A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

Largest excess of boundary crossings for martingales

Summary LetX ₁,X ₂, ... form a sequence of martingale differences and denote byZ(a, α) = sup n (S _n −an ^α)⁺ the largest excess forS _n =X ₁ + ... +X _n crossing the boundaryan ^α. We give a sufficient condition for the finiteness ofEZ(a, α)^β which is formulated in terms of bounds forE(X _i ^{+ p} andE(|X _i |γ|X ₁, ...,X _i-1), whereα, β, γ, p are suitably related. This general result is then applied to the case of independent random variables.     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

On the power function of the exact test for ther×c contingency table

Summary LetN=[n _ij ] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp _i =(p _i1, …,p _ic ).Freeman andHalton have proposed an exact conditional test for the hypothesisH ₀ :p _i =(p ₁, …p _c ) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964].     

A note on generalized wald’s method

Let {v _n(θ)} be a sequence of statistics such that whenθ =θ ₀,v _n(θ ₀) N _p(0,Σ), whereΣ is of rankp andθ εR ^d. Suppose that underθ =θ ₀, {Σ _n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv _n ^T (θ ₀)Σ _n ⁻¹ v _n(θ ₀) x ²(p). It often happens thatv _n(θ ₀) N _p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv _n ^T (θ ₀)Σ _n ⁻ v _n(θ ₀) x ²(k), wherek = rank (Σ) andΣ _n ⁻ is a generalized inverse ofΣ _n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ _n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions. Research partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

A bound on the expected overshoot for some concave boundaries

LetX ₁,X ₂,… be i.i.d. with finite meanμ>0,S _n =X ₁+…+X _n . Forf(n)=n ^β ,c>0 we consider the stopping timesT _c =inf{n:S _n >c+f(n)} with overshootR _c =S _{T c} −(c+f(T _c )). For 0<β<1 we give a bound for sup _c≥0 ER _c in the spirit of Lorden’s well-known inequality forf=0.     

Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables

Let X ₁, X ₂, ..., X _n be independent exponential random variables such that X _i has failure rate λ for i = 1, ..., p and X _j has failure rate λ^* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D _i:n (p,q) = X _i:n −X _i-1:n the ith spacing of the order statistics X _1:n ≤ X _2:n ≤ ... ≤ X _n:n , i = 1, ..., n, where X _0:n ≡ 0. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D _i:n (p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings. Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.: NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02).     

Response of double-auction markets to instantaneous Selling�CBuying signals with stochastic Bid�CAsk spread

Statistical properties of order-driven double-auction markets with Bid–Ask spread are investigated through the dynamical quantities such as response function. We first attempt to utilize the so-called Madhavan–Richardson–Roomans model (MRR for short) to simulate the stochastic process of the price-change in empirical data sets (say, EUR/JPY or USD/JPY exchange rates) in which the Bid–Ask spread fluctuates in time. We find that the MRR theory apparently fails to simulate so much as the qualitative behaviour (‘non-monotonic’ behaviour) of the response function R(l) (l denotes the difference of times at which the response function is evaluated) calculated from the data. Especially, we confirm that the stochastic nature of the Bid–Ask spread causes apparent deviations from a linear relationship between the R(l) and the auto-correlation function C(l), namely, R(l) μ -C(l){R(l) \propto -C(l)}. To make the microscopic model of double-auction markets having stochastic Bid–Ask spread, we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence. We also reveal that the minority game modeling with the adaptive (‘annealed’) look-up table reproduces the non-linear relationship R(l) μ -f(C(l)){R(l) \propto -f(C(l))} (f(x) stands for a non-linear function leading to ‘λ-shapes’) more effectively than the fixed (‘quenched’) look-up table does.     

A general class of univariate skew distributions considering Stein’s lemma and infinite divisibility

In this article, we consider a general form of univariate skewed distributions. We denote this form by GUS(λ; h(x)) or GUS with density s(x|λ, h(x)) = 2f(x)G(λ h(x)), where f is a symmetric density, G is a symmetric differentiable distribution, and h(x) is an odd function. A special case of this general form, normal case, is derived and denoted by GUSN(λ; h(x)). Some representations and some main properties of GUS(λ; h(x)) are studied. The moments of GUSN(λ; h(x)) and SN(λ), the known skew normal distribution of Azzalini (1985), are compared and the relationship between them is given. As an application, we use it to construct a new form for skew t-distribution and skew Cauchy distribution. In addition, we extend Stein’s lemma and study infinite divisibility of GUSN(λ; h(x)).     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

On Estimators of the Nearest Neighbour Distance Distribution Function for Stationary Point Processes

There are three approaches for the estimation of the distribution function D(r) of distance to the nearest neighbour of a stationary point process: the border method, the Hanisch method and the Kaplan-Meier approach. The corresponding estimators and some modifications are compared with respect to bias and mean squared error (mse). Simulations for Poisson, cluster and hard-core processes show that the classical border estimator has good properties; still better is the Hanisch estimator. Typically, mse depends on r, having small values for small and large r and a maximum in between. The mse is not reduced if the exact intensity λ (if known) or intensity estimators from larger windows are built in the estimators of D(r); in contrast, the intensity estimator should have the same precision as that of λ D(r). In the case of replicated estimation from more than one window the best way of pooling the subwindow estimates is averaging by weights which are proportional to squared point numbers.     

Limit theorems for the spacings of weak records

Let W(1), W(2), . . . be weak record values obtained from a sample of independent variables with common discrete distribution. In the present paper, we derive weak and strong limit theorems for the spacings W(n + m) − W(n), m ≥ 1, n → ∞.     

Tests und Konfidenzintervalle für exponentielle Verteilungen und deren Anwendung auf einige diskrete Verteilungen

Zusammenfassung Es werden Verteilungen betrachtet, die (bezüglich irgendeines Ma?es) eine Dichte der GestaltC(ϑ) exp [ϑ x] besitzen. Für solche Verteilungen werden (ein- und zweiseitige) Tests und Konfidenzintervalle mit gewissen Optimalit?tseigenschaften entwickelt, und zwar fürϑ, für die Differenzϑ ₁ −ϑ ₂, sowie für einige Versionen desk-Stichproben Problems. Sodann werden einige Hilfss?tze über den bedingten Erwartungswert und die bedingte Varianz von zweiparametrigen Verteilungen abgeleitet, die bezüglich des einen Parameters reproduktiv sind und eine bezüglich des zweiten Parameters ersch?pfende und vollst?ndige Funktion besitzen. Schlie?lich werden die allgemeinen Ergebnisse auf einige diskrete Verteilungen (Binomial, Poisson, negativ Binomial, Pascal) angewendet und der Zusammenhang mit verschiedenen bekannten Tests diskutiert.

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Summary Probability distributions are considered which (with respect to any measure) possess a density function of the typeC(ϑ) exp [ϑ x]. For distributions of this type (one and twosided) tests and confidence intervals with some optimal properties are given, namely forϑ, for the differenceϑ ₁ −ϑ ₂, and for several versions of thek-sample problem. Furthermore, some lemmas concerning the conditional expectation and the conditional variance are proved for two-parameter families of distributions which are reproductive in one parameter and possess a complete statistic, sufficient for the second parameter. Finally the general results are applied to some discrete distributions (binomial, Poisson, negative binomial, Pascal) and the relationship to several fairly known tests is discussed.

     

A uniqueness theorem for convex-ranged probabilities

A finitely additive probability measure P defined on a class Σ of subsets of a space Ω is convex-ranged if, for all P(A)>0 and all 0 < α < 1, there exists a set, Σ∋B⊆A, such that P(B)=αP(A).?Our main result shows that, for any two probabilities P and Q, with P convex-ranged and Q countably additive, P=Q whenever there exists a set A∈Σ, with 0 < P(A) < 1, such that (P(A)=P(B)?Q(A)=Q(B)) for all B∈Σ. Received: 18 December 1999 / Accepted: 17 July 2000     

Asymptotic normality of M-estimators in a semiparametric model with longitudinal data

Suppose that the longitudinal observations (Y _ij , X _ij , t _ij ) for i = 1, . . . ,n; j = 1, . . . ,m _i are modeled by the semiparamtric model where β ₀ is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and e _ij are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter β ₀ is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β ₀ in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β ₀ is derived. The asymptotic distributions of the local M-estimators of g(·) at both interior and boundary points are also established. Various applications of our main results are given. The research is supported in part by National Natural Science Foundation of China (Grant No. 10671089).     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.     

Asymptotic results and tests for the choice of approximative models in nonlinear two-phases regression models, heteroscedatic case

This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t _i)=f(t _i)+ε(t _i), i=1,2,…,n, where the (ε(t _i))_i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time we assume that the ε(t _i) have non constant and unknown variances σ²(t _i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation. Received: February 1996     

Linearity of regression for non-adjacent order statistics

Let X ₁,X ₂,…,X _n be a random sample from a continuous distribution with the corresponding order statistics X _1:n≤X _2:n≤…≤X _n:n. All the distributions for which E(X _{k+r: n}|X _k:n)=a X _k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998