Natural Exponential Families and Generalized Hypergeometric Measures

Let ν be a positive Borel measure on ?ⁿ and _pF_q(a₁,…, a_p; b₁,…, b_q; s) be a generalized hypergeometric series. We define a generalized hypergeometric measure, μ_p,q := _pF_q(a₁,…, a_p; b₁,…, b_q;ν), as a series of convolution powers of the measure ν, and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968 Kemp , A. W. ( 1968 ). A wide class of discrete distributions and the associated differential equations . Sankhyā, Ser. A 30 : 401 – 410 . [Google Scholar]) family of distributions is an example of μ_p,q in which ν is a Dirac measure on ?. For the case in which ν is a Dirac measure on ?ⁿ, we relate μ_p,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994 Bar-Lev , S. K. , Bshouty , D. , Enis , P. , Letac , G. , Lu , I. , Richards , D. ( 1994 ). The diagonal natural exponential families on ? ⁿ and their classification . J. Theoret. Probab. 7 : 883 – 929 .[Crossref] , [Google Scholar]). For p < q, we show that certain measures μ_p,q can be expressed as the convolution of a sequence of independent multi-dimensional Bernoulli trials. For p = q, q + 1, we show that the measures μ_p,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.     

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.     

Subset selection based on likelihood from uniform and related popupulations

Let π_i (i=1,2,…, k) be charceterized by the uniform distribution on (a_i;b_i), where exactly one of a_i and b_i is unknown. With unequal sample sizes, suppose that from the k (>=2) given populations, we wish to select a random-size subset containing the one with the smllest value of θ_i= b_i - a_i. RuleR_i selects π if a likelihood-based k-dimensional confidence region for the unknown (θ₁,… θ_k) contains at least one point having θ_i as its smallest component. A second rule, R , is derived through a likelihood ratio and turns out to be that of Barr and prabhu whenthe sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g ( z ; θ_i) =M(z)Q(θ_i) if a(θ_i) < z <b(θ_i). Extensions to the cases when both a_i and b_i are unknown and when θ_j isof interest are indicated. 1<=j<=k     

Lois limites pour les statistiques d'ordre dans le cas non identiquement distribue

Let X₁, X₂,… be a sequence of independent random variables with distribution functions F₁, where 1 ≤ i ≤ n, and for each n ≥ 1 let X_1,n ≤… ≤ X_n,n denote the order statistics of the first n random variables. Under suitable hypotheses about the F₁, we characterize the limit distribution functions H(x) for which P(X_k,n ? a_nx + b_n) → H(x), where a_n > 0 and b_n are real constants. We consider the cases where κ = κ(n) satisfies √n {κ(n)/n — λ} → 0 and √n {κ(n)/n — λ} → ∞ separately.     

On characterizing distributions via regression of functions of generalized order statistics

Let X(1,n,m₁,k),X(2,n,m₂,k),…,X(n,n,m,k) be n generalized order statistics from a continuous distribution F which is strictly increasing over (a,b),−∞≤a<b≤∞, the support of F. Let g be an absolutely continuous and monotonically increasing function in (a,b) with finite g(a⁺),g(b⁻) and E(g(X)). Then for some positive integer s,1<s≤n, we give characterization of distributions by means of

     

On An Intriguing Distributional Identity

For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω₁ → Ω₂ a continuous, strictly increasing function, such that Ω₁∩Ω₂?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g^{? 1}(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = ^dψ^{? 1}(U) ? U for U ～ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.     

A characterization of some {3v1 + v3, 3v0 + v2; 3, 3}-minihypers and its applications to error-correcting codes

Recently, Hamada et al. (Des. Codes Cryptogr. 2(1992), 225–229) showed that there exists an [87,5,57;3]-code meeting the Grismer bound. But it is unknown whether or not an [87,5,57;3]-code is unique up to equivalence. In order to characterize all [87,5,57;3]-codes, it is sufficient to characterize all {2v₂ + 2v₃,2v₁ + 2v₂; 4, 3}-minihypers and in order to characterize all {2v₂ + 2v₃,2v₁ + 2v₂;4,3}-minihypers, it is necessary to characterize all {3v₁ + v₃,3v₀ + v₂;3,3}-minihypers, where v₀ = 0, v₁ = 1, v₂ = 4 and v₃ = 13 (cf. Hamada and Helleseth, J. Statist. Plann. Inference, 56 (1996)). The purpose of this paper is to characterize all {3v₁ + v₃,3v₀ + v₂;3,3}-minihypers.     

Jump-detection and curve estimation methods for discontinuous regression functions based on the piecewise B-spline function

Jump-detection and curve estimation methods for the discontinuous regression function are proposed in this article. First, two estimators of the regression function based on B-splines are considered. The first estimator is obtained when the knot sequence is quasi-uniform; by adding a knot with multiplicity p + 1 at a fixed point x₀ on support [a, b], we can obtain the second estimator. Then, the jump locations are detected by the performance of the difference of the residual sum of squares DRSS(x₀) (x₀ ∈ (a, b)); subsequently the regression function with jumps can be fitted based on piecewise B-spline function. Asymptotic properties are established under some mild conditions. Several numerical examples using both simulated and real data are presented to evaluate the performance of the proposed method.     

Regression analysis using order statistics

In this article, we study the joint distribution of X and two linear combinations of order statistics, a ^T Y ₍₂₎ and b ^T Y ₍₂₎, where a = (a ₁, a ₂) ^T and b = (b ₁, b ₂) ^T are arbitrary vectors in R ² and Y ₍₂₎ = (Y ₍₁₎, Y ₍₂₎) ^T is a vector of ordered statistics obtained from (Y ₁, Y ₂) ^T when (X, Y ₁, Y ₂) ^T follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and Loperfido (Test 17:370–380, 2008).     

Fast simulation of truncated Gaussian distributions

We consider the problem of simulating a Gaussian vector X, conditional on the fact that each component of X belongs to a finite interval [a _i ,b _i ], or a semi-finite interval [a _i ,+??). In the one-dimensional case, we design a table-based algorithm that is computationally faster than alternative algorithms. In the two-dimensional case, we design an accept-reject algorithm. According to our calculations and numerical studies, the acceptance rate of this algorithm is bounded from below by 0.5 for semi-finite truncation intervals, and by 0.47 for finite intervals. Extension to three or more dimensions is discussed.     

Nonparametric estimation of survival functions for incomplete observations when the life time distribution is proportionally related ot the censoring time distribution

Let X₁, X₂…,X_n be a random sample from [ILM0001] and let Y₁, …,Y_n be a random sample from [ILM0002]. Then instead of observing a complete sample X₁,…X_n, we can only observe the pairs Z_i. = min(X_i.,Y_i) and [ILM0003] In this paper, we consider estimation of survival function [ILM0004] when [ILM0005], where β is an unknown positive real number.

     

A characterization of some {3v2 + v3, 3v1 + v2; 3, 3}-minihypers and some [15, 4, 9; 3]-codes with B2 = 0

It is known (cf. Hill and Newton (Ars Combin., 25A (1988), 61–72; Designs Codes Cryptography, 2 (1992), 137–157) and Remark A.2 in the Appendix) that (1) there is no [14, 4, 9; 3]-code meeting the Griesmer bound and (2) if C is a [15, 4, 9; 3]-code then B₂ = 0 or 2 and (3) there is a one-to-one correspondence between the set of all nonequivalent [15, 4, 9; 3]-codes with B₂ = 0 and the set of all {3v₂ + v₃, 3v₁ + v₂: 3, 3}-minihypers, where v₁ = 1, v₂ = 4, v₃ = 13 and B₂ denotes the number of codewords of weight 2 in its dual code. Recently it has been shown by Eupen and Hill (Designs Codes Cryptography, 4 (1994) 271–282) that a [15, 4, 9; 3]-code with B₂ = 2 is unique up to equivalence. The purpose of this paper is to characterize all [15, 4, 9; 3]-codes with B₂ = 0 using the geometrical structure of the corresponding {3v₂ + v₃, 3v₁ + v₂; 3, 3}-minihypers. Those results give a complete characterization of [15, 4, 9; 3]-codes.     

Orthogonal [k − 1, k + 1]-factorizations in graphs

Let G be a graph. Let F={F₁,F₂,...F_d} be a factorization of G and H be a subgraph of G. If H has exactly one edge in common with F_i for all i = 1,2,…,d, then we say that F is orthogonal to H. In this paper it is proved that for any d-matching M of a [kd − 1, kd + 1]-graphG, there is a [k − 1, k + 1]-factorization of G orthogonal to M where k ⩾ 2 is an integer.     

Some improved ratio type estimators of population mean and ratio in finite population sample surveys

In this paper we present a class of ratio type estimators of the population mean and ratio in a finite population sample surveys with without replacement simple random sampling design, where information on an auxiliary variate x positively correlated with the main variate y is available. Large sample approximations to mean square errors (MSE) of these estimatorsare evaluated and their MSE's are compared with the MSE of the usual ratio estimator [ybar]_R of [ybar] the population mean of y. It is shown that under certain conditions these estimators are more efficient than [ybar]_R. When a prior knowledge of the value of thecoefficient of variation, c_y, of y is at hand, ratio type estimator, say [ybar]₁ of [ybar] is proposed. It is shown, under certain conditions, that [ybar]₁ is more efficient than [ybar]_R. When values of c_y, c_x and the population correlation coefficient ρ is at hand, then we have proposed another estimator, say [ybar]₂ of [ybar], which is always better than [ybar]_R as far as the efficiency is concerned. In fact, is [ybar] ₂ is shown to be even better than [ybar]₁. Finally estimators better than the usual ratio estimator [ybar]/[xbar] of [Ybar] are given.     

A lower bound for the transition density function of a stochastic differential equation

Let W_t be a one-dimensional Brownian motion on the probability space (Ω,F,P), and let dx_t = a(x_t)dt + b(x_t)dw_t, b²(x) > 0, be a one-dimensional Ito stochastic differential equation. For a(x) = a₀ + a₁x + … + a_nxⁿ on a bounded interval we obtain a lower bound for p(t,x,y), the transition density function of the homogeneous Markov process x_t, depending directly on the coefficients a₀,a₁, …, a_n, and b(x).     

Principal differential analysis with a continuous covariate: low-dimensional approximations for functional data

Given a collection of n curves that are independent realizations of a functional variable, we are interested in finding patterns in the curve data by exploring low-dimensional approximations to the curves. It is assumed that the data curves are noisy samples from the vector space span <texlscub>f ₁, …, f _m </texlscub>, where f ₁, …, f _m are unknown functions on the real interval (0, T) with square-integrable derivatives of all orders m or less, and m<n. Ramsay [Principal differential analysis: Data reduction by differential operators, J. R. Statist. Soc. Ser. B 58 (1996), pp. 495–508] first proposed the method of regularized principal differential analysis (PDA) as an alternative to principal component analysis for finding low-dimensional approximations to curves. PDA is based on the following theorem: there exists an annihilating linear differential operator (LDO) ? of order m such that ?f _i =0, i=1, …, m [E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, Theorem 6.2]. PDA specifies m, then uses the data to estimate an annihilating LDO. Smooth estimates of the coefficients of the LDO are obtained by minimizing a penalized sum of the squared norm of the residuals. In this context, the residual is that part of the data curve that is not annihilated by the LDO. PDA obtains the smooth low dimensional approximation to the data curves by projecting onto the null space of the estimated annihilating LDO; PDA is thus useful for obtaining low-dimensional approximations to the data curves whether or not the interpretation of the annihilating LDO is intuitive or obvious from the context of the data. This paper extends PDA to allow for the coefficients in the LDO to smoothly depend upon a single continuous covariate. The estimating equations for the coefficients allowing for a continuous covariate are derived; the penalty of Eilers and Marx [Flexible smoothing with B-splines and penalties, Statist. Sci. 11(2) (1996), pp. 89–121] is used to impose smoothness. The results of a small computer simulation study investigating the bias and variance properties of the estimator are reported.     

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.     

Strassen type limit points for moving averages of a wiener process

Let {W(s); 8 ≥ 0} be a standard Wiener process, and let β_N = (2a_N (log (N/a_N) + log log N)^-1/2, 0 < α_N ≤ N < ∞, where α_N↑ and α_N/N is a non-increasing function of N, and define γ_N(t) = β_N[W(n_N + ta_N) ? W(n_N)), 0 ≥ t ≥ 1, with n_N = N – a_N. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γ_N(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With a_N = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.     

On occupation times for a risk process with reserve-dependent premium

Consider a risk reserve process under which the reserve can generate interest. For constants a and b such that a<b, we study the occupation time T _a,b (t), which is the total length of the time intervals up to time t during which the reserve is between a and b. We first present a general formula for piecewise deterministic Markov processes, which will be used for the computation of the Laplace transform of T _a,b (t). Explicit results are then given for the special case that claim sizes are exponentially distributed. The classical model is discussed in detail.