Trimmed mean or sample median?

It is shown that in the symmetric case trimmed means are better estimates of location than sample medians, unless the error distributions are sharply peaked at the centre.     

An estimate of deviation probabilities of the sample mean of variables with ψ-semimixing

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 1, pp. 107–119, January–March, 1994.     

Limit theorems for the median deviation

Summary In this paper, we obtain a strong law and central limit theorem for the median deviation under only very mild smoothness conditions on the underlying distribution. Under an additional condition implied by symmetry, we derive a weak Bahadur representation for the median deviation and establish the asymptotic equivalence of the median deviation and the semi-interquartile range.     

Minimizing weighted mean absolute deviation of job completion times from their weighted mean

We address a single-machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean. This problem and its precursors aim to achieve the maximum admissible level of service equity. It has been shown earlier that the unweighted version of this problem is NP-hard in the ordinary sense. For that version, a pseudo-polynomial time dynamic program and a 2-approximate algorithm are available. However, not much (except for an important solution property) exists for the weighted version. In this paper, we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median (rather than the mean) of the job completion times; this generalizes the 2-approximation result mentioned above. We proceed to give a pseudo-polynomial time dynamic program, establishing the ordinary NP-hardness of the problem in general. We then present a fully-polynomial time approximation scheme as well. Finally, we report the findings from a limited computational study on the heuristic solution of the general problem. Our results specialize easily to the unweighted case; they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time.     

Max control chart with adaptive sample sizes for jointly monitoring process mean and standard deviation

This paper develops the two-state and three-state adaptive sample size control schemes based on the Max chart to simultaneously monitor the process mean and standard deviation. Since the Max chart is a single variables control chart where only one plotting statistic is needed, the design and operation of adaptive sample size schemes for this chart will be simpler than those for the joint X? and S charts. Three types of processes including on-target initial, off-target initial and steady-state conditions are considered to evaluate the chart performance. The results of this study show that both two-state and three-state schemes are more efficient than the conventional non-adaptive joint X? and S charts. The three-state procedure is only slightly better than the two-state scheme, and the most dramatic improvement occurs when the two-state scheme is compared with the non-adaptive joint X? and S charts. Moreover, with the ease of implementation, the two-state scheme is likely adequate in most practical applications.     

The expansion for the error distribution of sample median and its random weighting statistic

In this paper, the expansion for the error distribution of the sample median is obtained. To mimic the error distribution of the sample median, random weighting statistic is introduced. It is proved in this paper that the precision of the approximation of the error distribution for sample median by random weighting method is of the order ofo(1/n).Partially supported by the Chinese Science Foundation and the Peking University Science Foundation.     

On a new mean and functions of bounded deviation

The main purpose of the present paper is to introduce a new type of mean of a function and to study some of its special properties. In the last section we make use of this mean to show that a function of bounded deviation is not necessarily a function of bounded variation.     

A characterization of the logistic distribution by a sample median

Summary A functional form of the characteristic function of the sample median of three variates is shown to characterize the logistic distribution. A consequence of this is a characterization of the logistic in terms of the Laplace distribution. Research sponsored in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant No. AFOSR-77-3360. The United States Govenment is authorized to reproduce and distribute reprints for governmental purposes notwith-standing any copyright notation hereon.     

Finite sample breakdown point of Tukey's halfspace median

Tukey's halfspace median(HM), servicing as the multivariate counterpart of the univariate median,has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property(the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point(FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one—the Tukey's halfspace median. This paper presents a precise result on FSBP for Tukey's halfspace median. The result here depicts the complete prospect of the global robustness of HM in the finite sample practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing asymptotic breakdown point result.     

On fisher's information contained in the sample mean

Within the framework of a parametric model, it is assumed that a number of first moments of the observed random variable are known. It is shown how fast the Fisher information contained in the sample mean decreases as the sample size tends to infinity. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part II.     

An informational analog of the theorem of independence of sample mean and sample variance

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 63–67, 1985.     

Constant regression of quadratic statistics on the sample mean

пОлУЧЕНО пРЕДстАВлЕ НИЕ хАРАктЕРИстИЧЕс кИх ФУНкцИИ Дль слУЧАь, кОгДА кВАД РАтИЧЕскАь стАтИстИкА ВыБОРкИ О БлАДАЕт пОстОьННОИ Р ЕгРЕссИЕИ ОтНОсИтЕльНО ВыБОРО ЧНОгО сРЕДНЕгО. НАшА тЕОРЕМА ДАЕт пРЕ ДстАВлЕНИЕ кВАДРАтИ ЧЕскИх стАтИстИк, кОтОРыЕ НЕ МОгУт ОБлАДАть пОстОьННОИ РЕгРЕссИ ЕИ.     

标准差和平均差的内在关系及应用研究

由于绝对值不是变量的解析函数,不具有方差的良好数学性质;但由于它对极少数极端事件不敏感,因而更具有抗干扰性。在实证研究波动时,更偏向于平均差。本文正是在此矛盾的基础上,通过考虑分布和不考虑分布的两种情况,从理论和应用的不同角度,深入地研究了两者间的内在关系及应用情况,从而为科学合理地利用离散度指标提供新思路。     

Approximation of expected returns and optimal decisions under uncertainty using mean and mean absolute deviation

We consider economic decision problems under uncertainty consisting of choosing an optimal decisionX, so as to maximize to expected value of an objective function depending on a stochastic parameterp. The paper establishes an optimal policy intervalX _A X ¹ X _B, where the boundsX _A,X _B are given in terms of simple parameters of the distribution ofp, in particular the mean, and the mean absolute deviationd=E ¦p– ¦. The convexity assumptions needed to establish such bounds are shown to hold naturally in some classical problems of production under uncertainty.

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Zusammenfassung Wir betrachten wirtschaftliche Entscheidungsprobleme mit Unsicherheit, in denen eine optimale EntscheidungX so getroffen werden soll, daß der Erwartungswert einer Zielfunktion, abhängig von einem stochastischen Parameterp, maximiert werden soll. In dieser Arbeit wird ein IntervallX _A X ¹ X _B für die optimale Politik angegeben, wobei die SchrankenX _A,X _B durch einfache Größen der Verteilung vonp ausgedrückt werden, im besonderen durch den Mittelwert und die mittlere absolute Abweichungd=E ¦p– ¦. Ferner wird gezeigt, daß die für die Herleitung der Schranken benötigten Konvexitätsannahmen in natürlicher Weise für einige klassische Produktionsprobleme mit Unsicherheit gelten.

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Supported by BARD Project No. I-10-79 and by Technion VPR Fund-Lawrence Deutsch Research Fund.     

A large deviation principle for bootstrapped sample means

A large deviation principle for bootstrapped sample means is established. It relies on the Bolthausen large deviation principle for sums of i.i.d. Banach space valued random variables. The rate function of the large deviation principle for bootstrapped sample means is the same as the classical one.

     

A Property of the mean deviation (Of some discrete distributions)

Summary This note describes an interesting property of the mean deviation which holds for a number of commonly known discrete distributions. The property is also examined for some of the well-known continuous distributions.