Processo empírico

Em teoria das probabilidades, um processo empírico é um processo estocástico que descreve a proporção de objetos em um sistema em um dado estado. Para um processo em um espaço de estados discreto, uma cadeia de Markov populacional de tempo contínuo[1][2] ou modelo populacional de Markov[3] é um processo que conta o número de objetos em um dado estado (sem reescalonamento). Na teoria de campo médio, teoremas do limite (conforme o número de objetos se torna grande) são considerados e generalizam o teorema central do limite para medidas empíricas.[4] Aplicações da teoria dos processos empíricos surgem na estatística não paramétrica.[5]

Definição

Para variáveis aleatórias independentes e identicamente distribuídas em com função distribuição acumulada comum , a função distribuição empírica é definida por:

em que é a função indicadora do conjunto .[6]

Para todo fixo, é uma sequência de variáveis aleatórias que converge a quase certamente pela lei forte dos grandes números, isto é, converge pontualmente a . O matemático ucraniano Valery Glivenko e o matemático italiano Francesco Paolo Cantelli fortaleceram este resultado ao provar a convergência uniforme de a pelo teorema de Glivenko–Cantelli.[7]

Uma versão centralizada e escalonada da medida empírica é a medida sinalizada:

Isto induz um mapa sobre as funções mensuráveis dado por:

Pelo teorema central do limite, converge em distribuição a uma variável aleatória normal para um conjunto mensurável fixo .[8] De forma semelhante, para uma função fixa , converge em distribuição a uma variável aleatória normal , desde que e .[9]

é um processo empírico indexado por , uma coleção de subconjuntos mensuráveis de .[10]

é um processo empírico indexado por , uma coleção de funções mensuráveis de a .[11]

Um resultado significante na área dos processos empíricos é o teorema de Donsker. Isto levou a um estudo das classes de Donsker: conjuntos de funções com a útil propriedade de processo empíricos indexados por estas classes que convergem fracamente a um certo processo gaussiano.[12] Ainda que se possa mostrar que classes de Donsker são classes de Glivenko–Cantelli, o contrário não é verdadeiro em geral.

Exemplo

Como um exemplo, considere funções distribuição empírica. Para variáveis aleatórias independentes e identicamente distribuídas de valores reais , elas são dadas por:

Neste caso, processos empíricos são indexados por uma classe . Mostrou-se que é uma classe de Donsker em particular.[13]

converge fracamente em a uma ponte browniana .

Referências

  1. Bortolussi, Luca; Hillston, Jane; Latella, Diego; Massink, Mieke. «Continuous approximation of collective system behaviour: A tutorial». Performance Evaluation. 70 (5): 317–349. doi:10.1016/j.peva.2013.01.001 
  2. Stefanek, Anton; Hayden, Richard A.; Gonagle, Mark Mac; Bradley, Jeremy T. (4 de junho de 2012). «Mean-Field Analysis of Markov Models with Reward Feedback». Springer, Berlin, Heidelberg. Analytical and Stochastic Modeling Techniques and Applications. Lecture Notes in Computer Science (em inglês): 193–211. ISBN 9783642307812. doi:10.1007/978-3-642-30782-9_14 
  3. Dayar, Tuǧrul; Hermanns, Holger; Spieler, David; Wolf, Verena (1 de novembro de 2011). «Bounding the equilibrium distribution of Markov population models». Numerical Linear Algebra with Applications (em inglês). 18 (6): 931–946. ISSN 1099-1506. doi:10.1002/nla.795 
  4. Dudley, R. M. (1978). «Central Limit Theorems for Empirical Measures». The Annals of Probability (em inglês). 6 (6): 899–929. ISSN 0091-1798. doi:10.1214/aop/1176995384 
  5. Mojirsheibani, Majid. «Nonparametric curve estimation with missing data: A general empirical process approach». Journal of Statistical Planning and Inference. 137 (9): 2733–2758. doi:10.1016/j.jspi.2006.02.016 
  6. Billingsley, Patrick (20 de janeiro de 2012). Probability and Measure (em inglês). [S.l.]: John Wiley & Sons. ISBN 9781118341919 
  7. Wolfowitz, J. (1954). «Generalization of the Theorem of Glivenko-Cantelli». The Annals of Mathematical Statistics (em inglês). 25 (1): 131–138. ISSN 0003-4851. doi:10.1214/aoms/1177728852 
  8. Dudley, R. M. (28 de julho de 1999). Uniform Central Limit Theorems (em inglês). [S.l.]: Cambridge University Press. ISBN 9780521461023 
  9. Dzhaparidze, K. O.; Nikulin, M. S. (1 de outubro de 1982). «Probability distributions of the Kolmogorov and omega-square statistics for continuous distributions with shift and scale parameters». Journal of Soviet Mathematics (em inglês). 20 (3): 2147–2164. ISSN 0090-4104. doi:10.1007/BF01239992 
  10. R., Kosorok, Michael (2008). Introduction to empirical processes and semiparametric inference. New York, N.Y.: Springer. ISBN 9780387749785. OCLC 233972325 
  11. Shorack, G.; Wellner, J. (1 de janeiro de 2009). Empirical Processes with Applications to Statistics. Col: Classics in Applied Mathematics. [S.l.]: Society for Industrial and Applied Mathematics. ISBN 9780898716849. doi:10.1137/1.9780898719017 
  12. vaart, Aad van der; Wellner, Jon (9 de março de 2013). Weak Convergence and Empirical Processes: With Applications to Statistics (em inglês). [S.l.]: Springer Science & Business Media. ISBN 9781475725452 
  13. Donsker, Monroe D. (1952). «Justification and Extension of Doob's Heuristic Approach to the Kolmogorov- Smirnov Theorems». The Annals of Mathematical Statistics (em inglês). 23 (2): 277–281. ISSN 0003-4851. doi:10.1214/aoms/1177729445 

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