We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we ...derive higher monotonicity properties of generalized weighted premiums; in particular, we deduce for weight functions that are totally positive of order three a monotonicity property of the variance-to-mean ratio, or index of dispersion, of the loss variable. We derive the higher order total positivity properties of some ratios that arise in actuarial and insurance analysis of combined risks. Further, we examine seven classes of weight functions that have appeared in the literature and we ascertain the higher order total positivity properties of those functions.
We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also ...analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties.
The Hankel matrix of type B Narayana polynomials was proved to be totally positive by Wang and Zhu, and independently by Sokal. Pan and Zeng raised the problem of giving a planar network proof of ...this result. In this paper, we present such a proof by constructing a planar network allowing negative weights, applying the Lindström-Gessel-Viennot lemma and establishing an involution on the set of nonintersecting families of directed paths.
•Providing a combinatorial proof of the Hankel total positivity of type B Narayana polynomials.•Allowing negative weights in the construction of planar networks.•Establishing an elegant sign-reversing involution on families of nonintersecting lattice paths.
Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array Tn,kn,k≥0, which satisfies the recurrence ...relation:Tn,k=λ(a1k+a2)Tn−1,k+(b1−da1)n−(b1−2da1)k+b2−d(a1−a2)Tn−1,k−1+d(b1−da1)λ(n−k+1)Tn−1,k−2, where T0,0=1 and Tn,k=0 unless 0≤k≤n. We derive some properties of Tn,kn,k≥0, including the explicit formulae of Tn,k and the exponential generating function of the generalized Eulerian polynomial Tn(q), and the ordinary generating function of Tn(q) in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of Tn(q), stability of the iterated Turán-type polynomial Tn+1(q)Tn−1(q)−Tn2(q). Furthermore, we also prove the q-Stieltjes moment property and 3-q-log-convexity of Tn(q) and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for γ-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get γ-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types A and B in a unified manner. We also derive γ-positivity for a symmetric sub-array of Tn,kn,k≥0, which in particular gives a unified proof of γ-positivity of Eulerian polynomials of types A and B.
Our results not only can immediately apply to Eulerian triangles of two kinds and arrays from staircase tableaux and tree-like tableaux, but also to segmented permutations and flag excedance numbers in colored permutations groups in a unified approach. In particular, we also confirm a conjecture of Nunge about the unimodality from segmented permutations.
Consider bivariate observations
(
X
1
,
Y
1
)
,
…
,
(
X
n
,
Y
n
)
∈
R
×
R
with unknown conditional distributions
Q
x
of
Y
, given that
X
=
x
. The goal is to estimate these distributions under the ...sole assumption that
Q
x
is isotonic in
x
with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution
L
(
X
,
Y
)
under the sole assumption that it is totally positive of order two. An algorithm is developed which estimates the unknown family of distributions
(
Q
x
)
x
via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data.
The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators Ln(f) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis ...functions bn,k(x)(0≤k≤n) of order n are constructed by replacing x in the ith factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function pi(x), where pi(0)=0 and pi(1)=1 for 1≤i≤n. These operators Ln(f) are positive linear operators that preserve constant functions, and a non-constant function γnp(x). If all the pi(x) are strictly increasing and strictly convex, then γnp(x) is strictly increasing and strictly convex as well. Iterates LnM(f) of Ln(f) are also considered. It is shown that LnM(f) converges to f(0)+(f(1)−f(0))γnp(x) as M→∞. Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every (1,γnp(x))-convex function f∈C0,1, we have Ln(f;x)≥f(x); and by invoking the total positivity of the system {bn,k(x)}0≤k≤n, we show that if f is (1,γnp(x))-convex, then Ln(f;x) is also (1,γnp(x))-convex. Finally we show that if all the pi(x) are monomial functions, then for every (1,γn+1p(x))-convex function f, Ln(f;x)≥Ln+1(f;x) if and only if p1(x)=⋯=pn(x)=x.
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