domain \(\mathcal {X}\), and y is response. For physical experiments with model unknown, the true model \(f(\cdot )\) is unknown and for computer experiments the function \(f(\cdot )\) is known,...

Most experimental designs, such as simple random design, random block design, Latin square design, fractional factorial design (FFD, for short), optimal design, and robust design are concerned with randomness, balance between factors and levels of each factor, orthogonality, efficiency, and robustness. From the previous chapters, we see that the un...

Motivated by the overall mean model and the famous Koksma–Hlawka inequality, the main idea of uniform experimental design is to scatter the experimental design points uniformly over the experimental domain. In this regard, uniformity measures constitute the main concept of the uniform experimental design. Discrepancy is a measure which is defined a...

In the previous chapter, the deterministic methods such as good lattice point method and its modifications were introduced. Generally, most existing uniform designs are obtained by numerical search on the U-type design space \(\mathcal {U}(n;n^s)\) or \(\mathcal {U}(n;q_1,\ldots ,q_s)\). This chapter introduces some powerful optimization methods an...

The research of developing efficient methodologies for constructing optimal experimental designs has been very active in the last decade. Uniform design is one of the most popular approaches, carried out by filling up experimental points in a determinately uniform fashion. Applications of coding theory in experimental design are interesting and pro...

Experimental designs and modeling are very important in chemometrics and chemical engineering. There are many kinds of experimental designs, which include the fractional factorial design (including the orthogonal design), the optimal regression design, and the uniform design. The uniform experimental design can be regarded as a fractional factorial...

The threshold accepting (TA) algorithm, a stochastic heuristic technique, has been proposed as an effective method for constructing uniform designs (UDs). However, some flaws of this algorithm have been detected recently, especially when optimizing designs with large sizes. This paper proposes an adjusted TA algorithm that can overcome the defects...

In this paper we propose a new method, based on R-C similar transformation method, to study classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging two rows and related two columns of a magic square. Many new results for classification of the magic squares of order 5 are obtained by the R-C similar t...

This paper presents an enumeration algorithm to generate all magic squares of order 5 based on the ideas of basic form (Schroeppel [7]) and generating vector which is extension of Frénicle Quads (Ollerenshaw and Bondi [6]). The results lead us to extend Frénicle-Amela patterns from the case of order 4 to the case of order 5, which we refer to Fréni...

Representative points obtained by number-theoretic method (RPs-NTM) are used to construct a discrete approximation to the arcsine distribution. An interesting property of the discrete approximation is presented. In addition, another type of representative points of minimizing mean squared error (RPs-MSE) is also considered as an approximation. The...

The paper gives a new approach to statistical simulation and resampling by the use of number-theoretic methods and representative points. Resempling techniques take samples from an approximate population. The bootstrap suggests to use a random sample to form an approximate population. We propose to construct some approximate population distribution...

a b s t r a c t Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Inspired by the construction of cyclic orthogonal designs with prime power run numbers in Plackett and Burman (1946), a method for constructing χ 2 ðDÞ optimal multi-level SSDs with cyclic structure is explored in this pa...

There are various discrepancies that are measures of uniformity for a set of points on the unit hypercube. The discrepancies have played an important role in quasi-Monte Carlo methods. Each discrepancy has its own characteristic and some weakness. In this paper we point out some unreasonable phenomena associated with the commonly used discrepancies...

This paper studies a two-machine flow shop scheduling problem with a supporting precedence relation. The model originates from a real production context of a chemical factory that produces foam-rubber products. We extend the traditional two-machine flow shop by dividing the operations into two categories: supporting tasks and regular jobs. In the m...

This paper develops a new empirical likelihood method for semiparametric linear regression with a completely unknown error distribution and right censored survival data. The method is based on the Buckley-James (1979) estimating equation. It inherits some appealing properties of the complete data empirical likelihood method. For example, it does no...

Many construction methods for (nearly) uniform designs have been proposed under the centered $L_2$ -discrepancy, and most of them are only suitable for constructing designs with small size. This paper proposes a new method, called mixture method (MM), to construct nearly symmetrical/asymmetrical uniform designs with large number of runs and/or la...

In this paper, the wrap-around L2-discrepancy (WD) of asymmetrical design is represented as a quadratic form, thus the problem of constructing a uniform design becomes a quadratic integer programming problem. By the theory of optimization, some theoretic properties are obtained. Algorithms for constructing uniform designs are then studied. When the...

This article proposes some probability plots are proposed to test spherical and elliptical symmetry in terms of some invariant statistics under orthogonal transformations. Some correlation coefficients as numerical measures of detecting deviation from spherical or elliptical symmetry are recommended, and the empirical percentiles of these correlati...

The uniform design is a kind of space filling design that is robust against the model specification. The uniform design has been widely applied to experiments with mixtures. In this paper, we propose a new discrepancy DM2-discrepancy as a new criterion to measure the uniformity of designs with mixtures. A computational formula of the new discrepanc...

The goal of uniform mixture design is to scatter the design points in the experimental region uniformly. The commonly used criteria, such as mean square distance, are based on the Euclidean distance. Based on the Lee distance, a new criterion is proposed in this article. An algorithm, called NTLBG (number theoretic Linde-Buzo-Gray), is also propose...

Motivated by a random coverage problem from a real-life case study, this paper discusses and compares several number-theoretic methods including the ELP-net, NT-net, and others in statistical simulations. The methods and related conclusions studied in this paper can be extended into more general cases in statistical simulations.

For a class of multivariate elliptically contoured distributions the maximum-likelihood estimators of the mean vector and covariance matrix are found under certain conditions. Likelihood-ratio criteria are obtained for a class of null hypotheses. These have the same form as in the normal case.On donne des conditions permettant de calculer les estim...

Let X1,…,Xn be exchangeable normal variables with a common correlation p, and let X(1) > … > X(n) denote their order statistics. The random variable σni=n—k+1xi, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on...

The uniform experimental design is one important kind of space filling design and has been applied in computer experiments, laboratory experiments under model uncertainty, and experiments with mixtures. The uniform design is robust against model misspecification.

While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few easily implemented statistics for testing spherical symmetry for the underlying distribution of high-dimensional data. Many existing statistics for this purpose were constructed by the theory of empirical processes and turn out...

Criterion is essential for measuring the goodness of an experimental design. In this paper, lower bounds of various criteria in experimental designs will be reviewed according to methodology of their construction. The criteria include most well-known ones which are frequently used as benchmarks for orthogonal array, uniform design, supersaturated d...

Computer experiments have been widely used in various fields of industry, system engineering, and others because many physical phenomena are difficult or even impossible to study by conventional experimental methods. Design and modeling of computer experiments have become a hot topic since late Seventies of the Twentieth Century. Almost in the same...

Uniform designs has been widely used in various fields. In this talk, two constructions for uniform designs from combinatorics are presented. One is obtained by re- solvable PPBDs which serves to unify some known combina- torial constructions. The other provides a recursive method to obtain a new three-level uniform design from old one.

Fault-tolerant design for memory production is getting to play an important role on increasing the yield rate of manufacturing. Nowadays, the laser cutting technology improves the yield of memories because of the enhancement of the use of spare lines interleaved in the memory. However, the issue of choosing a cutting location affects the utilizatio...

Fractional factorial designs are popular and widely used for industrial experiments. Generalized minimum aberration is an important criterion recently proposed for both regular and non-regular designs. This paper provides a formal optimization treatment on optimal designs with generalized minimum aberration. New lower bounds and optimality results...

This paper considers the construction of optimal mixed-level supersaturated designs under the E(χ2)-optimality criterion. According to a lower bound of the E(χ2) value, a combinatorial construction method is proposed and illustrated. Meanwhile, a new class of combinatorial designs is put forward. Furthermore, some methods for constructing these com...

Riccomagno et al. [1997. Lattice-based D-optimum design for Fourier regression. Ann. Statist. 25, 2313-2317] gave a comprehensive study on D-optimality under a symmetric Fourier regression model and gave a characterization on orthogonality, complete orthogonality and D-optimality of lattice designs under a symmetrical Fourier model. This paper exte...

In this paper we provide rather weak conditions on a distribution which would guarantee that the t-statistic of a random vector of order n follows the t-distribution with n-1 degrees of freedom. The results sharpen the earlier conclusions of Mauldon [Characterizing properties of statistical distributions, Quart. J. Math. 2(7) (1956) 155-160] and th...

A general Kriging model consists of two additive components: a parametric term and a stochastic error process. It is known that Kriging is an interpolating predictor and allows for a better fit to the data, but suffers from a decreasing ability to generalize to unseen data. By incorporating a disturbing or an independent random error term into Krig...

This paper introduces an optimality criterion for mixed-level supersaturated designs, called E(χ2), which covers E(s2) for two-level designs and ave(χ2) for symmetrical designs as two special cases. Some lower bounds of E(χ2) are obtained. Furthermore, it is shown that the E(χ2) criterion coincides with other existing criteria for nonregular mixed-...

Supersaturated design is essentially a fractional factorial design in which the number of potential effects is greater than the number of runs. In this article, the supersaturated design is applied to a computer experiment through an example of steady current circuit model problem. A uniform mixed-level supersaturated design and the centered quadra...

In recent years, there has been increasing interest in the study of asym-metrical fractional factorial designs. Various new optimality criteria have been proposed from different principles for design construction and comparison, such as generalized minimum aberration, minimum moment aberration, minimum projec-tion uniformity and the χ 2 (D) (for de...

The commonly used approach for building a structure-activity/property relationship consists of three steps. First, one determines the descriptors for the molecular structure, then builds a metamodel by using some proper mathematical methods, and finally evaluates the meta-model. Some existing methods only can select important variables from the can...

With reference to a general class of empirical-type likelihoods, we develop higher-order asymptotics for the frequentist coverage of Bayesian credible sets based on posterior quantiles and highest posterior density. These asymptotics, in turn, characterise members of the class that allow approximate frequentist validity of such sets. It is seen tha...

Given two covariance matricesR andS for a given elliptically contoured distribution, we show how simple inequalities between the matrix elements imply thatER(f)?ES(f), e.g., whenx=(xi1,i2,...,in) is a multiindex vector and $$f(x) = \mathop {\min }\limits_{i_1 } \mathop {\max }\limits_{i_2 } \mathop {\min }\limits_{i_3 } \max ...x_{i_{1,...,} i_n }...

When the number of runs is large, to search for uniform designs in the sense of low-discrepancy is an NP hard problem. The number of runs of most of the available uniform designs is small (≤ 50). In this article, the authors employ a kind of the so-called Hamming distance method to construct uniform designs with two- or three-level such that some r...

The uniform design is one of major designs in computer experiments. In this paper, we develop the connection among the computer experiments, uniform designs in statistics and resolvable partially pairwise balanced designs (RPPBDs) from design theory. A series of RPPBDs with special requirements are constructed and used to produce new uniform design...

A novel microarray strategy, three-color cDNA microarray experiment, is originally applied to assess drug effects on gene expression patterns of “target disease.” By adding Alexa 594 as a dye-label for the third target sample, it is made possible to monitor changes in gene expression in response to disease and generate clues to gene function with t...

A high-performance liquid chromatographic (HPLC) fingerprint of Chinese Angelica (CA) was developed basing on the consistent chromatograms of 40 CA samples (Angelica sinensis (Oliv.) Diels). The unique properties of this HPLC fingerprints were validated by analyzing 13 related herbs including 4 Japanese Angelicae Root samples (JA, A. acutiloba Kita...

New lower bounds for three- and four-level designs under the cen- tered L2-discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider sev- eral modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the...

A new approach of estimating parameters in multivariate models is introduced. A fitting function will be used. The idea is to estimate parameters so that the fitting function equals or will be close to its expected value. The function will be decomposed into two parts. From one part, which will be independent of the mean parameters, the dispersion...

The role of uniformity measured by the centered L 2-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L 2-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and thre...

When the number of runs is large, to search for uniform designs in the sense of low-discrepancy is an NP hard problem. The number of runs of most of the available uniform designs is small (≤ 50). In this article, the authors employ a kind of the so-called Hamming distance method to construct uniform designs with two- or three-level such that some r...

PART I AN OVERVIEW INTRODUCTION Experiments and Their Statistical Designs Some Concepts in Experimental Design Computer Experiments Examples of Computer Experiments Space-Filling Designs Modeling Techniques Sensitivity Analysis Strategies for Computer Experiments and An Illustration Case Study Remarks on Computer Experiments Guidance of Reading Thi...

Computer experiments are very useful for improving vehicle designs. Construction of computer experiments is of crucial importance. There are many space filling designs for computer experiments. The uniformity and low-discrepancy sets have played an important role in the construction of designs for computer experiments. To gain insight into why spac...

By using the theory of spherical distributions and some properties of invariant statistics, we develop a stabilized uniform Q-Q plot for checking the multi-normality assumption in high-dimensional data analysis. Acceptance regions associated with the plot are given. Empirical performance of the acceptance regions is studied by Monte Carlo simulatio...