THE USE OF A CLASS OF FOLDOVER DESIGNS AS SEARCH DESIGNS

It is shown that members of a class of two-level nonorthogonal resolution IV designs with n factors are strongly resolvable search designs when k, the maximum number of two-factor interactions thought possible, equals one; weakly resolvable when k = 2 except when the number of factors is 6; and may not be weakly resolvable when k≥ 3.     

A characterization based on conditional expectations

We obtain the necessary and sufficient conditions so that any real function (x) is the conditional expectation E(h(X)/X≥x) of a random variable X with continuous distribution function, where h is a given real, continuous and strictly monotonic function.     

A very simple proof of the multivariate Chebyshev's inequality

Abstract

In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary random vector X with finite mean μ = E(X) and a positive definite covariance matrix V = Cov(X) whose Mahalanobis distance with respect to V to the mean μ is less than a fixed value. The main advantage of the proof is that it is a simple exercise for a first year probability course. An alternative proof based on principal components is also provided. This proof can be used to study the case of a singular covariance matrix V.     

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L ₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

A method for constructing supersaturated designs and its Es2 optimality

A lower bound for the Es² value of an arbitrary supersaturated design is derived. A general method for constructing supersaturated designs is proposed and shown to produce designs with n runs and m = k(n — 1) factors that achieve the lower bound for Es² and are thus optimal with respect to the Es² criterion. Within the class of designs given by the construction method, further discrimination can be made by minimizing the pairwise correlations and using the generalized D and A criteria proposed by Wu (1993). Efficient designs of 12, 16, 20 and 24 runs are constructed by following this approach.     

ESTIMATING A RATIO OF MEANS FROM BIVARIATE COUNTS WITH APPLICATIONS IN STEREOLOGY

In survey sampling and in stereology, it is often desirable to estimate the ratio of means θ= E(Y)/E(X) from bivariate count data (X, Y) with unknown joint distribution. We review methods that are available for this problem, with particular reference to stereological applications. We also develop new methods based on explicit statistical models for the data, and associated model diagnostics. The methods are tested on a stereological dataset. For point‐count data, binomial regression and bivariate binomial models are generally adequate. Intercept‐count data are often overdispersed relative to Poisson regression models, but adequately fitted by negative binomial regression.     

A REVIEW OF BOUNDS FOR THE EFFICIENCY FACTOR OF BLOCK DESIGNS

This paper draws together bounds for the efficiency factor of block designs, starting with the papers of Conniffe & Stone (1974) and Williams & Patterson (1977). By extending the methods of Jarrett (1983), firstly to cover supercomplete block designs and then to cover resolvable designs, a set of bounds is obtained which provides the best current bounds for any block design with equal replication and equal block size, including resolvable designs and two-replicate resolvable designs as special cases. The bounds given for non-resolvable designs apply strictly only to designs which are either regular-graph (John & Mitchell, 1977) or whose duals are regular-graph. It is conjectured (John & Williams, 1982) that they are in fact global bounds. Similar qualifications apply to the bounds for resolvable designs.     

Marginally restricted sequential D‐optimal designs

In many experiments, not all explanatory variables can be controlled. When the units arise sequentially, different approaches may be used. The authors study a natural sequential procedure for “marginally restricted” D‐optimal designs. They assume that one set of explanatory variables (x₁) is observed sequentially, and that the experimenter responds by choosing an appropriate value of the explanatory variable x₂. In order to solve the sequential problem a priori, the authors consider the problem of constructing optimal designs with a prior marginal distribution for x₁. This eliminates the influence of units already observed on the next unit to be designed. They give explicit designs for various cases in which the mean response follows a linear regression model; they also consider a case study with a nonlinear logistic response. They find that the optimal strategy often consists of randomizing the assignment of the values of x₂.     

A New Test for New Better Than Used in Expectation Lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. A popular nonparametric model of aging is new better than used in expectation (NBUE), when M(t) ? M(0) for all t ? 0. The exponential distribution lies at the boundary. There is a large literature on testing exponentiality against NBUE alternatives. However, comparisons of tests have been made only for alternatives much stronger than NBUE. We show that a new Kolmogorov-Smirnov type test is much more powerful than its competitors in most cases.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

A note on some negative dependence notions

A random vector X = (X ₁,…,X _n ) is negatively associated if and only if for every pair of partitions X ₁ = (X _π(1),…,X _π(k)), X ₂ = (X _π(k+1),…,X _π(n)) of X , P( X ₁ ? A, X ₂ ? B) ≤ P( X ₁ ? A)P( X ₂ ? B) whenever A and B are open upper sets and π is any permutation of {1,…,n}. In this paper, we develop some of concepts of negative dependence, which are weaker than negative association but stronger than negative orthant dependence by requiring the above inequality to hold only for some upper sets A and B and applying the arguments in Shaked.     

Estimated Generalized Least Squares for a Heteroscedastic Regression Model

This paper considers the general linear regression model yc = X₁β+u_t under the heteroscedastic structure E(u_t) = 0, E(u²) =σ²- (X_tβ)², E(u_t u_s) = 0, tæs, t, s= 1, T. It is shown that any estimated GLS estimator for β is asymptotically equivalent to the GLS estimator under some regularity conditions. A three-step GLS estimator, which calls upon the assumption E(u_t²) =s?²(X,β)² for the estimation of the disturbance covariance matrix, is considered.     

Construction of group divisible designs and rectangular designs from resolvable and almost resolvable balanced incomplete block designs

In this paper, we present a general construction of group divisible designs and rectangular designs by utilising resolvable and “almost resolvable” balanced incomplete block designs. As special cases, we obtain the following E-optimal designs: (a) Group divisible (GD) designs with λ₂=λ₁+1 and (b) Rectangular designs with 2 rows and having λ₃=λ₂−1=λ₁+1. Many of the GD designs are optimal among binary designs with regard to all type 1 criteria.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

General upper bound for the number of blocks having a given number of treatments common with a given block

The purpose of this paper is systematically to derive the general upper bound for the number of blocks having a given number of treatments common with a given block of certain incomplete block designs. The approach adopted here is based on the spectral decomposition of N′N for the incidence matrix N of a design, where N' is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah 1964, 1966), Kapadia (1966)) of using well known relations between blocks of the designs and their association schemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further generalizations of the main theorem are discussed with some illustrations.     

A compound beta distribution with applications in finance

If X and Y are gamma distributed independent random variables then it is well known that the ratio X / (X + Y) has the beta distribution. In this note, the distribution of W = X / (X + Y) is considered when X and Y have the compound gamma distribution. We refer to the distribution of W as compound beta and describe an application to consumer price indices to show that compound beta is a better model than one based on the standard beta distribution. We derive various properties of W, including its probability density function, cumulative distribution function, hazard rate function and moments.     

αn–Designs

This paper defines a broad class of resolvable incomplete block designs called α_n–designs, of which the original α–designs are a special case with n = 1. The statistical and mathematical properties of α–designs extend naturally to these n –dimensional designs. They are a flexible class of resolvable designs appropriate for use in factorial experiments, in constructing efficient t –latinized resolvable block designs, and for enhancing the existing class of α–designs for a single treatment factor.     

√n-CONSISTENT ESTIMATION IN A RANDOM COEFFICIENT AUTOREGRESSIVE MODEL

This paper deals with √n-consistent estimation of the parameter μ in the RCAR(l) model defined by the difference equation X_j=(μ+U_j)X_j-l+ej (jε Z), where {e_j: jε Z} and {U_j: jε Z} are two independent sets of i.i.d. random variables with zero means, positive finite variances and E[(μ+U₁)²] < 1. A class of asymptotically normal estimators of μ indexed by a family of bounded measurable functions is introduced. Then an estimator is constructed which is asymptotically equivalent to the best estimator in that class. This estimator, asymptotically equivalent to the quasi-maximum likelihood estimator derived in Nicholls & Quinn (1982), is much simpler to calculate and is asymptotically normal without the additional moment conditions those authors impose.     

Optimal designs for approximating the path of a stochastic process

We consider a centered stochastic process {X(t):t ∈ T} with known and continuous covariance function. On the basis of observations X(t₁), …, X(t_n) we approximate the whole path by orthogonal projection and measure the performance of the chosen design d = (t₁, …, t_n)′ by the corresponding mean squared L²-distance. For covariance functions on T² = [0, 1]², which satisfy a generalized Sacks-Ylvisaker regularity condition of order zero, we construct asymptotically optimal sequences of designs. Moreover, we characterize the achievement of a lower error bound, given by Micchelli and Wahba (1981), and study the question of whether this bound can be attained.