Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

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.     

A fixed sample size selection procedure for negative binomial populations

Summary A fixed sample size procedure for selecting the ‘best’ ofk negative binomial populations is developed. Selection is made in such a way that the probability of correct selection is at leastP* whenever the distance between the probabilities of success is at leastδ*. The exponentr is assumed to be known and the same for all populations. Extensive computer calculations* were employed to obtain the exact least favorable configuration. The smallest sample sizes needed to meet specifications (P*,δ*) are tabulated forr=1 (1)5;δ*=0.05 (0.05) 0.55 andP*=0.75, 0.80, 0.90, 0.95, 0.98, 0.99 involvingk=3 (1) 6, 8, 10 populations. All the computations were carried out on the Alabama Supercomputer. Part of this work was completed when the authors were at the Department of Statistics, Oklahoma State University, Stillwater, OK 74078.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

Gamma-admissibility of estimators in the one-parameter exponential family

Summary Admissibility of estimators under vague prior information on the distribution of the unknown parameter is studied which leads to the notion of gamma-admissibility. A sufficient condition for an estimator of the formδ(x)=(ax+b)/(cx+d) to be gamma-admissible in the one-parameter exponential family under squared error loss is established. As an application of this result two equalizer rules are shown to be unique gamma-minimax estimators by proving their gamma-admissibility.     

A geometric property of indifference curves in the (mean-standard deviation) plane

We prove that the slope of the indifference curve through point (m σ) of the (meanstandard deviation) plane must be smaller than the ratio (1+m)/σ. We show that the indifference curves corresponding to a quadratic utility function do actually satisfy that inequality; and prove, by the way, that these curves are not quarters of circles (as it is usually asserted), but smaller parts of them.

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Riassunto Si prova che, nel piano (media-scarto quadratico medio), l'inclinazione della curva d'indifferenza per il punto (m, σ) deve essere minore del rapporto (1+m)/gs. Si mostra che le curve d'indifferenza corrispondenti ad una funzione d'utilià quadratica soddisfano di fatto tale limitazione; e si prova, a questo proposito, che tali curve non sono — come generalmente asserito — quarti di circonferenze, ma archi più piccoli delle medesime.

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The second part of this article was stimulated by an observation of E. Castagnoli. The author gratefully acknowledges this to him.     

Notes and Comments: Sup-convolutions of HARA utilities in the affine term structure

Abstract In the financial literature, the problem of maximizing the expected utility of the terminal wealth has been investigated extensively (for a survey, see, e.g., Karatzas and Shreve (1998), p. 153, and references therein) by using different approaches. In this paper, we extend the existing literature in two directions. First, we let the utility function U(.) of the financial agent (who is a price taker) be implicitly defined through I(.)=(U ^′ (.))^–1, which is assumed to be additively separable, i.e., I(.)=∑ _k=1 ^N I _k (.). Second, we solve the investment problem in the general affine term structure model proposed by Duffie and Kan (1996) in which the functions I _k (.), k=1,...,N are associated to HARA utility functions (with possibly different risk aversion parameters), and we show that the utility maximization problem leads to a Riccati ODE. Moreover, we extend to the multi-factor framework the stability result proved in Grasselli (2003), namely, the almost-sure convergence of the solution with respect to the parameters of the utility function. Mathematics Subject Classification (2000): 91B28 Journal of Economic Literature Classification: G11     

Mechanisms that efficiently verify the optimality of a proposed action

The best known achievement of the literature on resource-allocating mechanisms and their message spaces is the first rigorous proof of the competitive mechanism's informational efficiency. In an exchange economy withN persons andK+1 commodities (including a numeraire), that mechanism announcesK prices as well as aK-compenent trade vector for each ofN−1 persons, making a total ofNK message variables. Trial messages are successively announced and after each announcement each personprivately determines, usingprivate information, whether she finds the proposed trades acceptable at the announced prices. When a message is reached with which all are content, then the trades specified in that message take place, and they satisfy Pareto optimality and individual rationality. The literature shows that no (suitably regular) mechanism can achieve the same thing with fewer thanNK message variables. In the classic proof, all the candidate mechanisms have the privacy property, and the proof uses that property in a crucial way. ‘Non-private’ mechanisms are, however, well-defined. We present a proof that forN>K,NK remains a lower bound even when we permit ‘non-private’ mechanisms. Our new proof does not use privacy at all. But in a non-private mechanism, minimality of the number of message variables can hardly be defended as the hallmark of informational efficiency, since a non-private mechanism requires some persons to know something about the private information of othersin addition to the information contained in the messages. The new proof of the lower boundNK invites a new interpretation of the competitive mechanism's informational efficiency. We provide a new concept of efficiency which the competitive mechanism exhibits and which does rest on privacy even whenN>K. To do so, we first define a class ofprojection mechanisms, wherein some of the message variables are proposed values of the action to be taken, and the rest are auxiliary variables. The competitive mechanism has the projection property, with a trade vector as its action and prices as the auxiliary variables. A projection mechanism proposes an action; for each proposal, the agents then use the auxiliary variables, together with their private information, to verify that the proposed action meets the mechanism's goal (Pareto optimality and individual rationality for the competitive mechanism) if, indeed, it does meet that goal. For a given goal, we seek projection mechanisms for which theverification effort (suitably measured) is not greater than that of any other projection mechanism that achieves the goal. We show the competitive mechanism to be verification-minimal within the class of private projection mechanisms that achieve Pareto optimality and individual rationality; that proofdoes use the privacy of the candidate mechanisms. We also show, under certain conditions, that a verification-minimal projection mechanism achieving a given goal has smallest ‘total communication effort’ (which is locally equivalent to the classic ‘message-space size’) among all private mechanisms that achieve the goal, whether or not they have the projection property.     

Characterization of some types of completeness resp. Total completeness and their conservation under direct products

Some notions ofL _p (μ)-completeness resp. totally L _p (μ)-completeness (1≦p≦∞) are characterized for families of probability distributions dominated by aσ-finite measureμ and their conservation with respect to direct products is proved. Furthermore, it is shown that totallyL _∞(μ)-completeness does not implyL ₁(μ)-completeness and that there are families of probability distributions in the i.i.d. case induced by the order statistic, which are L₁(μ)-complete but not totallyL _∞(μ)-complete.     

Consistent estimation in the presence of incidental parameters

Summary Let (X,A) be a measurable space andP _ϑη |A (ϑη) ∈ Θ x H, ∥A, (θ, η) ∈ Θ×H, a parametrized family of probability measures (for short:p-measures). This paper is concerned with the problem of consistently estimatingθ from realizations governed by , where η^u ∈ H, v ∈ ℕ, are unknown.     

Median unbiased estimates for M.L.R.-families

Summary Lehmann [p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family. The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general case.     

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).     

Condorcet efficiency of positional voting rules with single-peaked preferences

Consider a three-alternative election with n voters and assume that preferences are single-peaked. LetC(λ,n) be the Condorcet efficiency of the rule that assigns 1,λ and 0 points (respectively) to each first, second and third place vote. An exact representation is obtained forC(λ,∞). This relation shows that Borda rule (λ=1/2) is not the most efficient rule. In addition to this result, exact closed form relations are provided forC(0,n),C(1/2,n) andC(1,n). All these relations are obtained by assuming that every admissible configuration of preferences is equally likely to occur. This research was supported in part by the Swedish Insitute     

A note on the maximum deviation of the scale-contaminated normal to the best normal distribution

In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but different component variances: (1−p)N(μ,σ²)+pN(μ,τ²) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for normality are pointed out. Received May 2001     

Alcune nuove nozioni di semicontinuità per multifunzioni

Si introducono nuove nozioni di semicontinuità per multifunzioni che estendono quelle classiche relative alle funzioni. Inoltre, utilizzando tali generalizzazioni si studia la semicontinuità superiore di una funzione del tipo inf _v∈N(u) f(u, v). Infine, si applicano i risultati ottenuti ad un problema di scelta del prezzo.

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Summary In this paper new notions of semicontinuity for multifunctions are introduced; this definitions extend the classical notions concerning the functions. Moreover, using this generalizations, the upper semicontinuity of a function like inff(u, v) is studied. In the last section the results obtained are applied to av∈N(u) problem of selection price.

     

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.     

Bootstrap based goodness-of-fit-tests

Summary Let ℱ={F _θ} be a parametric family of distribution functions, and denote withF _n the empirical d.f. of an i.i.d. sample. Goodness-of-fit tests of a composite hypothesis (contained in ℱ) are usually based on the so-called estimated empirical process. Typically, they are not distribution-free. In such a situation the bootstrap offers a useful alternative. It is the purpose of this paper to show that this approximation holds with probability one. A simulation study is included which demonstrates the validity of the bootstrap for several selected parametric families.     

Two families of kurtosis measures

Two families of kurtosis measures are defined as K ₁(b)=E[ab ^−|z|] and K ₂(b)=E[a(1−|z|^b)] where z denotes the standardized variable and a is a normalizing constant chosen such that the kurtosis is equal to 3 for normal distributions. K ₂(b) is an extension of Stavig's robust kurtosis. As with Pearson's measure of kurtosis β₂=E[z ⁴], both measures are expected values of continuous functions of z that are even, convex or linear and strictly monotonic in ℜ⁻ and in ℜ⁺. In contrast to β₂, our proposed kurtosis measures give more importance to the central part of the distribution instead of the tails. Tests of normality based on these new measures are more sensitive with respect to the peak of the distribution. K ₁(b) and K ₂(b) satisfy Van Zwet's ordering and correlate highly with other kurtosis measures such as L-kurtosis and quantile kurtosis. RID="*" ID="*"  The authors thank the referees for their insightful comments that significantly improved the clarity of the article.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

Testing whetherF is more IFR thanG

A distributionF is said to be “more IFR” than another distributionG ifG ⁻¹ F is convex. WhenF(0) =G(0) = 0, the problem of testingH ₀ :F(x) =G (θx) for someθ > 0 andx ⩾ 0, against the alternativeH _A:F is more IFR thanG, is considered in this paper. Both cases, whenG is completely specified (one-sample case) and when it is not specified but a random sample form it is available (two-sample case) are considered. The proposed tests are based onU-statistics. The asymptotic relative efficiency of the tests are compared with several other tests and the test statistics remain asymptotically normal under certain dependency assumptions. Research supported in part by a grant from the US Air Force Office of Scientific Research.