On a simple characterisation of threshold graphs

In their very interesting paper “Set-packing problems and threshold graphs” [1] V. Chvatal and P. L. Hammer have shown that the constraints $$\begin{gathered} \sum\limits_{j = 1}^n {a_{ij} x_j \leqslant 1(i = 1,2, . . . , m)} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$ are equivalent to the only inequality $$\begin{gathered} \sum\limits_{j - 1}^n {c_j x_j \leqslant d} \hfill \\ x_j \in (0,1)(j = 1,2, . . . , n) \hfill \\ \end{gathered} $$ if and only if the intersection graph associated with the matrix (a _ij ) — see § 1 — is a threshold graph i.e. a graph none of whose induced subgraphs are isomorphic to 2K ₂,P ₄,C ₄: As Chvatal and Hammer have shown [1], threshold graphs can be characterised in many different ways; the main result of this paper is to give a new, very simple characterisation which will enable us to test whether a graph is a threshold by a simple inspection of its incidence matrix.     

A new approach to distribution free tests in contingency tables

We suggest an extremely wide class of asymptotically distribution free goodness of fit tests for testing independence in two-way contingency tables, or equivalently, independence of two discrete random variables. The nature of these tests is that the test statistics can be viewed as definite functions of the transformation of \(\widehat{T}_n = (\widehat{T}_{ij})=\Big (\frac{\nu _{ij}- n\hat{a}_i\hat{b}_j}{\sqrt{n\hat{a}_i\hat{b}_j}}\Big )\) where \(\nu _{ij}\) are frequencies and \(\hat{a}_i, \hat{b}_j\) are estimated marginal distributions. Our method is also applicable for testing independence of two discrete random vectors. We make some comparisons on statistical powers of the new tests with the conventional chi-square test and suggest some cases in which this class is significantly more powerful.     

A remark on robustness of linear best estimates

Let s₀ = ||x- h₀ (x)||₀ , s(B₁ ) = ||x- h₁ (x)||₀ , s₁ (B₁ ) = ||x- h₁ (x)||₁ , \begin{gathered} \sigma _0 = \parallel \xi - \eta _0 (\xi )\parallel _0 , \hfill \\ \sigma (B_1 ) = \parallel \xi - \eta _1 (\xi )\parallel _0 , \hfill \\ \sigma _1 (B_1 ) = \parallel \xi - \eta _1 (\xi )\parallel _1 , \hfill \\ \end{gathered}     

A characterization of entropies of degree α

The functional equation has been studied by several authors under various assumptions onf and onk, l. Here we solve this equation iff is measurable andk 3,l 2 are fixed integers. Using the solution we characterize the entropies of degree for all real . Our results generalize the results ofBehara/Nath [1973],Kannappan [1974] andMittal [1976].     

Contributions to nonparametric generalized failure rate function estimation

For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG ^–1 F(x) may be estimated by replacingf andF byf _n and , wheref _n is an empirical density function based on a sample of sizen from the distribution functionF, and . Under regularity conditions we show and, under additional restrictions whereC is a subset ofR and _n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n^–1/2+c _n ^1/2 ) is established under mild conditions, wherec _n is a sequence of positive constants related tof _n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15.     

Moving averages of ergodic processes

A necessary and sufficient condition for the almost everywhere convergence of the moving ergodic averages is given. The result is then generalized to ergodic flows, and finally constrasted with earlier results ofPfaffelhuber andJain.     

On generalized transforms of distribution functions

In this note we discuss the following problem. LetX andY to be two real valued independent r.v.'s with d.f.'sF and ?. Consider the d.f.F*? of the r.v.X oY, being o a binary operation among real numbers. We deal with the following equation: $$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$ where \(\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 \) are real or complex functionals, т another binary operation ands a parameter. We give a solution, that under stronger assumptions (Aczél 1966), is the only one, of the problem. Such a solution is obtained in two steps. First of all we give a solution in the very special case in whichX andY are degenerate r.v.'s. Secondly we extend the result to the general case under the following additional assumption: $$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$ .     

Convergence inr-mean of normalized partial sums in a separable Banach space

Ver {V _n } be a sequence of random elements in a separable Banach space. Conditions are obtained under which .     

A note on minimum variance

Summary Minimizing is discussed under the unbiasedness condition: and the condition (A):f _i (x) (i=1, ..., p) are linearly independent , and .     

A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables

Let be a sequence of stationary positively associated random variables and a sequence of positive constants be monotonically approaching infinity and be not asymptotically equivalent to loglog n. Under some suitable conditions, a nonclassical law of the iterated logarithm is investigated, i.e.

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

Consistency of a nonparametric estimation of a density functional

LetX be a random variable with distribution functionF and density functionf. Let ? and ψ be known measurable functions defined on the real lineR and the closed interval [0, 1], respectively. This paper proposes a smooth nonparametric estimate of the density functional \(\theta = \int\limits_R \phi (x) \psi \left[ {F (x)} \right]f^2 (x) dx\) based on a random sampleX ₁, ...,X _n fromF using a kernel functionk. The proposed estimate is given by \(\hat \theta = (n^2 a_n )^{ - 1} \mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n \phi (X_i ) \psi \left[ {\hat F (X_i )} \right]k\left[ {(X_i - X_j )/a_n } \right]\) , where \(\hat F(x) = n^{ - 1} \mathop \sum \limits_{i = 1}^n K\left[ {(x - X_i )/a_n } \right]\) with \(K (w) = \int\limits_{ - \infty }^w {k (u) } du\) . The estimate \(\hat \theta \) is shown to be consistent both in the weak and strong sense and is used to estimate the asymptotic relative efficiency of various nonparametric tests, with particular reference to those using the Chernoff-Savage statistic.     

Arbitrage and completeness in financial markets with given <Emphasis Type="Italic">N</Emphasis>-dimensional distributions

Abstract In this paper, we focus on the following problem: given a financial market, modelled by a process , and a family of probability measures on , with N a positive integer and the time space, we search for financially meaningful conditions which are equivalent to the existence and uniqueness of an equivalent (local) martingale measure (EMM) Q such that the price process S has under Q the pre-specified finite-dimensional distributions of order N (N-dds) . We call these two equivalent properties, respectively, N -mixed no free lunch and market N -completeness. They are based on a classification of contingent claims with respect to their path-dependence on S and on the related notion of N-mixed strategy. Finally, we apply this approach to the Black-Scholes model with jumps, by showing a uniqueness result for its equivalent martingale measures set. Mathematics Subject Classification (2000): 60G48, 91B28 Journal of Economic Literature Classification: G12, D52     

On the maxima of heterogeneous gamma variables with different shape and scale parameters

In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let \(X_{1}\) and \(X_{2}\) be two independent gamma random variables with \(X_{i}\) having shape parameter \(r_{i}>0\) and scale parameter \(\lambda _{i}\) , \(i=1,2\) , and let \(X^{*}_{1}\) and \(X^{*}_{2}\) be another set of independent gamma random variables with \(X^{*}_{i}\) having shape parameter \(r_{i}^{*}>0\) and scale parameter \(\lambda _{i}^{*}\) , \(i=1,2\) . Denote by \(X_{2:2}\) and \(X^{*}_{2:2}\) the corresponding maxima, respectively. It is proved that, among others, if \((r_{1},r_{2})\) majorize \((r_{1}^{*},r_{2}^{*})\) and \((\lambda _{1},\lambda _{2})\) weakly majorize \((\lambda _{1}^{*},\lambda _{2}^{*})\) , then \(X_{2:2}\) is stochastically larger that \(X^{*}_{2:2}\) in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.     

Third-order optimum properties of estimator-sequences

SequencesT ⁽ⁿ⁾ ,n∈N, are considered, whereT ⁽ⁿ⁾ estimates a vector parameter ?∈R ^p from an i.i.d. sample of sizen, and such sequences are compared on the basis of their risks ∫L(n ^1/2(T ⁿ (x)?θ))P _θ ⁿ (dx) relative to loss functionsL:R ^p →R. A characterization is given for sequencesT ^*(n) ,n∈N, which generate an essentially complete class in the following sense: For any sequenceT ⁽ⁿ⁾ ,n∈N, there exist functions Φ _n ,n∈N, such that forn→∞ we have $$\begin{gathered} \smallint L (n^{1/2} (T^{*(n)} + n^{ - 1} \Phi _n (T^{*(n)} ) - \theta )) dP_\theta ^n \leqslant \hfill \\ \leqslant \smallint L (n^{1/2} (T^{(n)} - \theta )) dP_\theta ^n + o (n^{ - 1} ), \hfill \\ \end{gathered} $$ for every ? and everyL satisfying certain conditions. If the estimator-sequences are compared by their risks ∫W(T ⁽ⁿ⁾ d P _θ ⁿ ,θ) with respect to loss functionsW:R ^p ×Θ→R then a similar result on asymptotically complete classes is valid. The results are obtained under the assumption thatT ^*(n) ,n∈N, andT ⁽ⁿ⁾ ,n∈N, admit stochastic expansions which are sufficiently regular, that the loss functionsL andW are sufficiently smooth and bounded by polynomials, and that the estimator-sequences have asymptotically bounded moments; the latter condition is not needed for bounded functionsL andW.     

Runs in an ordered sequence of random variables

Let be independent and identically distributed random variables with continuous distribution function. Denote by the corresponding order statistics. In the present paper, the concept of -neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables