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model used to model volatility in financial time series. Stochastic equicontinuity helps the estimated parameters of the GARCH model converge to the true parameters as the sample size increases, despite the model’s nonlinear nature.
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596:{\displaystyle \limsup _{n\rightarrow \infty }\Pr \left(\sup _{\theta \in \Theta }\sup _{\theta '\in B(\theta ,\delta )}|H_{n}(\theta ')-H_{n}(\theta )|>\epsilon \right)<\eta .}
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relates to the model currently being postulated or fitted rather than to an underlying model which is supposed to represent the mechanism generating the data. Then
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For instance, stochastic equicontinuity, along with other conditions, can be used to show uniform weak convergence, which can be used to prove the
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under which a set of observed data is considered to be a realisation of a probabilistic or statistical model. However, in
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996:: In nonlinear time series models, stochastic equicontinuity ensures the stability and consistency of estimators.
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1012:: Stochastic equicontinuity is a key condition for proving the consistency of estimators in time series models.
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of sequences of random variables and requires that this rate is essentially the same within a region of the
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Newey, Whitney K. (1991). "Uniform
Convergence in Probability and Stochastic Equicontinuity".
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Asymptotic Theory of
Expanding Parameter Space Methods and Data Dependence in Econometrics
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Example: Consider an M-estimator defined by minimizing a sample objective function
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de Jong, Robert M. (1993). "Stochastic
Equicontinuity for Mixing Processes".
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might represent a sequence of estimators applied to datasets of size
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1130:"Uniform Convergence in Probability and Stochastic Equicontinuity"
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is needed to prove the consistency and asymptotic normality of
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32:(estimation procedures) that is useful in dealing with their
1148:"Stochastic equicontinuity in nonlinear time series models"
826:, stochastic equicontinuity is needed in establishing the
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614:) represents a ball in the parameter space, centred at
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1110:"Applications of ULLNs: Consistency of M-estimators"
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as the amount of data increases. It is a version of
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723:converges uniformly to its population counterpart
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687:. Stochastic equicontinuity helps in showing that
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193:{\displaystyle \Theta \rightarrow \mathbb {R} }
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172:be a family of random functions defined from
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1054:: CS1 maint: location missing publisher (
165:{\displaystyle \{H_{n}(\theta ):n\geq 1\}}
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1199:. New York: Springer. pp. 138–142.
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48:. The property relates to the rate of
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830:of nonparametric estimators. Like -
781:{\displaystyle {\hat {\theta }}_{n}}
40:used in the context of functions of
1197:Convergence of Stochastic Processes
1237:. You can help Knowledge (XXG) by
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309:{\displaystyle \{H_{n}(\theta )\}}
255:{\displaystyle \{H_{n}(\theta )\}}
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179:
14:
928:{\displaystyle {\hat {f}}_{n}(x)}
883:{\displaystyle {\hat {f}}_{n}(x)}
220:is any normed metric space. Here
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788:converges to the true parameter
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1284:Asymptotic theory (statistics)
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1034:. Amsterdam. pp. 53–72.
994:Nonlinear Time Series Models
618:and whose radius depends on
424:{\displaystyle \delta >0}
1193:"Stochastic Equicontinuity"
808:{\displaystyle \theta _{0}}
1305:
1216:
1146:Hagemann, Andreas (2014).
745:{\displaystyle Q(\theta )}
398:{\displaystyle \eta >0}
1010:Consistency of Estimators
832:kernel density estimators
641:Stochastic equicontinuity
346:{\displaystyle \{H_{n}\}}
26:stochastic equicontinuity
1152:The Econometrics Journal
824:nonparametric estimation
820:Nonparametric Estimation
1191:Pollard, David (1984).
272:data generating process
213:{\displaystyle \Theta }
1233:-related article is a
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34:asymptotic behaviour
1115:. 15 February 2007.
964:as the sample size
828:uniform convergence
65:extremum estimators
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988:Time Series Models
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836:spline regressions
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98:. You can help by
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1289:Probability stubs
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1135:. 30 August 2010.
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1185:Further reading
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107:September 2010
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86:This section
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1072:Econometrica
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645:M-estimators
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637:M-Estimators
636:
631:Econometrics
626:Applications
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100:adding to it
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25:
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1231:probability
1158:: 188–196.
984:increases.
431:such that:
90: with:
61:convergence
50:convergence
44:: that is,
1278:Categories
1017:References
71:Definition
30:estimators
22:statistics
1165:1206.2385
1050:cite book
905:^
860:^
797:θ
767:^
764:θ
737:θ
708:θ
672:θ
588:η
577:ϵ
563:θ
547:−
537:θ
510:δ
504:θ
495:∈
488:θ
477:Θ
474:∈
471:θ
453:∞
450:→
413:δ
387:η
361:ϵ
298:θ
244:θ
208:Θ
183:→
180:Θ
154:≥
142:θ
540:′
491:′
200:, where
1093:2938179
96:2938179
1203:
1091:
1038:
620:δ
616:θ
318:θ
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1229:This
1160:arXiv
1133:(PDF)
1113:(PDF)
1089:JSTOR
1002:GARCH
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606:Here
92:JSTOR
1235:stub
1201:ISBN
1056:link
1036:ISBN
585:<
574:>
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379:and
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