A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

In this paper we consider the TJW product-limit estimatorF_n(x) of an unknown distribution functionFwhen the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimatorF_n(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation ofF_n(x)−F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator toF. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given.     

A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).     

Approximation and estimation of some compound distributions

The distribution of the total amount claimed up to time t can often be written in the form of a compound distribution G_t(x) = Σp_n(t)F⁽ⁿ⁾(x) where p_n(t) is the probability of exactly n claims while F is the distribution of a single claim. In the actuarial literature one often finds approximations of G_t(x) when the time t is large. It seems more natural to take t fixed and to look for approximations for x large. This paper contains a number of such results for a Poisson process and for a Pascal process. Different hypotheses on the tail behaviour of F(t) yield different expressions to estimate 1 - G_t(x). The results obtained should prove to have wider applicability than suggested by the insurance context. Within it, however, applications to premium calculation principles are immediate.     

Two linear equations in formal power series

The equations [gradφ(x)]^TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x₁,…,x_n)^T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian F_x(0).     

A kernel-type estimator for generalized quantiles

Let H_F be the distribution function of h(X₁,…,X_m), where h is a real valued function of m variables and X₁,…,X_n (n ⩾ m) is an i.i.d. sample from a distribution function F. A kernel-type estimator is proposed for the generalized quantile H⁻¹_F (p) (0 < p < 1). Under fairly general conditions we establish asymptotic normality of the estimator and consistent estimation of its asymptotic variance.     

Convergence in distribution of quotients of order statistics

Let X₁, X₂,… be i.i.d. random variables with continuous distribution function F < 1. It is known that if 1 - F(x) varies regularly of order - p, the successive quotients of the order statistics in decreasing order of X₁,…,X_n are asymptotically independent, as n→∞, with distribution functions x^kp, k = 1, 2, …. A strong converse is proved, viz. convergence in distribution of this type of one of the quotients implies regular varation of 1 - F(x).     

A multidimensional Newton-Raphson method and its applications to the existence of asymptotic Fn-estimators and their stochastic expansions

Given a suitable function F_n we define a class of estimators called asymptotic F_n-estimators (i.e., estimators which approximate the solution of F_n(θ) = 0). It is proved that this class is nonvoid if appropriate regularity conditions are fulfilled and if one has at hand a suitable initial estimator. Furthermore, it is shown that F_n-estimators admit a stochastic expansion (which enables to give results on asymptotic expansions for the distribution of these estimators).     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ? we say that x ∈ [0, 1] is the p-limit point of a sequence (x _n ) _n∈? ? [0, 1] (in symbols, x = p -lim _n∈? x _n ) if for every neighbourhood V of x, {n ∈ ?: x _n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f ^p : [0, 1] → [0, 1] is defined by f ^p (x) = p -lim _n∈? f ⁿ (x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f ^p . For a filter F we also define the ω ^F -limit set of f at x. We consider a question about continuity of the multivalued map x → ω _f ^F (x). We point out some connections between the Baire class of f ^p and tame dynamical systems, and give some open problems.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

Identifiability of the multinormal and other distributions under competing risks model

Let X₁, X₂ ,…, X_p be p random variables with joint distribution function F(x₁ ,…, x_p). Let Z = min(X₁, X₂ ,…, X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,…, x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

The number of polynomial functions which permute the matrices over a finite field

Let F = GF(q) denote the finite field of order q, and let F_n×n denote the algebra of n × n matrices over F. A function f:F_n×n → F_n×n is called a scalar polynomial function if there exists a polynomial f(x) ?F[x] which represents f when considered as a matrix function under substitution. In this paper a formula is obtained for the number of permutations of F_n×n which are scalar polynomial functions.     

Limit distribution of a roundoff error

Let [X] and {X} be the integer and the fractional parts of a random variable X. The conditional distribution function F_n(x)=P({X}≤x|[X]=n) for an integer n is investigated. F_n for a large n is regarded as the distribution of a roundoff error in an extremal event. For most well-known continuous distributions, it is shown that F_n converges as n→∞ and three types of limit distributions appear as the limit distribution according to the tail behavior of F.     

On the cumulative quantile regression process

Let (X, Y) be a bivariate random vector and F(x) the marginal distribution function of X. The quantile regression (QR) function of Y on X is defined as r(u) = E[Y | F(X) = u] and the cumulative QR function (CQR) M(u) as its integral over [0, u]. The empirical counterpart based on a sample of size n is M _n (u). In this paper, we construct strong Gaussian approximations of the associated CQR process under appropriate assumptions. The construction provides a firm basis for the study of functional statistics based on M _in (u). A law of the iterated logarithm for the CQR process follows from our result.     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

On the neutrix composition of the distributions x −s ln m |x| and x r

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F _n (f)}, where F _n (x) = F(x) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^?s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^?rs ln ^m |x| for r, s, m = 2, 3….     

Density estimation in the simple proportional hazards model

In the simple proportional hazards model of random right censorship the limiting variance v_ACL²(x) at x of the kernel density estimator based on the Abdushukurov-Cheng-Lin estimator is shown to be equal to the corresponding variance pertaining to the Kaplan-Meier estimator times the expected proportion p of uncensored observations. More surprisingly, for appropriate p, v_ACL²(x) is smaller than the asymptotic variance of the classical kernel estimator based on a complete sample, for any x below the (1 − e⁻¹)-quantile.     

A product quadrature algorithm by Hermite interpolation

This paper is concerned with numerical integration of ∫¹₋₁f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = e^iτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of p_n(W; x) or of (1 − x²)p_n−2(W; x), where p_n(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.