A q-Pólya urn model and the q-Pólya and inverse q-Pólya distributions

A q-Pólya urn model is introduced by assuming that the probability of drawing a white ball at a drawing varies geometrically, with rate q, both with the number of drawings and the number of white balls drawn in the previous drawings. Then, the probability mass functions and moments of (a) the number of white balls drawn in a specific number of drawings and (b) the number of black balls drawn until a specific number of white balls are drawn are derived. These two distributions turned out to be q-analogs of the Pólya and the inverse Pólya distributions, respectively. Also, the limiting distributions of the q-Pólya and the inverse q-Pólya distributions, as the number of balls in the urn tends to infinity, are shown to be a q-binomial and a negative q-binomial distribution, respectively. In addition, the positive or negative q-hypergeometric distribution is obtained as conditional distribution of a positive or negative q-binomial distribution, given its sum with another positive or negative q-binomial distribution, independent of it.     

Multistage bandit problems

Consider two or more treatments with dichotomous responses. The total number N of experimental units are to be allocated in a fixed number r of stages. The problem is to decide how many units to assign to each treatment in each stage. Responses from selections in previous stages are available and can be considered but responses in the current stage are not available until the next group of selections is made. Information is updated via the Bayes theorem after each stage. The goal is to maximize the overall expected number of successes in the N units.Two forms of prior information are considered: (i) All arms have beta priors, and (ii) prior distributions have continuous densities. Various characteristics of optimal decisions are presented. For example, in most cases of (i) and (ii), the rate of the optimal size of the first stage cannot be greater than √N when r = 2.     

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.     

Discrete q-distributions on Bernoulli trials with a geometrically varying success probability

Consider a sequence of independent Bernoulli trials and assume that the odds of success (or failure) or the probability of success (or failure) at the ith trial varies (increases or decreases) geometrically with rate (proportion) q, for increasing i=1,2,…. Introducing the notion of a geometric sequence of trials as a sequence of Bernoulli trials, with constant probability, that is terminated with the occurrence of the first success, a useful stochastic model is constructed. Specifically, consider a sequence of independent geometric sequences of trials and assume that the probability of success at the jth geometric sequence varies (increases or decreases) geometrically with rate (proportion) q, for increasing j=1,2,…. On both models, let X_n be the number of successes up the nth trial and T_k (or W_k) be the number of trials (or failures) until the occurrence of the kth success. The distributions of these random variables turned out to be q-analogues of the binomial and Pascal (or negative binomial) distributions. The distributions of X_n, for n→∞$n\to \infty$, and the distributions of W_k, for k→∞$k\to \infty$, can be approximated by a q  -Poisson distribution. Also, as k→0$k\to 0$, a zero truncated negative q  -binomial distribution _Uk=_Wk|_Wk>0$U_k=W_k|W_k>0$ can be approximated by a q-logarithmic distribution. These discrete q-distributions and their applications are reviewed, with critical comments and additions. Finally, consider a sequence of independent Bernoulli trials and assume that the probability of success (or failure) is a product of two sequences of probabilities with one of these sequences depending only the number of trials and the other depending only on the number of successes (or failures). The q-distributions of the number X_n of successes up to the nth trial and the number T_k of trials until the occurrence of the kth success are similarly reviewed.     

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.     

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.     

Existence of certain class of duadic group algebra codes

Duadic codes are defined in terms of idempotents of a group algebra GF(q)G, where G is a finite group and gcd(q,|G|)=1. Under the conditions of (1) q=2^m, and (2) the idempotents are taken to be central and (3) the splitting is μ₋₁, we show that such duadic codes exist if and only if q has odd-order modulo |G|.     

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.     

On the Breakdown Properties of Some Multivariate M‐Functionals*

Abstract. For probability distributions on ? ^q, a detailed study of the breakdown properties of some multivariate M‐functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution‐free’ M‐functional of scatter is given. These include a symmetrized version of Tyler's M‐functional of scatter, and the multivariate t M‐functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M‐functional of scatter and of its symmetrized version are 1/q and , respectively. For the multivariate t M‐functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/( q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low‐dimensional subspaces.     

Maximum likelihood estimation under a finite mixture of generalized exponential distributions based on censored data

In this paper, the identifiability of a finite mixture of generalized exponential distributions (GE(τ, α)) is proved and the maximum likelihood estimates (MLE’s) of the parameters are obtained using EM algorithm based on a general form of right-censored failure times. The results are specialized to type-I and type-II censored samples. A real data set is introduced and analyzed using a mixture of two GE(τ, α) distributions and also using a mixture of two Weibull(α, β) distributions. A comparison is carried out between the mentioned mixtures based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the GE(τ, α) mixture model fits the data better than the other mixture model.     

Simulating from polynomial-normal distributions

The family of polynomial-normal distributions includes a number of widely used distributions, such as the Gram–Charlier and Edgeworth distributions. In this note, we present three simple algorithms, (i) CDF Inversion, (ii) Acceptance–Rejection, (iii) and Ratio–of–Uniforms, for simulating variates from a polynomial-normal distribution. Details on the efficiency of the Acceptance–Rejection and the Ratio–of–Uniforms algorithms and a comparison across the various implementations are provided.     

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.     

Partitioned difference families and almost difference sets

Partitioned difference families (PDFs) were first studied by Ding and Yin in conjunction with the construction of constant composition codes (CCCs). In 2008, Yin et al. presented the constructions of a number of infinite classes of PDFs based on known difference sets in GF(q). In this paper, we further investigate the constructions of PDFs by using known almost difference sets in GF(q), and establish some recursive constructions of PDFs. As their applications, we also get a number of perfect difference systems of sets (DSSs) over Z_q² with q odd prime.     

Efficient estimates in linear and nonlinear regression with heteroscedastic errors

In this paper we consider the heteroscedastic regression model defined by the structural relation Y = r(V, β) + s(W)ε, where V is a p-dimensional random vector, W is a q-dimensional random vector, β is an unknown vector in some open subset B of R^m, r is a known function from R^p × B into R, s is an unknown function on R^q, and ε is an unobservable random variable that is independent of the pair (V, W). We construct asymptotically efficient estimates of the regression parameter β under mild assumptions on the functions r and s and on the distributions of ε and (V, W).     

Families of power series distributions,with particular reference to the Lerch family

The paper revisits the concept of a power series distribution by defining its series function, its power parameter, and hence its probability generating function. Realization that the series function for a particular distribution is a special case of a recognized mathematical function enables distributions to be classified into families. Examples are the generalized hypergeometric family and the q-series family, both of which contain generalizations of the geometric distribution. The Lerch function (a third generalization of the geometric series) is the series function for the Lerch family. A list of distributions belonging to the Lerch family is provided.     

Pseudo confidence regions for the solution of a multivariate linear functional relationship

The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ₁, Λ₂,…, Λ_q) are unknown population means of an m(q+1)-variate normal distribution $\text{N}{}_{\mathrm{m(q+1)}}\text{(ζΩ?Σ)}$, where ζ′ = (μ′, Λ₁′, Λ₂′,…, Λ_q′Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ? denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means.A CR for γ with level of confidence 1 ? α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 ? α. Some simulation results are presented.     

On the q-differences of the generalized q-factorials

The q-differences of the generalized q-factorial of t of order n and increment h, at t = 0, are examined. These q-numbers are the coefficients of the expansion of the generalized q-factorial of t of order n and increment h into q-factorials of t with unit increment. A combinatorial interpretation of these coefficients as q-rook numbers of a constant jump Ferrers board is provided. Further, explicit expressions, recurrence relations, limiting expressions, orthogonality relation and other properties of these q-numbers are derived.     

Testing optimality of experimental designs for a regression model with random variables

Tsukanov (Theor. Probab. Appl. 26 (1981) 173–177) considers the regression model $\text{E(}\text{y}\text{|}\text{Z}\text{)=}\text{Fp}\text{+}\text{Zq}$, $\text{D(}\text{y}\text{|}\text{Z}\text{)=σ}{}^2\text{I}{}_{\mathrm{n}}$, where $\text{y}\text{(n×1)}$ is a vector of measured values,$\text{F}\text{(n×k)}$ contains the control variables, $\text{Z}\text{(n×l)}$ contains the observed values, and $\text{p}\text{(k×1)}$ and $\text{q}\text{(l×1)}$ are being estimated. Assuming that $\text{Z}\text{=}\text{FL}\text{+}\text{R}$, where $\text{L}\text{(k×l)}$ is non-random, and the rows of $\text{R}\text{(n×l)}$ are i.i.d. $\text{N(}\text{0}\text{,Σ)}$, we extend Tsukanov's results by (i) computing E(detH_p), where H_p is the covariance matrix of p?, the l.s.e. of p, (ii) considering ‘optimality in the mean’ for the largest root criterion, (iii) discussing these equations when the matrix R has a left-spherical distribution.     

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.