The q-Bernstein basis as a q-binomial distribution

The q-Bernstein basis, used in the definition of the q-Bernstein polynomials, is shown to be the probability mass function of a q-binomial distribution. This distribution is defined on a sequence of zero–one Bernoulli trials with probability of failure at any trial increasing geometrically with the number of previous failures. A modification of this model, with the probability of failure at any trial decreasing geometrically with the number of previous failures, leads to a second q-binomial distribution that is also connected to the q-Bernstein polynomials. The q-factorial moments as well as the usual factorial moments of these distributions are derived. Further, the q-Bernstein polynomial B_n(f(t),q;x) is expressed as the expected value of the function f([X_n]_q/[n]_q) of the random variable X_n obeying the q-binomial distribution. Also, using the expression of the q-moments of X_n, an explicit expression of the q-Bernstein polynomial B_n(f_r(t),q;x), for f_r(t) a polynomial, is obtained.     

Flocks and partial flocks of quadrics: a survey

If O is an ovoid of PG(3,q), then a partition of all but two points of O into q−1 disjoint ovals is called a flock of O. A partition of a nonsingular hyperbolic quadric Q⁺(3,q) into q+1 disjoint irreducible conics is called a flock of Q⁺(3,q). Further, if O is either an oval or a hyperoval of PG(2,q) and if K is the cone with vertex a point x of PG(3,q)⧹PG(2,q) and base O, then a partition of K⧹{x} into q disjoint ovals or hyperovals in the respective cases is called a flock of K. The theory of flocks has applications to projective planes, generalized quadrangles, hyperovals, inversive planes; using flocks new translation planes, hyperovals and generalized quadrangles were discovered. Let Q be an elliptic quadric, a hyperbolic quadric or a quadratic cone of PG(3,q). A partial flock of Q is a set P consisting of β disjoint irreducible conics of Q. Partial flocks which are no flocks, have applications to k-arcs of PG(2,q), to translation planes and to partial line spreads of PG(3,q). Recently, the definition and many properties of flocks of quadratic cones in PG(3,q) were generalized to partial flocks of quadratic cones with vertex a point in PG(n,q), for n⩾3 odd.     

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.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

On cone representations of translation planes

Let F be a field of order q. It is known that an orthogonal array of the same order q has rank n over F if and only if it is represented as a cone cut by hyperplanes in n-dimensional space over F. Here we show that translation planes have a cone representation in (n + 1)-dimensional space over F, where n is the dimension of the plane over its kernel. If the plane is a semifield plane then the representation takes a particularly nice form. Rank 3 representations of Moulton planes are also briefly discussed.     

On the lengths of t-based confidence intervals

Confidence interval is a basic type of interval estimation in statistics. When dealing with samples from a normal population with the unknown mean and the variance, the traditional method to construct t-based confidence intervals for the mean parameter is to treat the n sampled units as n groups and build the intervals. Here we propose a generalized method. We first divide them into several equal-sized groups and then calculate the confidence intervals with the mean values of these groups. If we define “better” in terms of the expected length of the confidence interval, then the first method is better because the expected length of the confidence interval obtained from the first method is shorter. We prove this intuition theoretically. We also specify when the elements in each group are correlated, the first method is invalid, while the second can give us correct results in terms of the coverage probability. We illustrate this with analytical expressions. In practice, when the data set is extremely large and distributed in several data centers, the second method is a good tool to get confidence intervals, in both independent and correlated cases. Some simulations and real data analyses are presented to verify our theoretical results.     

A simple approximation relation between the symmetric generalized logistic and the students t distributions

In this paper, we obtain a new approximation of the Student's t distribution by using the symmetric generalized logistic (SGL) distribution function. The error of this approximation is shown to be 0(1/n² )where nis the degrees of freedom of thetdistribution. In comparison to similar approximations by George and Ojo and George et al. (1986), this new approximation is much simpler and more accurate. It is also shown that under some conditions, the tdistribution is a good approximation of the SGL distribution. Therefore, the complicated expressions for the cumulants and moments of the SGL can be approximated by those of the t, distribution. Finally, numerical results are given.     

Construction of orthogonal Latin hypercube designs with flexible run sizes

Latin hypercube designs (LHDs) have recently found wide applications in computer experiments. A number of methods have been proposed to construct LHDs with orthogonality among main-effects. When second-order effects are present, it is desirable that an orthogonal LHD satisfies the property that the sum of elementwise products of any three columns (whether distinct or not) is 0. The orthogonal LHDs constructed by Ye (1998), Cioppa and Lucas (2007), Sun et al. (2009) and Georgiou (2009) all have this property. However, the run size n of any design in the former three references must be a power of two (n=2^c) or a power of two plus one (n=2^c+1), which is a rather severe restriction. In this paper, we construct orthogonal LHDs with more flexible run sizes which also have the property that the sum of elementwise product of any three columns is 0. Further, we compare the proposed designs with some existing orthogonal LHDs, and prove that any orthogonal LHD with this property, including the proposed orthogonal LHD, is optimal in the sense of having the minimum values of ave(|t|), t_max, ave(|q|) and q_max.     

The number of solutions to the alternate matrix equation over a finite field and a q-identity

Let F_q be a finite field with q elements, where q is a power of a prime. In this paper, we first correct a counting error for the formula N(K_2ν,0^(m)) occurring in Carlitz (1954. Arch. Math. V, 19–31). Next, using the geometry of symplectic group over F_q, we have given the numbers of solutions X of rank k and solutions X to equation XAX′=B over F_q, where A and B are alternate matrices of order n, rank 2ν and order m, rank 2s, respectively. Finally, an elementary q-identity is obtained from N(K_2ν,0⁽⁰⁾), and the explicit results for N(K_n,2ν,K_m,2s) is represented by terminating q-hypergeometric series.     

Elation generalized quadrangles of order (p,t), p prime,are classical

Elation generalized quadrangles of order (p, t), p prime, are classical.     

On the quantile process based on the autoregressive residuals

Let {X_t} be the stationary AR(p) process satisfying the difference equation X_t=β₁X_t−1 + … + β_pX_t−p+ε_t, where {ε_t} is a sequence of iid random variables with mean zero and finite variance. Motivated by a goodness of fit test on the true errors {ε_t}, we are led to study the asymptotic behavior of the quantile process based on residuals (the residual quantile process). Particularly, we concentrate on the deviations between the residual quantile process and the empirical process based on the true errors. In this asymptotic study, it is shown that the deviations converge to zero in probability uniformly over certain intervals with specific order as sample size increases. Here, these intervals are allowed to vary with the sample size n and converge to the unit interval as n goes to infinity. Then, based on our result and the strong approximation result of Csörgö and Révész (1978), we propose a goodness of fit test statistic of which limiting distribution is the same as of a functional form of a standard Brownian bridge.     

Weight hierarchies of linear codes of dimension 3

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d₁,d₂,…,d_k), where d_r is the smallest support of an r-dimensional subcode of C. The weight hierarchies of [n,3;q] codes are studied. In particular, for q⩽5 the possible weight hierarchies of [n,3;q] codes are determined.     

Implications of survey design for generalized regression estimation of linear functions

The paper presents a general randomization theory approach to point and interval estimation of Q linear functions T_q = Σ^N₁c_kqY_k(q = 1,…,Q), where Y₁,…,Y_N are values of a variable of interest Y in a finite population. Such linear functions include population and domain means and totals, population regression coefficients, etc. We assume that some auxiliary information can be exploited. This suggests the generalized regression technique based on the fit of a linear model, whereby is created approximately design unbiased estimators T?_q. The paper focuses on estimation of the variance-covariance matrix of the T?_q for single stage and two stage designs. Two techniques based on Taylor expansions are compared. Results of Monte-Carlo experiments (not reported here) show that the coverage properties are good of normal-theory confidence intervals flowing from one or the other variance estimate.     

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.     

The Berry-Esseen bound for Studentized U-statistics

Callaert and Veraverbeke (1981) recently obtained a Berry-Esseen-type bound of order n^–1/2 for Studentized nondegenerate U-statistics of degree two. The condition these authors need to obtain this order bound is the finiteness of the 4.5th absolute moment of the kernel h. In this note it is shown that this assumption can be weakened to that of a finite (4 + ?)th absolute moment of the kernel h, for some ? > 0. Our proof resembles part of Helmers and van Zwet (1982), where an analogous result is obtained for the Student t-statistic. The present note extends this to Studentized U-statistics.     

A construction for generalized hadamard matrices GH(4q, EA(q))

We give a construction for a generalized Hadamard matrix GH(4q, EA(q)) as a 4 × 4 matrix of q × q blocks, for q an odd prime power other than 3 or 5. Each block is a GH(q, EA(q)) and certain combinations of 4 blocks form GH(2q, EA(q)) matrices. Hence a GH(4q, EA(q)) matrix exists for every prime power q.     

Inference Based on Order Statistics from the Generalized Pareto Distribution and Application

ABSTRACT

In this article, we derive exact explicit expressions for the single, double, triple, and quadruple moments of order statistics from the generalized Pareto distribution (GPD). Also, we obtain the best linear unbiased estimates of the location and scale parameters (BLUE's) of the GPD. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the generalized Pareto parameters using Edgeworth approximation and compare them with those based on Monte Carlo simulations. To show the usefulness of our results, we also present a numerical example. Finally, we give an application to real data.     

Generalized linear mixed models for correlated binary data with t-link

A critical issue in modeling binary response data is the choice of the links. We introduce a new link based on the Student’s t-distribution (t-link) for correlated binary data. The t-link relates to the common probit-normal link adding one additional parameter which controls the heaviness of the tails of the link. We propose an interesting EM algorithm for computing the maximum likelihood for generalized linear mixed t-link models for correlated binary data. In contrast with recent developments (Tan et al. in J. Stat. Comput. Simul. 77:929–943, 2007; Meza et al. in Comput. Stat. Data Anal. 53:1350–1360, 2009), this algorithm uses closed-form expressions at the E-step, as opposed to Monte Carlo simulation. Our proposed algorithm relies on available formulas for the mean and variance of a truncated multivariate t-distribution. To illustrate the new method, a real data set on respiratory infection in children and a simulation study are presented.