Large deviations in estimation of an Ornstein-Uhlenbeck model

1999 ◽  
Vol 36 (1) ◽  
pp. 60-77 ◽  
Author(s):  
Danielle Florens-Landais ◽  
Huyên Pham
Keyword(s):  
Large Deviation ◽  
Score Function ◽  
Rate Function ◽  
Estimating Functions ◽  
Uhlenbeck Process ◽  
Large Deviation Principles ◽  
Dependent Change ◽  
Parameter Dependent ◽  
Explicit Rate ◽  
Drift Parameter

A large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process. We establish an LDP for two estimating functions, one of them being the score function. The first one is derived by applying the Gärtner–Ellis theorem. But this theorem is not suitable for the LDP on the score function and we circumvent this key point by using a parameter-dependent change of measure. We then state large deviation principles for the maximum likelihood estimator and another consistent drift estimator.

Large deviations in estimation of an Ornstein-Uhlenbeck model

1999 ◽  
Vol 36 (01) ◽  
pp. 60-77 ◽  
Author(s):  
Danielle Florens-Landais ◽  
Huyên Pham
Keyword(s):  
Large Deviation ◽  
Score Function ◽  
Rate Function ◽  
Estimating Functions ◽  
Uhlenbeck Process ◽  
Large Deviation Principles ◽  
Dependent Change ◽  
Parameter Dependent ◽  
Explicit Rate ◽  
Drift Parameter

A large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process. We establish an LDP for two estimating functions, one of them being the score function. The first one is derived by applying the Gärtner–Ellis theorem. But this theorem is not suitable for the LDP on the score function and we circumvent this key point by using a parameter-dependent change of measure. We then state large deviation principles for the maximum likelihood estimator and another consistent drift estimator.

scholarly journals Large deviations for the Ornstein-Uhlenbeck process with shift

2015 ◽  
Vol 47 (03) ◽  
pp. 880-901 ◽  
Author(s):  
Bernard Bercu ◽  
Adrien Richou
Keyword(s):  
Maximum Likelihood ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Shift Parameter ◽  
Uhlenbeck Process ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Ornstein Uhlenbeck Process

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

scholarly journals Large deviations for the Ornstein-Uhlenbeck process with shift

2015 ◽  
Vol 47 (3) ◽  
pp. 880-901 ◽  
Author(s):  
Bernard Bercu ◽  
Adrien Richou
Keyword(s):  
Maximum Likelihood ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Shift Parameter ◽  
Uhlenbeck Process ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Ornstein Uhlenbeck Process

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

Economies of scale in queues with sources having power-law large deviation scalings

1996 ◽  
Vol 33 (3) ◽  
pp. 840-857 ◽  
Author(s):  
N. G. Duffield
Keyword(s):  
Power Law ◽  
Generating Functions ◽  
Economies Of Scale ◽  
Shape Function ◽  
Large Deviation ◽  
Rate Function ◽  
Uhlenbeck Process ◽  
Asymptotic Decay ◽  
Potential Structure ◽  
Asymptotic Decay Rate

We analyse the queue QL at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[Wt/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions limL→xL–1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta (such as occur in FBM) we analyse the asymptotics of the shape function limb→xb–u/a(I(b) – δbv/a) = vu for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL > Lb] ≈ exp[−δLbv/a] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

Economies of scale in queues with sources having power-law large deviation scalings

1996 ◽  
Vol 33 (03) ◽  
pp. 840-857 ◽  
Author(s):  
N. G. Duffield
Keyword(s):  
Power Law ◽  
Generating Functions ◽  
Economies Of Scale ◽  
Shape Function ◽  
Large Deviation ◽  
Rate Function ◽  
Uhlenbeck Process ◽  
Asymptotic Decay ◽  
Potential Structure ◽  
Asymptotic Decay Rate

We analyse the queue QL at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[W t/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions lim L→x L –1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta (such as occur in FBM) we analyse the asymptotics of the shape function lim b→x b –u/a (I(b) – δbv/a ) = vu for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL > Lb] ≈ exp[−δLbv/a ] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

Functional Large Deviation Principles for Waiting and Departure Processes

1998 ◽  
Vol 12 (4) ◽  
pp. 479-507 ◽  
Author(s):  
Anatolii A. Puhalskii ◽  
Ward Whitt
Keyword(s):  
Large Deviation ◽  
Rate Function ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Departure Process ◽  
Large Deviation Principles ◽  
Special Cases ◽  
Skorohod Topologies ◽  
Departure Processes

We establish functional large deviation principles (FLDPs) for waiting and departure processes in single-server queues with unlimited waiting space and the first-in first-out service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition, and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit expressions for the rate function of the departure process, but not more generally. In general, the rate function for the departure process evidently must be calculated numerically. We also obtain an FLDP for the departure process of completed work, which has important application to the concept of effective bandwidths for admission control and capacity planning in packet communication networks.

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