Usual Regularity Conditions Research Articles

A class of convergent primal-dual subgradient algorithms for decomposable convex programs

In this paper we develop a primal-dual subgradient algorithm for preferably decomposable, generally nondifferentiable, convex programming problems, under usual regularity conditions. The algorithm employs a Lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties, in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal-dual optimal solutions. Several classical types of penalty functions are shown to satisfy these specified properties. A geometric convergence rate is established for the algorithm under some additional assumptions. This approach has three principal advantages. Firstly, both primal and dual solutions are available which prove to be useful in several contexts. Secondly, the choice of step sizes, which plays an important role in subgradient optimization, is guided more determinably in this method via primal and dual information. Thirdly, typical subgradient algorithms suffer from the lack of an appropriate stopping criterion, and so the quality of the solution obtained after a finite number of steps is usually unknown. In contrast, by using the primal-dual gap, the proposed algorithm possesses a natural stopping criterion.

Successive improvement of the order of ancillarity

It is shown by an argument similar to one of Welch (1947) that it is, under usual regularity conditions, in general possible to improve the order of ancillarity successively. This means that when an asymptotic ancillary has been constructed, such that its distribution is independent of the parameter apart from terms of order Othen a new ancillary may be constructed with a distribution which depends on the parameter only through terms of order O. It is shown how the new ancillary may be constructed explicitly when the expansion is of the Edgeworth type, more specifically when it is a normal density multiplied by linear combinations of Hermite polynomials.

Rational reflection coefficients and inverse scattering on the line

Inverse scattering for the Schrodinger equation on the line is studied for reflection and transmission coefficients that satisfy the usual regularity conditions and are rational functions ofk. The origin is still a particular point, but the potentials do not need to be cut at this point like in previous studies. Giving up this restriction corresponds to the existence of poles for both reflection coefficients in both upper and lower halfk-planes. It is shown that the problem reduces to solving a linear algebraic system. A different algorithm, made of a sequence of Darboux-Backlund transforms, gives also the solution in closed form and enables to study separately modifications of both sides of the potential due to the introduction of poles. Thus it paves the way for approximation studies. Generalizations and particular problems will be studied in forthcoming papers.

On Upper and Lower Bounds for the Variance of a Function of a Random Variable

Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.

Interpreting the Likelihood Ratio Statistic in Factor Models When Sample Size is Small

Abstract The use of the likelihood ratio statistic in testing the goodness of fit of the exploratory factor model has no formal justification when, as is often the case in practice, the usual regularity conditions are not met. In a Monte Carlo experiment it is found that the asymptotic theory seems to be appropriate when the regularity conditions obtain and sample size is at least 30. When the regularity conditions are not satisfied, the asymptotic theory seems to be misleading in all sample sizes considered.

The Use of the Ergodic Theorems in Random Geometry

Suppose that we have a stationary, orderly point process of rate A on the line, satisfying the usual regularity condition that the number Y(A) of points in a bounded set A has finite expectation. We all have an intuitive idea of what is meant by a 'typical' interval of the process, but how do we sample such an interval? It is well known that we obtain quite an 'atypical' interval if we select the one in progress at time t which has length Z(t), say; such sampling is biased in favour of the longer intervals. If G is the common distribution function of L(t) and L*(t), the forward and backwards recurrence times, then it is known that P{L(t) < x, L*(t) y} = G(x) + G(y) G(x + y). So the distribution F of Z(t) can be shown to be F(z)= G(z)zg(z), where g = G' always exists. However, if one were lucky enough for L*(t) to be zero, then one feels that Z(t) should be a more typical length. This is the philosophy underlying the

Note on minimum contrast estimates forMarkov processes

In the present paper it is shown that the concept of minimum contrast estimates (m.c.e.) considered inPfanzagl [1969a] for independent and identically distributed (iid) observations can be modified to cover stationary discrete timeMarkov processes admitting a unique stationary distribution which dominates the transition probabilities (Condition (S)). Sufficient conditions on the measurability and strong consistency of m.c.e. stated inPfanzagl [1969a] for the idd case are reformulated to give sufficient conditions for the existence of measurable m.c.e. and their strong consistency for such processes (section 1). The proofs of the main theorems are only sketched, because they are nearly the same as those given inPfanzagl [1969a] for the iid case. The concept of m.c.e. covers maximum likelihood estimates (m.l.e.) as a special case; therefore an application of the results to m.l.e. yields sufficient conditions for the existence of measurable m.l.e. and their strong consistency if the parameter space is compact metrizable or locally compact with countable base (Section 2). These conditions are weaker than the usual regularity conditions (see for exampleBillingsley [1961b] and the references cited there) and under the assumption that the transition probabilities as well as the stationary distribution are absolutely continuous with respect to a σ-finite measure they can be expressed in terms of the corresponding equivalence classes of transition densities. This seems to be more transparent than the conditions given byRoussas [1965]. In Section 3 asymptotic normality of m.c.e. is proved under conditions which correspond to those used inRoussas [1968] for the case of m.l.e.

Gravitational Collapse of Homogeneous Spheres

IN the study of adiabatic contraction of perfect fluid spheres, the problem of gravitational bounce or collapse to a singularity has been studied in the literature under the following assumptions 1,2: first, that the motion is isotropic; second, that the density is uniform. In addition, to make the models physically meaningful, the usual regularity conditions at the centre and the joining conditions at the interface between the region occupied by matter and the surrounding empty space are assumed to hold. But we show in this letter that the assumption of isotropy is redundant in the sense that the conditions of uniform density and regularity at the centre necessarily lead to the first condition.

A Pair of Dual Integral Equations Involving Bessel Functions of the First and the Second Kind

In this paper we give a method for the solution of the dual integral equationswhere Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation forsubject to the usual regularity conditions and the following boundary conditions:

Aggregation of Utility Functions

We are primarily concerned here with the question of integrability of the total demand in a model in which each consumer acts according to a cardinal utility function and has a fixed monetary income. It is well known that concavity of the various utilities is not sufficient to guarantee integrability, nor even to ensure rationality of the revealed preference. We show that if each personal utility function is homogeneous, in addition to satisfying the usual regularity conditions, then an aggregate utility function can be defined explicitly in terms of the given utilities. Furthermore, under the same assumptions we give a new characterization of equilibrium and show that equilibrium satisfactions are unique.

Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters

It is shown that, under usual regularity conditions, the maximum likelihood estimator of a structural parameter is strongly consistent, when the (infinitely many) incidental parameters are independently distributed chance variables with a common unknown distribution function. The latter is also consistently estimated although it is not assumed to belong to a parametric class. Application is made to several problems, in particular to the problem of estimating a straight line with both variables subject to error, which thus after all has a maximum likelihood solution.

A Remark on the Roots of the Maximum Likelihood Equation

The statistical literature combines two types of investigations concerning the consistency of maximum likelihood (M.L.) estimates. A few of these, such as the most excellent paper of A. Wald [1], do prove directly the consistency of M.L. estimates. However, most investigators seem to have concentrated their efforts on proving the existence and consistency of suitably selected roots of the successive likelihood equations. Some authors, see [2], for example, add the supplementary remark that such consistent roots will eventually be unique in suitably small neighborhoods of the true value and will achieve a local maximum. It is the purpose of the present note to point out by means of examples that this second mode of attack is not adequate. In the examples given below, the usual regularity conditions of Cramer [3] or Wald [4] are satisfied, but the M.L. estimates are not consistent. It should also be pointed out that the direct proofs of existence of roots, simple in the case of a unidimensional parameter, become unwieldy in more than one dimension. On the other hand, if one has proved the consistency of the M.L. estimates, the existence of roots follows trivially from the fact that when a differentiable function reaches its maximum in an open set, the derivatives vanish at that point.

Derivation of a Broad Class of Consistent Estimates

Given a chance vector $\mathbf{X}$ with distribution function $F(\mathbf{X}, \theta_T)$, where $\mathbf{theta}_T$ denotes the true unknown parameter vector, a broad class of estimates of $\mathbf{theta}_T$ is derived which is shown to be identical with the class of all consistent estimates of $\mathbf{\theta}_T$. A sub-class is obtained each member of which has the following properties: a.) Its construction depends upon the solution of an equation involving a single vector function of the parameter vector $\theta$ and the members of a sequence $\{\mathbf{X}_n\}$ of independent and identically distributed chance vectors; b.) the estimate so obtained converges almost certainly to $\mathbf{\theta}_T$; c.) it is a symmetric function of the members of the sequence $\{\mathbf{X}_n\}$. In order to obtain this subclass it is postulated that a function of $\mathbf{X}$ and $\mathbf{\theta}$ exists (continuous in $\mathbf{\theta}$ for a certain neighborhood of the true parameter $\mathbf{\theta}_T$ and existing for each $\mathbf{X}$ in a subset of the sample space) which satisfies a Lipschitz condition in $\mathbf{\theta}$. In particular if a density function $f(\mathbf{X}, \theta_T)$ exists satisfying certain conditions, the consistency of the maximum likelihood estimate can be established under regularity conditions quite different from those usually assumed [1]. This is not to be interpreted as a weakening of the usual regularity conditions but rather as an extension of the class of consistent likelihood estimates obtained under the usual regularity conditions.