Sensitivity of Bayes decisions for the success probability for samples from a nonbinomial population

Summary A family of generalised negative binomial distributions is employed to investigate inference robustness of the Bayes estimator of the unknown parameter of the binomial distribution. A zone of sensitivity for the test of significance is constructed to forewarn the pro-Jeffreys Bayesians against indiscriminate choice of the probability in favour of the null hypothesis. A few selected tables are presented to illustrate the effect of relaxation of the ‘binomiality’ assumption.     

Bayes estimation of number of signals

Bayes estimation of the number of signals, q, based on a binomial prior distribution is studied. It is found that the Bayes estimate depends on the eigenvalues of the sample covariance matrix S for white-noise case and the eigenvalues of the matrix S ₂ (S ₁+A)^–1 for the colored-noise case, where S ₁ is the sample covariance matrix of observations consisting only noise, S ₂ the sample covariance matrix of observations consisting both noise and signals and A is some positive definite matrix. Posterior distributions for both the cases are derived by expanding zonal polynomial in terms of monomial symmetric functions and using some of the important formulae of James (1964, Ann. Math. Statist., 35, 475–501).     

Bounds on Spectra of Codes with Known Dual Distance

We estimate the interval where the distance distribution of a code of length n and of given dual distance is upperbounded by the binomial distribution. The binomial upper bound is shown to be sharp in this range in the sense that for every subinterval of size about √n ln n there exists a spectrum component asymptotically achieving the binomial bound. For self-dual codes we give a better estimate for the interval of binomiality.     

Binomial Matrices

Every s×s matrix A yields a composition map acting on polynomials on R ^s . Specifically, for every polynomial p we define the mapping C _A by the formula (C _A p)(x):=p(Ax), xR ^s . For each nonnegative integer n, homogeneous polynomials of degree n form an invariant subspace for C _A . We let A ⁽ⁿ⁾ be the matrix representation of C _A relative to the monomial basis and call A ⁽ⁿ⁾ a binomial matrix. This paper studies the asymptotic behavior of A ⁽ⁿ⁾ as n. The special case of 2×2 matrices A with the property that A ²=I corresponds to discrete Taylor series and motivated our original interest in binomial matrices.     

Estimation of parameters in the beta binomial model

This paper contains some alternative methods for estimating the parameters in the beta binomial and truncated beta binomial models. These methods are compared with maximum likelihood on the basis of Asymptotic Relative Efficiency (ARE). For the beta binomial distribution a simple estimator based on moments or ratios of factorial moments has high ARE for most of the parameter space and it is an attractive and viable alternative to computing the maximum likelihood estimator. It is also simpler to compute than an estimator based on the mean and zeros, proposed by Chatfield and Goodhart (1970,Appl. Statist.,19, 240–250), and has much higher ARE for most part of the parameter space. For the truncated beta binomial, the simple estimator based on two moment relations does not behave quite as well as for the BB distribution, but a simple estimator based on two linear relations involving the first three moments and the frequency of ones has extremely high ARE. Some examples are provided to illustrate the procedure for the two models.     

Computation of exact confidence limits from discrete data

Suppose that the data have a discrete distribution determined by (∞, ψ) where θ is the scalar parameter of interest and ψ is a nuisance parameter vector. The Buehler 1 - α upper confidence limit for θ is as small as possible, subject to the constraints that (a) its coverage probability is at least 1 - α and (b) it is a nondecreasing function of a pre-specified statisticT. This confidence limit has important biostatistical and reliability applications. The main result of the paper is that for a wide class of models (including binomial and Poisson), parameters of interest 9 and statisticsT (which possess what we call the “logical ordering” property) there is a dramatic increase in the ease with which this upper confidence limit can be computed. This result is illustrated numerically for θ a difference of binomial probabilities. Kabaila & Lloyd (2002) also show that ifT is poorly chosen then an assumption required for the validity of the formula for this confidence limit may not be satisfied. We show that for binomial data this assumption must be satisfied whenT possesses the “logical ordering” property.     

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45     

Empirical Bayes Estimation in Regression Model

This paper considers the empirical Bayes （EB） estimation problem for the parameter β of the linear regression model y = Xβ＋ ε with ε- N（0, σ^2I） given β. Based on Pitman closeness （PC） criterion and mean square error matrix （MSEM） criterion, we prove the superiority of the EB estimator over the ordinary least square estimator （OLSE）.     

Asymptotically minimum variance unbiased estimation for a class of power series distributions

Summary The problem of finding an asymptotically minimum variance unbiased estimator (A.M.V.U.E.) for the parameter of certain truncated power series distributions, is discussed, when the generating function of their coefficients are i) polynomials of binomial type ii) generalized ascending factorials iii) polynomials with coefficients the well known Eulerian numbers.     

Bernshtein's regularity conditions in a problem of empirical bayesian approach

It is proved that the posterior distribution of the random success parameter in a binomial distribution may be approximated by a beta distribution, if the second parameter of the binomial lawn, goes to infinity and the prior density of the success parameter belongs to some subset of, 1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 38–43, 1978.In conclusion, I would like to thank L. N. Bol'shev for suggesting the problem in this form in one of the lectures at the Steklov Mathematical Institute.     

Monogenic pseudo‐complex power functions and their applications

The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D‐elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd.     

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.     

On characterization of power series distributions by a marginal distribution and a regression function

The conditional distribution of Y given X=x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribution when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial.     

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

We consider orthogonal polynomials , where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight W_N which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit and link this limit to an external field for an equilibrium problem in logarithmic potential theory.     

A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Let F _q[X] denote a polynomial ring over a finite field F _q with q elements. Let 𝒫_n be the set of monic polynomials over F _q of degree n. Assuming that each of the qⁿ possible monic polynomials in 𝒫_n is equally likely, we give a complete characterization of the limiting behavior of P(Ω_n=m) as n→∞ by a uniform asymptotic formula valid for m≥1 and n−m→∞, where Ω_n represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in 𝒫_n. The distribution of Ω_n is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q⁻¹. Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerning P(Ω_n=m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Rényi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 17–47, 1998     

Percolation and limit theory for the poisson lilypond model

The lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d > 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Empirical Bayes test for scale exponential family

In this paper, we consider the empirical Bayes (EB) test problem for the scale parameters in the scale exponential family with a weighted linear loss function. The EB test rules are constructed by the kernel estimation method. The asymptotical optimality and convergence rates of the EB test rules are obtained. The main results are illustrated by applying the proposed test to type II censored data from the exponential distribution and to the test problem for the dispersion parameter in the linear regression model. __________ Translated from Journal of University of Science and Technology of China, 2004, 34(1): 1–10     

Estimating the Hurst Parameter

Let be a fractional Brownian motion with parameter 0 < H < 1. We are interested in the estimation of this parameter. To achieve this goal, we consider certain functionals of the second order increments of b _H (·), using variation technics. Based on an almost-sure convergence theorem for general functionals, we single out particular functionals that allows to construct certain regression models for the parameter H. We show that this regression based estimator for H is asymptotically unbiased, consistent and that it satisfies a Central Limit Theorem.      

二项分布参数多层Bayes和E Bayes估计的性质

讨论无失效数据下二项分布参数E Bayes估计和多层Bayes估计的性质,证明二项参数的多层Bayes估计和E Bayes估计渐近相等,且E Bayes估计值小于多层Bayes估计值.     

Negative binomial version of the Lee–Carter model for mortality forecasting

Mortality improvements pose a challenge for the planning of public retirement systems as well as for the private life annuities business. For public policy, as well as for the management of financial institutions, it is important to forecast future mortality rates. Standard models for mortality forecasting assume that the force of mortality at age x in calendar year t is of the form exp(α_x + β_xκ_t). The log of the time series of age-specific death rates is thus expressed as the sum of an age-specific component α_x that is independent of time and another component that is the product of a time-varying parameter κ_t reflecting the general level of mortality, and an age-specific component β_x that represents how rapidly or slowly mortality at each age varies when the general level of mortality changes. The parameters are usually estimated via singular value decomposition or via maximum likelihood in a binomial or Poisson regression model. This paper demonstrates that it is possible to take into account the overdispersion present in the mortality data by estimating the parameter in a negative binomial regression model. Copyright © 2007 John Wiley & Sons, Ltd.