Concerning the vector v(m,t)

Let q = mt + 1 be a prime power, and let v(m, t) be the (m + 1)-vector (b₁, b₂, …, b_{m + 1}) of elements of GF(q) such that for each k, 1 ⩽ k ⩽ m + 1, the set {b_i−b_j:i∈{1,2,…m+1} − {m + 2 − k}, j  i + k(mod m + 2) and 1⩽j⩽m+1} forms a system of representatives for the cyclotomic classes of index m in GF(q). In this paper, we investigate the existence of such vectors. An upper bound on t for the existence of a v(m, t) is given for each fixed m unless both m and t are even, in which case there is no such a vector. Some special cases are also considered.     

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.     

DECOMPOSITION OF A SEMI-MARKOV PROCESS UNDER A MARKOVIAN RULE

Consider a semi-Markov process {X(t), t>0} with transition epochs T₀ T₁, T₂…. Suppose that at each one of the epochs {T_n} one of R possible events, E₁, E₂,…, E_R can happen, where the occurrences of successive events form a Markov chain. for a fixed r, let the times the event E_r happens be U_o U₁, U₂,…. In this paper we are interested in the process {Y(t), t>0)} where Y(t)=X(U_k) if and only if U_k≤tk+1. It will be shown that {Y(t)} is a semi-Markov process, and its properties with respect to those of {X(t)} will be examined.     

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and c_i(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[¹¹¹⁰₀₁₂₁], c₁=[¹₀], c₂=[⁰₂].) Let T_i (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the s^n?r solutions of the equations At=c_i, wheret′=(t₁,…,t_n) is a vector of variables.(Thus, EG(4, 3) consists of the 3⁴=81 points of the form (t₁,t₂,t₃,t₄), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=c_i is s^n?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T₁ and T₂ are 2-flats consisting of 3^4?2=9 points each. The flats T₁,T₂,…,T_f are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T₁ are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T₂ consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sⁿ symmetric factorial experiment obtained by taking T₁,T₂,…,T_f together. (Thus, in the example, 3⁴=81 treatments of the 3⁴ factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T₁ and T₂ enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X._j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′._jX._j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′._jX._j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X._j into the X_ij (i=1,…,f). Furthermore, the X_ij are in turn partitioned into smaller matrices (which we shall here call the) X_ijh. We show that each X_ijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X._j and X′._jX._j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.     

Nonparametric location-scale models for censored successive survival times

Let (T₁,T₂) be gap times corresponding to two consecutive events, which are observed subject to (univariate) random right-censoring. The censoring variable corresponding to the second gap time T₂ will in general depend on this gap time. Suppose the vector (T₁,T₂) satisfies the nonparametric location-scale regression model T₂=m(T₁)+σ(T₁)?, where the functions m and σ are ‘smooth’, and ? is independent of T₁. The aim of this paper is twofold. First, we propose a nonparametric estimator of the distribution of the error variable under this model. This problem differs from others considered in the recent related literature in that the censoring acts not only on the response but also on the covariate, having no obvious solution. On the basis of the idea of transfer of tail information (Van Keilegom and Akritas, 1999), we then use the proposed estimator of the error distribution to introduce nonparametric estimators for important targets such as: (a) the conditional distribution of T₂ given T₁; (b) the bivariate distribution of the gap times; and (c) the so-called transition probabilities. The asymptotic properties of these estimators are obtained. We also illustrate through simulations, that the new estimators based on the location-scale model may behave much better than existing ones.     

Construction of main effect plus two plans for 2m factorials

This paper gives a main effect plus two plan (MEP.2 plan) for 2^m factorials (m⩾4) in the same setup as in Mukerjee and Chatterjee (Utilitas Math. 45 (1994) 213) in a smaller number of treatments. For a certain design T₁, combinatorial properties on a design T₂ are presented so that the design T=T₁+T₂ is a MEP.2 plan, where “+” stands for a union of two designs. Our results are more flexible for the choice of T₂ than the results of Mukerjee and Chatterjee (Utilitas Math. 45 (1994) 213).     

A Restricted Subset Selection Rule for Selecting at Least One of the t Best Normal Populations in Terms of Their Means When Their Common Variance is Known,Case II

Consider k( ? 2) normal populations with unknown means μ₁, …, μ_k, and a common known variance σ². Let μ_[1] ? ??? ? μ_[k] denote the ordered μ_i.The populations associated with the t(1 ? t ? k ? 1) largest means are called the t best populations. Hsu and Panchapakesan (2004) proposed and investigated a procedure R_HPfor selecting a non empty subset of the k populations whose size is at most m(1 ? m ? k ? t) so that at least one of the t best populations is included in the selected subset with a minimum guaranteed probability P* whenever μ_{[k ? t + 1]} ? μ_{[k ? t]} ? δ*, where P*?and?δ* are specified in advance of the experiment. This probability requirement is known as the indifference-zone probability requirement. In the present article, we investigate the same procedure R_HP for the same goal as before but when k ? t < m ? k ? 1 so that at least one of the t best populations is included in the selected subset with a minimum guaranteed probability P* whatever be the configuration of the unknown μ_i. The probability requirement in this latter case is termed the subset selection probability requirement. Santner (1976) proposed and investigated a different procedure (R_S) based on samples of size n from each of the populations, considering both cases, 1 ? m ? k ? t and k ? t < m ? k. The special case of t = 1 was earlier studied by Gupta and Santner (1973) and Hsu and Panchapakesan (2002) for their respective procedures.     

On computer generation of balanced arrays

This paper provides an algebraic (and hence computing) procedure for generation of balanced arrays having two symbols, m rows, specified minimum and maximum column weights, arbitrary strength t≦m, and index set parameters μ^t₁, μ^t₂,…, μ^t_t. μ^t₀ is unspecified, and calculated as part of the algorithm, although the procedure for specifying it is straightforward and can be used if desired. Array generation is herein reduced to finding integral solutions to a linear programming problem. It is shown that the integral solutions of the system of equations comprise all balanced arrays with the given set of parameters.A computing algorithm is provided which constructs the system of equations to be solved; it has been interfaced with a standard linear programming package to provide some preliminary results.Additional algorithms whose development should result in substantial decreases in computing costs are discussed.     

Combination-Based Permutation Tests: Equipower Property and Power Behavior in Presence of Correlation

Multivariate combination-based permutation tests have been widely used in many complex problems. In this paper we focus on the equipower property, derived directly from the finite-sample consistency property, and we analyze the impact of the dependency structure on the combined tests. At first, we consider the finite-sample consistency property which assumes that sample sizes are fixed (and possibly small) and considers on each subject a large number of informative variables. Moreover, since permutation test statistics do not require to be standardized, we need not assume that data are homoscedastic in the alternative. The equipower property is then derived from these two notions: consider the unconditional permutation power of a test statistic T for fixed sample sizes, with V ? 2 independent and identically distributed variables and fixed effect δ, calculated in two ways: (i) by considering two V-dimensional samples sized m₁ and m₂, respectively; (ii) by considering two unidimensional samples sized n₁ = Vm₁ and n₂ = Vm₂, respectively. Since the unconditional power essentially depends on the non centrality induced by T, and two ways are provided with exactly the same likelihood and the same non centrality, we show that they are provided with the same power function, at least approximately. As regards both investigating the equipower property and the power behavior in presence of correlation we performed an extensive simulation study.     

Estimation in dynamic regression with an integrated process

This paper deals with a dynamic regression model y_t = αy_t−1 + βz_t + u_t, where z_t is an integrated process of order one abbreviated as z_t ∼ I(1). Generally speaking, nonstandard asymptotic theory is required to investigate asymptotic properties of statistics related to an integrated process and the asymptotic results are very different from standard ones. There are two distinctive properties in nonstandard asymptotics: the so-called ‘super-consistency’ or T-consistency (where T is a sample size) and the weak convergence to a functional of the Wiener process. In spite of z_t being involved in our model, however, it is shown that our asymptotic results are the same as in the standard asymptotics in classical dynamic regression models, or if the disturbance u_t is serially correlated the OLS estimators of α and β have √T-inconsistency. This is due to the cointegration between y_t−1 and z_t. Although this point was clarified by Park and Phillips (1989) in a general context, we examine this explicitly through our specific model and connect the standard asymptotic theory with the nonstandard one in our case. Furthermore we investigate the limiting properties of other statistics such as t-ratio, the Durbin-Watson test and h-test. We also propose a consistent estimator of α and β by making use of Durbin's 2-step method. Finally, we carry out simulation studies which support our theoretical results.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

Minimal 2-coverings of a finite affine space based on GF(2)

Let EG(m, 2) denote the m-dimensional finite Euclidean space (or geometry) based on GF(2), the finite field with elements 0 and 1. Let T be a set of points in this space, then T is said to form a q-covering (where q is an integer satisfying 1?q?m) of EG(m, 2) if and only if T has a nonempty intersection with every (m-q)-flat of EG(m, 2). This problem first arose in the statistical context of factorial search designs where it is known to have very important and wide ranging applications. Evidently, it is also useful to study this from the purely combinatorial point of view. In this paper, certain fundamental studies have been made for the case when q=2. Let N denote the size of the set T. Given N, we study the maximal value of m.     

Robustness of MML estimators based on censored samples and robust test statistics

We investigate the efficiences of Tiku's (1967) modified maximum likelihood estimators μ_c and σ_c (based on symmetrically censored normal samples) for estimating the location and scale parameters μ and σ of symmetric non-normal distributions. We show that μ_c and σ_c are jointly more efficient than x? and s for long-tailed distributions (kurtosis $\text{β}{}_2{}^1\text{=}\text{μ}{}_4\text{μ}{}_2{}^2\text{>4.2, β}{}_2{}^1\text{= 4.2}$ for the Logistic), and always more efficient than the trimmed mean μT and the matching sample estimate σ_T of σ. We also show that μ_c and σ_c are jointly at least as efficient as some of the more prominent “robust” estimators (Gross, 1976). We show that the statistic $\text{t}{}_{\mathrm{c}}\text{= μ}{}_{\mathrm{c}}\text{m}\text{σ}{}_{\mathrm{c}}\text{, m = n ?2r + 2rβ}$ (r is the number of observations censored on each side of the sample and β is a constant), is robust and powerful for testing an assumed value of μ. We define a statistic T_c (based on μ_c andσ_c) for testing that two symmetric distributions are identical and show that T_c is robust and generally more poweerful than the well-known nonparametric statistics (Wilcoxon, normal-score, Kolmogorov-Smirnov), against the important location-shift alternatives. We generalize the statistic T_c to test that k symmetric distibutions are identical. The asymptotic distributions of t_c and T_c are normal, under some very general regularity conditions. For small samples, the upper (lower) percentage points of t_c and T_c are shown to be closely approximated by Student's t-distributions. Besides, the statistics μ_c and σ_c (and hence t_c and T_c) are explicit and simple functions of sample observations and are easy to compute.     

Distribution of order statistics from selected subsets of concomitants

For a random sample of size n from an absolutely continuous bivariate population (X, Y), let X_i:n be the i th X-order statistic and Y_[i:n] be its concomitant. We study the joint distribution of (V_s:m, W_t:n−m), where V_s:m is the s th order statistic of the upper subset {Y_[i:n], i=n−m+1,…,n}, and W_t:n−m is the t th order statistic of the lower subset {Y_[j:n], j=1,…,n−m  } of concomitants. When m=⌈_np0⌉$m=\lceil {\mathit{np}}_0\rceil$, s=⌈_mp1⌉$s=\lceil {\mathit{mp}}_1\rceil$, and t=⌈(n−m)_p2⌉$t=\lceil (n-m)p_2\rceil$, 0<_pi<1,i=0,1,2$0<p_i<1,i=0,1,2$, and n→∞$n\to \infty$, we show that the joint distribution is asymptotically bivariate normal and establish the rate of convergence. We propose second order approximations to the joint and marginal distributions with significantly better performance for the bivariate normal and Farlie–Gumbel bivariate exponential parents, even for moderate sample sizes. We discuss implications of our findings to data-snooping and selection problems.     

Pmc-optimal nonparametric quantile estimator

According to Pitman's Measure of Closeness, if T₁and T₂are two estimators of a real parameter $[d], then T₁is better than T₂if Po[d]{\T₁-o[d] < \T₂-0[d]\} > 1/2 for all 0[d]. It may however happen that while T₁is better than T₂and T₂is better than T₃, T₃is better than T₁. Given q ? (0,1) and a sample X₁, X₂, ..., X_nfrom an unknown F ? F, an estimator T* = T*(X₁,X₂...X_n)of the q-th quantile of the distribution F is constructed such that P_F{\F(T*)-q\ <[d] \F(T)-q\} >[d] 1/2 for all F?F and for all T€T, where F is a nonparametric family of distributions and T is a class of estimators. It is shown that T* =X_j:n'for a suitably chosen jth order statistic.     

On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm $\text{6A6 = {tr(A′A)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for 6A6 will be given for a balanced fractional 2^m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ₀, λ₁,…, λ_m). This array is identical with a balanced array of strength m, m constraints and index set {λ₀, λ₁,…, λ_m}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize 6A6 will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ? N ? 16, (ii) m = 5, 16 ? N ? 32, and (iii) m = 6, 22 ? N ? 40.     

Bayes Estimation of Shift Point in Geometric Sequence

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,… X _n were observed from geometric population with parameter q ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in parameter q ₂. The Bayes estimates of m, q ₁, q ₂, reliability R ₁ (t) and R ₂ (t) at time t are derived for symmetric and asymmetric loss functions under informative and non informative priors. A simulation study is carried out.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.