On tail parameter estimation in certain point process models

Let μ(ds, dx) denote Poisson random measure with intensity dsG(dx) on (0, ∞) × (0, ∞), for a measure G(dx) with tails varying regularly at ∞. We deal with estimation of index of regular variation α and weight parameter ξ if the point process is observed in certain windows K_n = [0, T_n] × [Y_n, ∞), where Y_n → ∞ as n → ∞. In particular, we look at asymptotic behaviour of the Hill estimator for α. In certain submodels, better estimators are available; they converge at higher speed and have a strong optimality property. This is deduced from the parametric case G(dx) = ξαx^−α−1 dx via a neighbourhood argument in terms of Hellinger distances.     

A note on the construction of optimal linear codes

Let us denote by (n,k,d)-code, a binary linear code with code length nk information symbols and the minimum distance d. It is well known that the problem of obtaining a binary linear code whose code length n is minimum among (n,k,d)-codes for given integers k and d, is equivalent to solve a linear programming whose solutions correspond to a minimum redundancy error-correcting code. In this paper it will be shown that for some given integers d, there exists no solution of the linear programming except a solution which is obtained using a flat in a finite projective geometry.     

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.     

On a problem of counting by weighing

Suppose it is desired to obtain a large number N_s of items for which individual counting is impractical, but one can demand a batch to weigh at least w units so that the number of items N in the batch may be close to the desired number N_s. If the items have mean weight ωTH, it is reasonable to have w equal to ωTHN_s when ωTH is known. When ωTH is unknown, one can take a sample of size n, not bigger than N_s, estimate ωTH by a good estimator ω_n, and set w equal to ω_nN_s. Let R_n = Kp²N²_s/n + K_sn be a measure of loss, where K_e and K_s are the coefficients representing the cost of the error in estimation and the cost of the sampling respectively, and p is the coefficient of variation for the weight of the items. If one determines the sample size to be the integer closest to pCN_s when p is known, where C is (K_e/K_s)^1/2, then R_n will be minimized. If p is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given ω and ω has a prior inverted gamma distribution, the optimal sample size can be found to be the nonnegative integer closest to pCN_s + p²A(pC – 1), where A is a known constant given in the prior distribution.     

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.     

Weighted A-optimality for fractional 2m factorial designs of resolution V

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.     

Embedding of orthogonal arrays of strength two and deficiency greater than two

Let x ≥ 0 and n ≥ 2 be integers. Suppose there exists an orthogonal array $\text{A(n, q, μ}{}^1\text{)}$ of strength 2 in n symbols with q rows and $\text{n}{}^2\text{μ}{}^1$ columns where $\text{q = q}{}^1\text{? d, q}{}^1\text{= n}{}^2\text{x + n + 1}$, $\text{μ}{}^1\text{= (n ?}\text{1}\text{)x +}\text{1}$ and d is a positive integer. Then d is called the deficiency of the orthogonal array. The question of embedding such an array into a complete array $\text{A(n, q}{}^1\text{, μ}{}^1\text{)}$ is considered for the case d ≥ 3. It is shown that for d = 3 such an embedding is always possible if n ≥ 2(d ? 1)²(2d² ? 2d + 1). Partial results are indicated if d ≥ 4 for the embedding of a related design in a corresponding balanced incomplete block design.     

A lower bound to the measure of optimality for main effect plans in the general asymmetric factorial experiments

A design d is called D-optimal if it maximizes det(M _d ) and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this paper, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s ₁×s ₂×···×s _m levels asymmetric orthogonal array of strength at least 1. Nonisomorphic asymmetrical MS-optimal orthogonal arrays of strength 1 with N=6, 8 and 12 runs are also presented.     

On block-transitive designs with affine automorphism group

Detailed necessary and sufficient conditions for a k-subset of AG(d, 3) to generate the block set of a block-transitive t-design with automorphism group AGL(d, 3) are derived for t = 3, 4, 5. Similar necessary conditions are found for the existence of a block-transitive design with automorphism group AGL(d, p) when p is an arbitrary odd prime. A search was carried out to find feasible parameter sets satisfying the implied divisibility conditions. The only ‘small’ feasible parameter sets found with k or v − k not exceeding 1000 were for t = 4 and (d, p) = (7, 3), (8, 3), and (3, 7). Examples of block-transitive 4-designs admitting AGL(7, 3) are found for each of the values k = 115, 116, 230, 437, and 552.     

Coupon collector's problem and generalized Pareto distributions

Let N_n={1,2,…,n}. We sample with replacement from the set N_n assuming that each element has probability 1/n of being drawn. Let M_n be the waiting time determined by certain stoping rules in the coupon collector's problem. We investigate models for the asymptotic behavior of the excesses of M_n over the high thresholds.     

Optimal designs for covariates’ models

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Idempotent and multivariate copulas with fractal support

Using special iterated function systems (IFS) Fredricks et al. (2005) constructed two-dimensional copulas with fractal supports and showed that for every s∈(1,2)$s\in (\mathrm{1,2})$ there exists a copula A whose support has Hausdorff dimension s. In the current paper we present a stronger version and prove that the same result holds for the subclass of idempotent copulas. Additionally we show that every doubly stochastic idempotent matrix N (having neither minimum nor maximum rank) induces a family of idempotent copulas such that, firstly, the corresponding Markov kernels transform according to N   and, secondly, the set of Hausdorff dimensions of the supports of elements of the family covers (1,2). Furthermore we generalize the IFS approach to arbitrary dimensions d≥2$d\ge 2$ and show that for every s∈(1,d)$s\in (1,d)$ we can find a d-dimensional copula whose support has Hausdorff dimension s.     

Arrays of strength s on two symbols

This paper gives necessary and sufficient conditions on σ, s, t and on μ, s, t for an array with s+t rows to have strength s and weight σ, or to be balanced and have strength s and weight μ. If a balanced array can exist, the conditions provide a construction. The solutions for t=1,2 are also given in an alternate form useful for the study of trim arrays. The balanced solution for t=1 is more detailed than that known so far, and permits one to determine whether or not a solution exists in possibly fewer steps.     

Regular automorphism groups on partial geometries

Recently, Ghinelli (Geom Dedicata (1992) 165–174) had studied the generalized quadrangles which admits automorphism groups acting regularly on the points. In this paper, we generalize her idea to partial geometries, pg(s + 1, t + 1, α). Some examples and basic properties are given. In particular, we prove that under certain conditions on the automorphism group and the lines, such a geometry is a translation net. Applying the results to the case when s = t and the automorphism group G is abelian, we find that either the geometry is a translation net or all the lines of the geometry are generated by a subset of G. Also, for this case, we conjecture that the parameter α is either s or s + 1, except (s, α) = (5, 2), and we have checked that it is true for s ⩽ 500.     

Cyclic designs for a covariate model

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. We restrict our attention to classes of designs d for which the number of observations N to be taken is a multiple of V, i.e. N = V × R with R ≥2, and each treatment level is observed R times. Among these designs, called here equireplicated, there is a subclass characterized by the following: the allocation matrix of each treatment level (for short, allocation matrix) is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. We call these designs cyclic. Besides having easy representation, the most efficient cyclic designs are often D-optimal in the class of equireplicated designs. A known upper bound for the determinant of the information matrix M(d) of a design, in the class of equireplicated ones, depends on the congruences of N and V modulo 4. For some combinations of parameter moduli, we give here methods of constructing families of D-optimal cyclic designs. Moreover, for some sets of parameters (N, V,K = V), where the upper bound on ∣M(d)∣ (for that specific combination of moduli) is not attainable, it is also possible to construct highly D-efficient cyclic designs. Finally, for N≤24 and V≤6, computer search was used to determine the most efficient design in the class of cyclic ones. They are presented, together with their respective efficiency in the class of equireplicated designs.     

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.     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.     

An inequality for random replacement sampling plans

A sample (X₁ …, X_n) is drawn from a population of size N. Karlin (1974) conjectured that for any function ? in a certain class of real-valued functions on the sample space, ? is at least as large for sampling with replacement as for any other random replacement sampling plan. This conjecture is proved under the assumption that ?     

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.