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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 18-22 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 23-26 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 27-35 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper studies the transient behaviour of a single-channel queueing problem wherein (i) the input, following aPoisson distribution, is in batches of variable size (ii) the queue discipline is first-come-first-served; it being assumed that the batches are pre-ordered for service purposes (iii) the output, following a general distribution, is in batches of variable size. TheLaplace transform of the probability generating function of the waiting line size is obtained and the corresponding results are derived when the service time distribution is (i) hyper-exponential with m branches (ii) phase-type and (iii) exponential. Finally, some particular cases are discussed.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 48-53 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 54-67 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary Blackwell's studies concerning comparisons of experiments yield a concept of sufficiency which in general is not equivalent to the popular notion of a sufficient statistic made precise byHalmos andSavage. In fact regularity conditions can be given under which the two concepts coincide. Properties ofMarkov kernels and ofBlackwell's sufficiency relation are examined first. After the proof of the equivalence theorem counterexamples are presented to indicate the significance of the regularity conditions.
    Notes: Zusammenfassung Der vonBlackwell mittelsMarkoffscher Kerne eingeführte Begriff der Erschöpftheit eines klassischen Experiments für ein zweites mit gleicher Indexmenge wird mit dem Begriff der erschöpfenden Statistik vonHalmos undSavage verglichen. In der Tat gibt es Bedingungen, unter denen beide Begriffe zusammenfallen. Der Äquivalenzsatz wird durch EigenschaftenMarkoffscher Kerne sowie derBlackwellschen Erschöpftheitsrelation vorbereitet. Gegenbeispiele ermöglichen es, die Notwendigkeit der Regularitätsbedingungen einzuschen.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 76-76 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 72-75 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Es wird gezeigt, daß man beik-stufigen Tests zwar für die Gestalt vonBayes-Lösungen ein Analogon zu den unbeschränkten Sequenztests hat, daß aber der Satz vonWald undWolfowitz über die gleichmäßige Optimalität bzgl. des Stichprobenumfangs keine Entsprechung besitzt.
    Notes: Summary This note deals with sequential tests requiring at mostk observations. First a lemma is proved concerning the form of Bayes solutions. Then it is demonstrated by giving a counter-example that the theorem ofWald andWolfowitz has no correspondence fork-stage tests.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 77-77 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 19 (1972), S. 68-71 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary The well-known “law of rare events” deals with a “large” number of independent events, each of which can occur with the same probability. Under suitable assumptions the number of events which occur has approximately a Poisson distribution. [CompareMorgenstern, p. 39, 1968;Waerden, p. 47, 1957]. An example which is often used for illustration is the frequency distribution of accidents, in which only one car or one person is involved. In this paper we refer to accidents in which two parties are involved. This leads to a generalization of the “law of rare events” where cases of dependent events are concerned. In addition a new proof of the general formula as to the moments of Poisson distribution is given. This proves a limit theorem of “weakly dependent” random variables of a certain sequence.
    Notes: Zusammenfassung Beim bekannten “Gesetz seltener Ereignisse” betrachtet mane “große” Zahl unabhängiger Ereignisse, die jeweils mit der gleichen „kleinen” Wahrscheinlichkeit eintreten können. Unter geeigneten Annahmen ist die Zahl der eingetretenen Ereignisse näherungsweise poissonverteilt. [Vgl. z.B. Morgenstern, p. 39, 1968;Waerden, p. 47, 1957]. Ein oft angegebenes Beispiel hierfür ist die Häufigkeitsverteilung von Unfällen, die jeweils nur eine Person oder ein Fahrzeug betreffen. In dieser Arbeit gehen wir von Unfällen mit zwei Beteiligten aus. Auf diese Weise erhält man eine Verallgemeinerung des „Gesetzes seltener Ereignisse” auf gewisse abhängige Ereignisse. Außerdem wird ein neuer Beweis für die allgemeine Formel der Momente der Poissonverteilung gegeben. Damit ist ein Grenzwertsatz für eine spezielle Folge „schwach abhängiger” Zufallsgrößen bewiesen.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Metrika 25 (1978), S. 77-93 
    ISSN: 1435-926X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Exponential smoothing procedures, in particular those recommended byBrown [1962] are used extensively in many areas of economics, business and engineering. It is shown in this paper that: i) Brown's forecasting procedures are optimal in terms of achieving minimum mean square error forecasts only if the underlying stochastic process is included in a limited subclass of ARIMA (p, d, q) processes. Hence, it is shown what assumptions are made when using these procedures. ii) The implication of point (i) is that the users ofBrown's procedures tacitly assume that the stochastic processes which occur in the real world are from the particular restricted subclass of ARIMA (p, d, q) processes. No reason can be found why these particular models should occur more frequently than others. iii) It is further shown that even if a stochastic process which would lead toBrown's model occurred, the actual methods used for making the forecasts are clumsy and much simpler procedures can be employed.
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