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}     

Confidence intervals for two-dimensional data with circular tolerances in a gauge R&R study

In this paper, we discuss our application of the Bootstrap method to construct the confidence interval of the diameter for two-dimensional data with circular tolerances in a gauge repeatability and reproducibility study. Factors simulated to validate performance include: the variance component, and sample size. The simulation results show that the Bootstrap method can cover the stated nominal coefficient in most scenarios. There exists a positive correlation between width of confidence intervals and variance components; the width of confidence intervals for diameters is increased when the variance components ([^(s)]_x², [^(s)]_y² or [^(s)]_xy²){(\hat{{\sigma}}_x^2, \hat{{\sigma}}_y^2\,{\rm or}\,\hat{{\sigma}}_{xy}^2)} are increased. The coverage proportion is not significantly affected by variance-components. Also, the width of confidence interval for the diameter and coverage proportion is not significantly affected by sample size. One real example based on a nested design is used to demonstrate the application of the proposed method.     

Distribution of run statistics in partially exchangeable processes

Let { X_i} _{i 3 1}{\{ X_{i}\} _{i\geq 1}} be an infinite sequence of recurrent partially exchangeable binary random variables. We study the exact distributions of two run statistics (total number of success runs and the longest success run) in { X_i} _{i 3 1}{\{ X_{i}\} _{i\geq1}} . Since a flexible class of models for binary sequences can be obtained using the concept of partial exchangeability, as a special case of our results one can obtain the distribution of runs in ordinary Markov chains, exchangeable and independent sequences. The results also enable us to study the distribution of runs in particular urn models.     

Estimation of the location parameter under LINEX loss function: multivariate case

The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, ${\beta\in\Re^r}The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, b ? ?^r{\beta\in\Re^r} may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing is known about the mean vector θ).     

Estimation of quantities determined as implicit functions of unknown parameters

Consider a family of distribution functions ${\{F(x, \theta),\,\theta \in \Theta\}}Consider a family of distribution functions {F(x, q), q ? Q}{\{F(x, \theta),\,\theta \in \Theta\}} . Suppose that there exists an estimator of the unknown parameter vector θ based on given data set. Then it is readily to obtain an estimator of any quantity given as an explicit function g(θ). Particularly, it is the case when the maximum likelihood estimator of θ is available. However, often some quantities of interest can not be expressed as an explicit function, rather it is determined as an implicit function of θ. The present article studies this problem. Sufficient conditions are given for deriving estimators of these quantities. The results are then applied to estimate change point of failure rate function, and change point of mean residual life function.     

A mechanism for reducing small-business customer waiting-line dissatisfaction

This paper presents a technique whereby a small business (i.e. a one-cash register operation) can reduce customer waiting-line time dissatisfaction in the purchase stage of the consumer decision process. When the queue length reaches or exceeds critical value N^*, another employee is temporarily transferred to the role of ‘server assistant’ to increase the effective service rate; when the queue length eventually decreases to a second critical value N_*, the server assistant returns to primary duties. An optimal customer-reneging decision model is utilized to model the reneging character of the queue. Simulation experiments confirm key hypotheses concerning the behaviour of the queue and compare the effectiveness of a computed (N^*, N_*) policy with that of alternatives.     

THE USE OF THE LORENZ CURVE AND THE PARETO DISTRIBUTION IN INTERNAL PAY STRUCTURING: A RESEARCH NOTE

This note presents some findings from the early stages of research into the effectiveness of company pay structures. ¹ 1 A two-year project sponsored by the S.S.R.C. The authors gratefully acknowledge the assistance of the S.S.R.C. in this work.
The use of the Lorenz curve as a measure of inequality of distribution of a firm's payroll is described. Values for L, the Lorenz coefficient, are presented for the total payroll of a chemical processing firm and for the earnings of manual workers in a medium-heavy engineering company. The use of the Pareto distribution for management salary structuring is demonstrated. Values for α, the Pareto coefficient, are presented for two chemical companies and two engineering firms. Some possible practical uses for such Lorenz and Pareto values are suggested.     

Vergleich zweier Operationscharakteristiken

Summary We compare the OC-curvesL _n.c (p) (1) andL _n.c ^* (p) (2). The first is founded on the binomial distribution, the latter relates to the Poisson distribution and is often used as approximation. These OC-curves occur in Statistical Quality Control as probabilities for the acception of a lot as approximations for such probabilities; they are regarded as functions of the fraction defectivep. It is shown that the two OC-curves have exactly one intersection point between 0 and 1, if the acceptance numberc is 1 and the sample sizen is >c+1.Forp between 0 and the intersection pointp _s we have thenL _n.c.(p)>L _n.c ^* (p); from p _s <p1 followsL _n.c(p)n.c ^* (p).An interval is given which coversp _s and with an example it is shown how one might use the results of this paper for the construction of sampling plans.     

Transformationen,die die Normalverteilung charakterisieren

Zusammenfassung Es sei A: R ⁿ R ⁿ eine Abbildung mit für jedes sei einn-dimensionaler Zufallsvektor. Wir beschreiben die Klasse aller TransformationenA, für die unabhängige, nachN(0, 1) verteilte Komponenten hat, sofern nur die KomponentenX ₁,...,X _n des Zufallsvektors ebenfalls unabhängig und identish Gaußisch verteilt sind mit Erwartungswert Null und Varianz 1. Weiter sind Bedingungen angegeben, die sicherstellen, daß nachN(O, ²) verteilte KomponentenX ₁,...,X _n hat, sofern dieX ₁,...,X _n unabhängig und und identisch verteilt sind. Zwei vonBeer undLukacs behandelte Transformationen sind Spezialfälle der hier untersuchten Transformationen.

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Summary Let A: R ⁿ R ⁿ be a transformation with the property for every . We consider a random vector and characterize the class of all transformationsA such that has independentN (0, 1) distributed componentsY ₁,...,Y _n if has the same distribution. Furthermore in the paper there are given conditions which ensure that hasN(O, ² distributed components if and are identically distributed and the componentsX ₁,...,X _n are independent, identically distributed random variables. Two of the transformations tried byBeer andLukacs are special cases of our transformations.

     

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 .     

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.     

Nonparametric regression function estimation using interaction least squares splines and comlexity regularization

Let (X, Y) be a pair of random variables withsupp(X)⊆[0,1] ^l andEY ²<∞. Letm ^* be the best approximation of the regression function of (X, Y) by sums of functions of at mostd variables (1≤d≤l). Estimation ofm ^* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at mostd variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X, Y) the weak and strongL ₂-consistency of the estimate is shown. Furthermore, for everyp≥1 and every distribution of (X, Y) withsupp(X)⊆[0,1] ^l ,y bounded andm ^* p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate      

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.     

Sulla convergenza di un algoritmo di direzioni ammissibili per problemi di ottimo a vincoli lineari con funzione oggetto convessa non differenziabile

Se, essendof la funzione obiettivo del problema, {x ^k } e {f(x ^k )} sono le successioni delle approssimazioni rispettivamente di una soluzione ottimax ^* e dell' ottimof(x ^*) generate da un noto algoritmo di direzioni ammissibili a parametri antizigzag _k , mostriamo che per avere (a) lim _k f(x ^*)=f(x ^*) basta assumere lim _{k k} =0. Inoltre, ove si assuma in più la stretta convessità dif, si ha anche (b) lim _k x ^k =x ^*. Da quest'ultima condizione deriviamo infine specifiche ipotesi, in ordine alla (b), per il caso particolare del problema di trasporto stocastico.

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Summary The aim of the present paper is to analyze, without differentiability of the objective functionf, the convergence of a known «feasible directions» algorithm for constrained optimization problems having the constraints linear [8], 6.5.2.In these circumstances (i.e. iff is not differentiable) one must, almost in general, verify some preliminary conditions to obtain convergence [4]. Nevertheless, this work is not always easy to accomplish particularly in absence of differentiability.Here, we establish that under the convexity assumption forf, the only condition lim _{k k} =0, where the _k are the antizigzag parameters, suffices to obtain the convergence of the algorithm, i.e. lim _k f(x ^k )=opt., thex ^k being the approximate solutions to problem. The proof is obtained by application of the Th. 24.5, [6]. Successively, we consider the question if one has also the convergence of {x ^k } to optimal solution. By using now the Cor. 27.2.2, [6], we establish, for this purpose, that under an additional general qualification forf — precisely the strict convexity — the convergence of {x ^k } is also stated. Finally, we examine the above property for the stochastic transportation problem [1] for which we indicate special conditions in order to verify the latter convergence property.

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pervenuto il 28-4-82     

Tests of significance for the average square deviation from target in a finite lot

Consider lots of discrete items 1, 2, …,N with quality characteristicsx ₁,x ₂, …,x _N . Leta be a target value for item quality. Lot quality is identified with the average square deviation from target per item in the lot (lot average square deviation from target). Under economic considerations this is an appropriate lot quality indicator if the loss respectively the profit incurred from an item is a quadratic function ofx _i −a. The present paper investigates tests of significance on the lot average square deviationz under the following assumptions: The lot is a subsequence of a process of production, storage, transport; the random quality characteristics of items resulting from this process are i.i.d. with normal distributionN(μ, σ ²); the target valuea coincides with the process meanμ.     

<Emphasis Type="SmallCaps">Notes and Comments:</Emphasis> On the uniqueness of convex-ranged probabilities

Abstract In Marinacci (2000), the following theorem was proved. Theorem 1. (Marinacci (2000) Let P and Q be two finitely additive probabilities on a λ -system Σ . Suppose that P is convex-ranged and that Q is countably additive. If there exists an A ⁺ ∈ Σ with 0<P(A ⁺ )<1 such that whenever B∈ Σ , then P=Q. Mathematics Subject Classification (2000): 28A10, 91B06 Journal of Economic Literature Classification: C60, D81     

Variation diminishing property of densities of uniform generalized order statistics

Let f _*,r , r ≥ 1, denote the density function of rth uniform generalized order statistics as defined by Kamps (1995) or Cramer and Kamps (2003). We prove the following variation diminishing property: the number of zeros in (0,1) of any linear combination does not exceed the number of sign changes in the sequence (a ₁, . . . ,a _r ). This result is applied to study monotonicity and convexity properties of f _*,r .     

Representing preferences with nontransitive indifference by a single real-valued function

Let be an interval order on a topological space (X, τ), and let x _˜^* y if and only if [y z x z], and x _˜^** y if and only if [z x z y]. Then _˜^* and _˜^** are complete preorders. In the particular case when is a semiorder, let x _˜⁰ y if and only if x _˜^* y and x _˜^** y. Then _˜⁰ is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing _˜^*, _˜^** and _˜⁰, by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathematical Economics, 1987, 16, 139–146). Finally, we discuss the existence of a pair of continuous functions u, υ representing a strongly separable interval order on a measurable topological space (X, τ, μ, ).     

A generalized maximum likelihood characterization of the normal distribution

LetP be a probability measure on ℝ andI _x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝⁿ and θ ∈ ℝ the inequality holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization of the normal density to a characterization ofN(0, σ²) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980).     

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.