Mathematical Theory of Statistics

Strasser, Helmut

doi:10.1515/9783110850826

Cited by 309 publications

(187 citation statements)

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“…In this section we describe the notion of local asymptotic normality and its significance in statistics [28,49,45,50]. Suppose that we observe a sample X 1 , .…”

Section: Classical Statistical Experimentsmentioning

confidence: 99%

“…In this section we will present some basic ideas of this theory, the converging point being the notion of local asymptotic normality. For more information we refer to the monographs [28,49,45,50].…”

Section: Statistical Experiments Sufficiency Randomizationsmentioning

confidence: 99%

“…Finally we mention another useful characterization of sufficiency known as the Factorization Theorem [45] which later will be extended to the quantum case. Theorem 2.6 Let E = (P θ : θ ∈ Θ) be a dominated experiment on (Ω, Σ) with P θ ∼ µ, and let Σ 0 ⊂ Σ be a sub-σ-field.…”

Section: Statistical Experiments Sufficiency Randomizationsmentioning

confidence: 99%

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Local Asymptotic Normality in Quantum Statistics

Guţǎ

Jenčová

2007

Commun. Math. Phys.

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The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕ n θ 0 +u/ √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all "local parameters" u ∈ R m such that θ = θ0 + u/ √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ R m . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For reader's convenience and completeness we review the relevant results of the classical as well as the quantum theory.

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“…In this section we describe the notion of local asymptotic normality and its significance in statistics [28,49,45,50]. Suppose that we observe a sample X 1 , .…”

Section: Classical Statistical Experimentsmentioning

confidence: 99%

Section: Statistical Experiments Sufficiency Randomizationsmentioning

confidence: 99%

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Local Asymptotic Normality in Quantum Statistics

Guţǎ

Jenčová

2007

Commun. Math. Phys.

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“…The loss of information is measured in terms of the deficiency of statistical experiments in the sense of Le Cam (1986), see also Strasser (1985). It is well known that the deficiency L1 is a pseudo distance on the set tS (T) of all experiments given a parameter set T.…”

Section: Id Random Variables With Common Distributionmentioning

confidence: 99%

Global sufficiency of extreme order statistics in location models of Weibull type

Marohn

1991

Probab. Th. Rel. Fields

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“…For a theory of such approximations, see Strasser (1985), Le Cam (1985) and (1986). The next section will elaborate on such approximations in a context derived from Wald's assumptions.…”

Section: Gaussian Experiments and Distancesmentioning

confidence: 99%

On the Standard Asymptotic Confidence Ellipsoids of Wald

Cam¹

1990

International Statistical Review / Revue Internationale de Stat

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Mathematical Theory of Statistics

Cited by 309 publications

References 0 publications

Local Asymptotic Normality in Quantum Statistics

Local Asymptotic Normality in Quantum Statistics

Global sufficiency of extreme order statistics in location models of Weibull type

On the Standard Asymptotic Confidence Ellipsoids of Wald

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