Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

Best Interpolatory Approximation in Normed Linear Spaces

A theory of best approximation with interpolatory contraints from a finite-dimensional subspaceMof a normed linear spaceXis developed. In particular, to eachxX, best approximations are sought from a subsetM(x) ofMwhichdependson the elementxbeing approximated. It is shown that this “parametric approximation” problem can be essentially reduced to the “usual” one involving a certainfixedsubspaceM₀ofM. More detailed results can be obtained when (1) Xis a Hilbert space, or (2) Mis an “interpolating subspace” ofX(in the sense of [1]).     

Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

An n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213–233) to have an α-symmetric distribution, α > 0, if its characteristic function is of the form φ(|ξ₁|^α + … + |ξ_n|^α). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on ⁿ. A new class of “zonally” symmetric stable laws on ⁿ is defined, and series expansions are derived for their characteristic functions and densities.     

Some Asymptotic Formulae for Gaussian Distributions

This paper considers asymptotic expansions of certain expectations which appear in the theory of large deviation for Gaussian random vectors with values in a separable real Hilbert space. A typical application is to calculation of the “tails” of distributions of smooth functionals,p(r)=P{Φ(r⁻¹ξ)0},r→∞, e.g., the probability that a centered Gaussian random vector hits the exterior of a large sphere surrounding the origin. The method provides asymptotic formulae for the probability itself and not for its logarithm in a situation, where it is natural to expect thatp(r)=c′r^D exp{−c″r²}. Calculations are based on a combination of the method of characteristic functionals with the Laplace method used to find asymptotics of integrals containing a fast decaying function with “small” support.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

On semidirectly closed pseudovarieties of aperiodic semigroups

The aim of this work is to study the unknown intervals of the lattice of aperiodic pseudovarieties which are semidirectly closed and answer questions proposed by Almeida in his book “Finite Semigroups and Universal Algebra”. The main results state that the intervals [V^*(B₂),ER∩LR] and [V^*(B₂¹),ER∩A] are not trivial, and that both contain a chain isomorphic to the chain of real numbers. These results are a consequence of the study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B₂.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

Inequalities for predictive ratios and posterior variances in natural exponential families

The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x₁ and x₂ is defined to be p_g(x₁)p_g(x₂)/[p_g( )]² = h_g(x₁, x₂), where p_g(x₁) = ∫ ƒ(x₁θ) g(θ) dθ is the predictive distribution of x₁ with respect to the prior g. We prove that h_g(x₁, x₂) ≥ h_g^*(x₁, x₂) for all x₁ and x₂ if ƒ(xθ) is in the natural exponential family and Cov_gx(θ) ≥ Cov_g^*x(θ) in the Loewner sense, for all x on a straight line from x₁ to x₂. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g^* differ in that the “sample size”     

Fast pattern-matching on indeterminate strings

In a string x on an alphabet Σ, a position i is said to be indeterminate iff x[i] may be any one of a specified subset {λ₁,λ₂,…,λ_j} of Σ, 2j|Σ|. A string x containing indeterminate positions is therefore also said to be indeterminate. Indeterminate strings can arise in DNA and amino acid sequences as well as in cryptological applications and the analysis of musical texts. In this paper we describe fast algorithms for finding all occurrences of a pattern p=p[1..m] in a given text x=x[1..n], where either or both of p and x can be indeterminate. Our algorithms are based on the Sunday variant of the Boyer–Moore pattern-matching algorithm, one of the fastest exact pattern-matching algorithms known. The methodology we describe applies more generally to all variants of Boyer–Moore (such as Horspool's, for example) that depend only on calculation of the δ (“rightmost shift”) array: our method therefore assumes that Σ is indexed (essentially, an integer alphabet), a requirement normally satisfied in practice.     

Topological group criterion for in compact-open-like topologies, II

We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C(Y), which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ interacts with the point-wise addition + on C(Y) as: inversion is a homeomorphism and + is separately continuous. When is + jointly continuous, i.e. σ is a group topology? This is so if Y is Lindelöf and Čech-complete, and the converse generally fails. We show in the present paper: under the Continuum Hypothesis, for Y separable metrizable, if σ is a group topology, then Y is (Lindelöf and) Čech-complete, i.e. Polish. The proof consists in showing that if Y is not Čech-complete, then there is a family of compact sets in βY which is maximal in a certain sense.     

Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss

Let X ≡ (X₁, …, X_t) have a multinomial distribution based on N trials with unknown vector of cell probabilities p ≡ (p₁, …, p_t). This paper derives admissibility and complete class results for the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). Let and f(x¦p) denote the (t − 1)-dimensional simplex, the support of X and the probability mass function of X, respectively. First it is shown that δ is Bayes w.r.t. EL for prior P if and only if δ is Bayes w.r.t. SEL for P. The admissible rules under EL are proved to be Bayes, a result known for the case of SEL. Let Q denote the class of subsets of of the form T = _j=1^kF_j where k ≥ 1 and each F_j is a facet of which satisfies: F a facet of such that F naF_jF ncT. The minimal complete class of rules w.r.t. EL when N ≥ t − 1 is characterized as the class of Bayes rules with respect to priors P which satisfy P( ⁰) = 1, ξ(x) ≡ ∫ f(x¦p) P(dp) > 0 for all x in {x : sup ^₀ f(x¦p) > 0} for some ⁰ in Q containing all the vertices of . As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL when the estimation problem has parameter space Θ = but it is shown to be inadmissible for the problem with parameter space Θ = ( minus its vertices). This is a severe form of “tyranny of boundary.” Finally it is shown that when N ≥ t − 1 any estimator δ which satisfies δ(x) > 0 x is admissible under EL if and only if it is admissible under SEL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa.     

Effacement des dérivations et spectres premiers des algèbres quantiques

Given any (commutative) field k and any iterated Ore extension R=k[X₁][X₂;σ₂,δ₂][X_N;σ_N,δ_N] satisfying some suitable assumptions, we construct the so-called “Derivative-Elimination Algorithm.” It consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X₁,…,X_N) and terminating with new variables (T₁,…,T_N). These new variables generate some quantum-affine space such that . This algorithm induces a natural embedding which satisfies the following property:

Full-size image

. We study both the derivative-elimination algorithm and natural embedding and use them to produce, for the general case, a (common) proof of the “quantum Gelfand–Kirillov” property for the prime homomorphic images of the following quantum algebras: , (wW), R_q[G] (where G denotes any complex, semi-simple, connected, simply connected Lie group with associated Lie algebra and Weyl group W), quantum matrices algebras, quantum Weyl algebras and quantum Euclidean (respectively symplectic) spaces. Another application will be given in [G. Cauchon, J. Algebra, to appear]: In the general case, the prime spectrum of any quantum matrices algebra satisfies the normal separation property.     

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N.G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^card(Ω) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality card(Ω), the group Sym(Ω) contains a coproduct of 2^card(Ω) copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set V. From this one easily obtains analogs of the results stated above with “group” and Sym(Ω) replaced by “monoid” and the monoid Self(Ω) of endomaps of Ω, by “associative K-algebra” and the K-algebra End_K (V) of endomorphisms of a K-vector-space V with basis Ω, and by “lattice” and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K(V) contains a coproduct of 2^card(Ω) copies of itself.That paper also gave an example of a group of cardinality 2^card(Ω) that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_K(V), and Equiv(Ω), and also for the monoid Rel(Ω) of binary relations on Ω.Many open questions and areas for further investigation are noted.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.     

Normal Approximation Rate of the Kernel Smoothing Estimator in a Partial Linear Model

By establishing the asymptotic normality for the kernel smoothing estimatorβ_nof the parametric componentsβin the partial linear modelY=X′β+g(T)+, P. Speckman (1988,J. Roy. Statist. Soc. Ser. B50, 413–456) proved that the usual parametric raten^−1/2is attainable under the usual “optimal” bandwidth choice which permits the achievement of the optimal nonparametric rate for the estimation of the nonparametric componentg. In this paper we investigate the accuracy of the normal approximation forβ_nand find that, contrary to what we might expect, the optimal Berry–Esseen raten^−1/2is not attainable unlessgis undersmoothed, that is, the bandwidth is chosen with faster rate of tending to zero than the “optimal” bandwidth choice.     

Addendum to the Paper “The Generalized Integer Gamma Distribution—A Basis for Distributions in Multivariate Statistics”

A brief remark on the paper “The Generalized Integer Gamma Distribution— A Basis for Distributions in Multivariate Statistics,” (1998,J. Multivariate Anal.64, 86–102) and an additional result concerning the distribution of the product of some particular independent beta random variables, which broadens the scope of the results in that paper, are presented.     

Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

We consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H^1+n/p_p( ⁿ) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.     

Uniqueness of Minimal Projections in Smooth Matrix Spaces

Let X=(M(n, m), ·), where · fulfills Condition 0.3 and W=M(n, 1)+M(1, m). A formula for a minimal projection from X onto W is given in (E. W. Cheney and W. A. Light, 1985, “Approximation Theory in Tensor Product Spaces,” Lecture Notes in Mathematics, Springer-Verlag, Berlin; E. J. Halton and W. A. Light, 1985, Math. Proc. Cambridge Philos. Soc.97, 127–136; and W. A. Light, 1986, Math. Z.191, 633–643). We will show that this projection is the unique minimal projection (see Theorem 2.1).     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

The n-width of the unit ball of H

Let E be a compact subset of the open unit disc Δ and let H^q be the Hardy space of analytic functions f on Δ for which stf¦^q has a harmonic majorant. We determine the value of the Kolmogorov, Gel'fand, and linear n-widths in L^p(E, μ) of the restriction to E of the unit ball of H^q when p q or when 1 q < p < ∞ and E is “small”.