Expertises : proc\'{e}dures statistiques d'aide \`{a} la d\'{e}cision

Citeres); Guy Morel (LMPT

REVIEW

Expertises : proc\'{e}dures statistiques d'aide \`{a} la d\'{e}cision

Guy Morel (LMPT, Citeres)

Statistics Theory · Submitted 27 Feb 2006 · 22 references

Pith's verdict: A genuinely novel decision-theoretic frame that reduces p-values and flat-prior posteriors to a vote over conditionally unbiased experts—but the central identification is fixed by an unaxiomatized 1/2 atom-split convention.

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CitationCiteres), Guy Morel (LMPT. "Expertises : proc\'{e}dures statistiques d'aide \`{a} la d\'{e}cision." arXiv:math/0602611 (2006). https://pith.science/paper/3N6GRP5U

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Editor's note

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Dear colleague,

The paper is worth a look. What is actually new: it defines “experts” as decision rules that are unbiased conditionally on every event, proves (Prop 2.3.1) that these are exactly the deterministic Neyman tests sandwiched between consecutive simple test functions of the likelihood ratio K = p0/p1, then probabilizes the space of experts and calls the resulting probability on decisions a “vote.” That characterization is clean, and I verified the core of Chapters 2–4; it holds. In the two-point problem the vote reproduces the mid-p-value, and in the normal one-sided family the compatible votes reproduce the flat-prior posterior. This is a genuine conceptual unification: p-values and non-informative posterior surfaces appear as outputs of one deliberation mechanism.

The soft spots are real but localized. The vote measure is built from Eθ, but when K has an atom at {K=k}, the paper splits that atom evenly between φ(k,0) and φ(k,1) because it “seems legitimate” (Definition 2.4.1). That 1/2 is exactly what makes the vote equal the mid-p. With a general split α, the vote becomes Pθ(K>k) + α Pθ(K=k); only α=1/2 gives mid-p, while α=0 and α=1 give the two boundary p-values. The same convention propagates through the essential statistic in Section 4.3 and through the probability on Θ in Section 5.2. The paper itself acknowledges non-uniqueness in Section 4.3 and again in Section 5.2 (“Il y a bien d’autres manières…”). So the headline identification is not a theorem about experts alone; it is a theorem about experts plus a hand-chosen weighting rule. The mathematics is coherent and the conditional-unbiasedness notion is a legitimate primitive, but the framework does not uniquely select the mid-p or the flat-prior-like posterior.

Two smaller issues: the claim that “most classical MLR models” satisfy the conditions of Proposition 5.2.3 is asserted without a catalogue, and there is no comparison with the confidence-distribution literature, which heavily overlaps. The author leans on his own [Mor.1] and [Mor.2] for level-0 UPP tests and minimax rules; I did not find circularity, but those references should be checked. The review copy truncates part of Section 5.3 and the annexes, so my confidence in the tail is moderate.

Who this is for: statistical decision theorists and people working on foundations of p-values. Not yet for applied users. It deserves a serious referee: Proposition 2.3.1 and the vote machinery are worth extracting and re-deriving with explicit axioms. My recommendation is to engage, but treat the 1/2 split as an open modeling choice, not a derived result.

Referee report

Significance

Proposition 2.3.1 is a clean and checkable characterization, and the vote construction offers a novel interpretation of p-values as frequencies of conditionally unbiased decision rules. The extension to stable hypotheses and the construction of a global essential statistic in Section 5.2 are substantial and mostly coherent. The main limitation is that the headline equalities are not uniqueness theorems: the vote equals the mid-p only because of the equitable atom-splitting convention in Definition 2.4.1, and the posterior-like probability on Θ depends on the choice of a compatible family of votes. The manuscript itself acknowledges non-uniqueness in Sections 4.3 and 5.2, but the abstract and Section 2.4 present the identification more strongly than is justified. With a clear statement of the conventions and their consequences, the paper would be a useful contribution.

Major objections (2)

[Section 2.4, Definition 2.4.1] Definition 2.4.1 fixes the vote by splitting the atom {K=k} equally between φ(k,0) and φ(k,1). This is a modeling choice, not a consequence of the definition of an expert. If one assigns mass α∈[0,1] to φ(k,1) and 1−α to φ(k,0), the vote on {K=k} becomes Q^ω_{θ,α}({1}) = P_θ(K<k)+α P_θ(K=k), and only α=1/2 gives the mid-p-value. Since the identification with the mid-p, and through it the essential statistic of Section 4.3 and the parameter-space probabilities of Section 5.2, all use α=1/2, the central claim that the framework yields p-values is not derived from the expert axioms alone. The manuscript should state this explicitly and either justify α=1/2 by an additional axiom (e.g. symmetry or neutrality) or present the result as one member of a family of votes.
[Section 5.2, Proposition 5.2.4 and normal example] The probability on the parameter space is not unique. Proposition 5.2.4 explicitly constructs weighted votes that extend to a different probability, N(x, σ²/n+c²) in the normal example, while the most-favorable votes give N(x, σ²/n). Both are compatible in the sense of Definition 5.2.1. Therefore the comparison with the flat-prior Bayes posterior is a property of one particular choice, not a general consequence of the framework. The text should clarify that the axioms of experts plus compatibility do not single out the reported probability, and the abstract's wording should be adjusted accordingly.

Minor objections (4)

[Section 2.4] The phrase 'il semble légitime de partager équitablement la masse' deserves a fuller discussion of alternative splits and of the dependence of the vote on the split parameter α; as written, a reader may mistake a convention for a theorem.
[Section 4.3, Definition 4.3.1] The definition of K(t) excludes the simultaneous case (at,ut)=(−∞,?) and (bt,vt)=(+∞,?) only through a brief parenthetical remark; the displayed cases should explicitly cover or exclude all possible configurations of at and bt.
[General presentation] The manuscript contains many OCR and spacing artifacts (e.g. 'd ´ecision', 'a ′ t', 'θ ′p.s.'), which make the text harder to read; a clean typeset version would improve the presentation.
[Annexes and Section 5.3] The review copy truncates the tail of Section 5.3 and does not include Annexes II–IV; since several proofs rely on the normalization lemmas of those annexes, the final manuscript should ensure they are complete and available.

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The paper

The paper claims

The paper claims that an 'expert' decision rule is exactly a common Neyman test, and the vote of all experts is the mid-p-value.

The reading

This paper tries to ground two-hypothesis decision procedures in an intrinsic property: a rule is an 'expert' exactly when its diagnosis $d$ is more likely under the hypothesis it names, even after conditioning on any non-negligible event. It proves that such experts are precisely the common Neyman tests of the two hypotheses, i.e. rules almost surely sandwiched between consecutive simple test functions of the likelihood ratio $K=p_0/p_1$. Instead of selecting one expert, the paper lets all experts vote, weighting each expert by how much its decision probabilities differ under the model, and claims the vote for decision 1 is exactly the mid-p-value. This matters because it produces a probability on the two decisions, and for monotone-likelihood families a probability on the parameter space, with no loss function and no prior. If correct, the framework is not a new inferential quantity but a democratic interpretation of the standard Neyman testing machinery.

What carries the argument

The machinery is the totally ordered ladder $\Phi_s=\{\varphi(0,1);\ \varphi(k,\beta)=1_{\{K<k\}}+\beta 1_{\{K=k\}},\ k\in\mathbb{R}_+^*, \beta\in\{0,1\};\ \varphi(\infty,0)\}$, with lexicographic order on $(k,\beta)$. Proposition 2.3.1 shows every expert is almost surely inside this ladder, and Definition 2.4.1 turns the ladder into a ballot box: the two measures $m_\theta(]\leftarrow,\varphi[)=E_\theta(\varphi)$ and $m'_\theta(]\leftarrow,\varphi])=E_\theta(\varphi)$ agree everywhere except on the atoms $\{K=k\}$, and the equitable split of each atom gives the vote $Q^\omega_\theta(\{1\})=1-E_\theta(\varphi(k,1/2))$. The same ladder, re-indexed through the essential statistic $K(T)$ of Definition 4.3.1, carries the extension to stable hypotheses, one-sided and two-sided hypotheses, and finally to a probability on the parameter space.

Extended reading notes

Core claim

The central discovery is Proposition 2.3.1: a deterministic rule $\varphi$ is an expert for the choice between $P_0$ and $P_1$, meaning $P_1(\varphi=1)\ge P_0(\varphi=1)$ and the same inequality holds conditionally on every event, if and only if it is almost surely sandwiched between consecutive simple Neyman test functions $\varphi(k,0)\le\varphi\le\varphi(k,1)$ of the likelihood ratio $K=p_0/p_1$, with boundary cases at $k=0$ and $k=\infty$. When the experts vote, using the probability $m_\theta$ on the ordered space of simple test functions defined by $m_\theta(]\leftarrow,\varphi[)=E_\theta(\varphi)$ and splitting the mass at any atom $\{K=k\}$ evenly between the two adjacent test functions, the vote for decision 1 is $1-E_\theta(\varphi(k,1/2))$ when $K(\omega)=k\in(0,\infty)$, with boundary values 1 and 0. Thus the vote is the mid-p-value. In monotone-likelihood models the compatible votes of Section 5.2 extend to a probability on the parameter space; in the normal example that probability is $N(\bar{x},\sigma^2/n)$, the same object a Bayesian obtains from the improper flat prior.

Load-bearing premise

The load-bearing premise is the convention that when the data lands exactly on a boundary value of the likelihood ratio that has positive probability, the experts on that atom split their votes equally between the two adjacent decisions; if that fifty-fifty split is replaced, the vote stops being the mid-p-value.

Editorial extensions

If this is right

For two simple hypotheses, the output a user receives from the vote is the mid-p-value, a quantity already familiar from testing practice, so the vote can be reported and compared with standard rejection thresholds.
In monotone-likelihood models the vote for $\{\theta<\theta_0\}$ against $\{\theta\ge\theta_0\}$, evaluated at the boundary $\theta_0$, is both the minimum rejection threshold of the UMP test and a neutral vote: its mean under $P_{\theta_0}$ is $1/2$.
The compatible-vote construction yields a genuine probability on the order $\sigma$-algebra of $\Theta$; for one-sided hypotheses this probability is not a Bayes posterior from a prior on $\Theta$, and in the normal illustration it is the flat-prior posterior.
Pure two-sided hypotheses admit only trivial experts, so the practical route for two-sided problems is to reconstruct votes from compatible one-sided votes, not to search for a two-sided expert.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equitable $1/2$ split is the single convention that makes the vote coincide with the mid-p-value. This is my inference: any other split of the atom defines a family of beta-votes that are not mid-p-values, so the paper's novelty lies in the democratic interpretation of the mid-p rather than in a new numerical quantity.
The compatibility axioms can be read as a general recipe for constructing evidence measures on ordered parameter spaces: a coherent vote map is a probability on the order $\sigma$-algebra, and the paper's conditions are continuity and boundary conditions for that probability. This recipe could be transferred to partially ordered or product parameter spaces by building transition probabilities, tho
For discrete models the difference between $m_\theta$ and $m'_\theta$ at atoms means the vote is deliberately non-conservative, using the mid-p rather than the usual upper p. A user who prefers conservative testing would choose the atom split $\beta=1$ or $\beta=0$, so the paper's choice is a substantive small-sample decision, not merely a technical device.

Original abstract

In this study, we introduce a new approach to statistical decision theory. Without using a loss function, we select good decision rules to choice between two hypotheses. We call them "experts". They are globally unbiased but also conditionally unbiased on a family of events. We do not try to define the best expert. We define a probability distribution on the space of "experts". The measure of evidence for a hypothesis is the inductive probability of experts that decide this hypothesis, we call this measure: a "vote". We compare this point of view with the p-values. For some family of hypotheses, the "votes" can define a probability on the space of parameters. We compare these results with the Bayes posterior distributions. We study in detail real-parameter families of distributions with monotone likelihood ratio and multiparameter exponential families.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,201 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.64 · human participants 0.03 · controlled comparison 0.08 · randomized assignment 0.02 · states quantitative result 0.06 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this study, we introduce a new approach to statistical decision theory. Without using a loss function, we select good decision rules to choice between two hypotheses. We call them "experts". They are globally unbiased but also conditionally unbiased on a family of events. We do not try to define the best expert. We define a probability distribution on the space of "experts". The measure of evidence for a hypothesis is the inductive probability of experts that decide this hypothesis, we call this measure: a "vote". We compare this point of view with the p-values. For some family of hypotheses, the "votes" can define a probability on the space of parameters. We compare these results with the Bayes posterior distributions. We study in detail real-parameter families of distributions with monotone likelihood ratio and multiparameter exponential families.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this study, we introduce a new approach to statistical decision theory.	contribution 1.00	0.90	0.06	asserted 0.99
s1	Without using a loss function, we select good decision rules to choice between two hypotheses. We call them "experts".	method 0.64	0.92	0.15	asserted 1.00
s2	They are globally unbiased but also conditionally unbiased on a family of events.	result 0.71	0.79	0.09	asserted 1.00
s3	We do not try to define the best expert.	limitation 0.66	0.87	0.07	asserted 0.95
s4	We define a probability distribution on the space of "experts".	method 0.82	0.94	0.06	asserted 1.00
s5	The measure of evidence for a hypothesis is the inductive probability of experts that decide this hypothesis, we call this measure: a "vote".	method 0.79	0.88	0.20	asserted 1.00
s6	We compare this point of view with the p-values.	contribution 0.47	0.75	0.04	asserted 0.68
s7	For some family of hypotheses, the "votes" can define a probability on the space of parameters.	result 0.87	0.85	0.15	hedged possibility 0.87
s8	We compare these results with the Bayes posterior distributions.	method 0.40	0.84	0.07	asserted 0.76
s9	We study in detail real-parameter families of distributions with monotone likelihood ratio and multiparameter exponential families.	method 0.61	0.90	0.05	asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.94 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (22)

[1]

Nous devons montrer que K et K ′ sont ´ egalesP0 et P1 presque sˆ urement. C’est-` a-dire que les ´ ev´ enementsB ={K >K ′} et C ={K <K ′} sont de probabilit´ es nulles pourP0 et P1. D´ emontrons queP1(B) est nulle. Il est ´ equivalent d’avoir P1(B′) = 0 avec B′ = B∩{p1 > 0}. Sur B′, la statistique K est ﬁnie, on a K ′<K < +∞ et donc p0 =K.p 1, p′ 0 =K ′....

View on Pith
[2]

d´ epend deX2. Cette remarque reste vraie pour toutes les hypoth` eses qui font intervenir des param` etres expertisables ` a partir de statistiques ind´ ependantes. 142 ANNEXE I. Soient P0 et P1, deux probabilit´ es d´ eﬁnies sur (Ω,A). Elles admettent toujours des densit´ esp0 et p1 par rapport ` a une mesure µ sur (Ω ,A) (cf. [Leh.] p. 74). En 2.2 nous...

View on Pith
[3]

vaut +∞. Lemme 2 Soit (Ω,A, (pθ.µ )θ∈Θ 0∪Θ 1 ) un probl` eme de d´ ecision ` a hypoth` eses stables. On note ∆ s l’ensemble des fonctions de test simples d´ eﬁnies ` a partird’une famille {h(θ0,θ 1)}(θ0,θ 1)∈Θ 0×Θ 1 normalis´ ee. Pour tout θ1∈ Θ 1 ; on a inf{φ (θ0,θ 1) (0, 1) }θ0∈Θ 0 =f(t0,u 0) =inf ∆ s D´ emonstration La d´ emonstration est semblable ` a...

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[4]

vaut 0. 147 ANNEXE III. Soit (Ω ,A, (Pθ =pθ.µ )θ∈Θ 0∪Θ 1=Θ ), un probl` eme de d´ ecision ` a hypoth` eses stables par rapport ` a la statistique r´ eelle T . D’apr` es la d´ eﬁnition 4.1.2, ceci implique l’existence, pour tout couple (θ0,θ 1)∈ Θ 0× Θ 1, d’une fonction croissante h(θ0,θ 1) : IR→IR+ v´ eriﬁantpθ0/p θ1 =h(θ0,θ 1)(T ) sur le domaine de d´ eﬁ...

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[5]

Il existe une fonction normalis´ ee diﬀ´ erente de h(θ0,θ 1) en t si et seulement si : t∈IR−(Di∪Ds), t est ind´ etermin´ e pour(θ0,θ 1) et h(θ0,θ 1)(I − t )<h (θ0,θ 1)(I + t ) (les limites h(θ0,θ 1)(I − t ) eth(θ0,θ 1)(I + t ) ne d´ ependent pas de la fonction normalis´ ee choisie)

View on Pith
[6]

Si t∈IR−(Di∪Ds) est ind´ etermin´ e pour(θ0,θ 1), il est totalement ind´ etermin´ e :∀ω ∈ T −1(t) ∀θ ∈ Θ pθ (ω ) = 0 , lorsque h(θ0,θ 1)(I − t ) > 0 ou h(θ0,θ 1)(I + t )< +∞. D´ emonstration I — Condition n´ ecessaire de 1). La condition t∈IR−(Di∪Ds) est n´ ecessaire puisque pour t∈ Ds (resp. t∈ Di−Ds) toutes les fonctions normalis´ ees valent +∞ (resp. 0...

View on Pith
[7]

il existe ω ′∈T −1(t) et θ′∈ Θ 0 tels que pθ ′ (ω ′)> 0

View on Pith
[8]

il existe ω ′′∈T −1(t) et θ′′∈ Θ 1 tels que pθ ′′ (ω ′′)> 0. Nous n’en avons suppos´ e qu’une seule vraie au d´ epart, maisnous allons montrer que l’on ne peut pas avoir l’une sans l’autre. Si on avait la premi` ere condition sans la seconde, on aurait pour tout θ1 de Θ 1 : pθ1 (ω ′) = 0 et pθ ′ (ω ′) > 0, donc h(θ ′,θ 1)(t) = + ∞ ; ce qui implique : ∀θ1 ...

View on Pith
[9]

tel que : e(θ0,θ 1) t p.s. ≤ ϕ (θ ′ 0,θ ′ 1) t et ϕ (θ0,θ 1) t p.s. ≤ e(θ ′ 0,θ ′ 1) t . Ceci est en particulier r´ ealis´ e lorsquee(θ0,θ 1) t p.s. = ϕ (θ0,θ 1) t . Nous allons distinguer deux cas. 154 1er cas : les densit´ espθ0 et pθ1 ne sont pas s´ epar´ ees. Les densit´ espθ0 et pθ1 sont dites non s´ epar´ ees lorsque : IR−(Dθ0 i ∪Dθ1 s )⁄=∅ ; Dθ0 i ...

View on Pith
[10]

= (θ0,θ 1) convient. ii) t∈B. D’apr` es a),e(θ0,θ 1) t (resp. ϕ (θ0,θ 1) t ) est ´ egale ` a1IA (T ) ou f(−∞, 1). Nous avons vu en a-ii) que t appartient ` a IR−(Di∪ Ds) et qu’il est ind´ etermin´ e pour (θ0,θ 1) ; mais t n’est pas totalement ind´ etermin´ e puisque t /∈ N ; d’apr` es la ﬁn de la partie III de la d´ emonstration du lemme pr´ ec´ edent, il...

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[11]

tel que : g(θ0,θ 1) t p.s. ≥ ψ (θ ′ 0,θ ′ 1) t et ψ (θ0,θ 1) t p.s. ≥ g(θ ′ 0,θ ′ 1) t . Ceci est en particulier r´ ealis´ e lorsqueg(θ0,θ 1) t p.s. = ψ (θ0,θ 1) t . Nous allons encore distinguer deux cas. 1er cas : les densit´ espθ0 et pθ1 ne sont pas s´ epar´ ees. On va d´ emontrer :g(θ0,θ 1) t p.s. = ψ (θ0,θ 1) t . La partie a) du 1 er cas de I reste v...

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[12]

tel que : g(θ0,θ 1) t p.s. ≥ ψ (θ ′ 0,θ ′ 1) t et ψ (θ0,θ 1) t p.s. ≥ g(θ ′ 0,θ ′ 1) t . Soit t∈IR−(Di∪Ds∪N ). On consid` ere les trois possibilit´ es de la partition IR=A +B +C d´ eﬁnie, ` a partir de deux densit´ es s´ epar´ ees, dans le 2 ` eme cas de la partie I. i) t∈A. g(θ0,θ 1) t (resp. ψ (θ0,θ 1) t ) est ´ egale ` a1IA (T ) ou 1− 1IC (T ). Pour ob...

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[13]

= (θ0,θ 1) convient. 160 ANNEXE IV. Soit (Ω ,A, (pθ.µ )θ∈Θ ) un mod` ele statistique ` a rapport de vraisemblance monotone par rapport ` a la statistique r´ eelleT , Θ ´ etant muni de la relation d’ordre totale : ≼ . Ceci suppose l’existence, pour θ′ ≺ θ′′, d’une fonction croissante h(θ ′′,θ ′) :IR→ IR+ v´ eriﬁantpθ ′′/p θ ′ = h(θ ′′,θ ′)(T ) sur le domai...

View on Pith
[14]

S’il existe ω t∈T −1(t) tel que : pθa(ω t)> 0 et pθb(ω t)> 0 (resp. pθb(ω t)> 0 et pθc(ω t)> 0), alors k2 =k3.k 1

View on Pith
[15]

Dans le premier cas : pθa (ω t)> 0 et pθb(ω t)> 0, on a φ (θc,θ b) (k3, 0) p.s. ≤ sup{φ (θb,θ a) (k1, 0) ,φ (θc,θ a) (k2, 0) , 1ID{θ ≻θ a} i (T )} et φ (θc,θ b) (k3, 1) p.s. ≥ inf{φ (θb,θ a) (k1, 1) ,φ (θc,θ a) (k2, 1) , 1− 1ID {θ ≺θ b} s (T )}

View on Pith
[16]

Dans le second cas : pθb (ω t)> 0 et pθc (ω t)> 0, on a φ (θb,θ a) (k1, 0) p.s. ≤ sup{φ (θc,θ a) (k2, 0) ,φ (θc,θ b) (k3, 0) , 1ID {θ ≻θ b} i (T )} et φ (θb,θ a) (k1, 1) p.s. ≥ inf{φ (θc,θ a) (k2, 1) ,φ (θc,θ b) (k3, 1) , 1− 1ID{θ ≺θ c} s (T )}. Ces ´ ecritures peuvent contenir des fonctions de test de la f orme φ (θ ′′,θ ′) (0, 0) (resp. φ (θ ′′,θ ′) (∞,...

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Cours de statistique math´ ematique. Economica, Paris, 1982. [Mor.] G. Morel

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Proc´ edures statistiques pour espace de d´ ecisions totalement ordonn´ e et famille de lois ` a vraisemblance monotone (th` ese). Universit´ e de Rouen, France, 1987

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D´ ecisions li´ ees aux intervalles d’une partition : le probl` eme du choix dans la pratique de la recherche. Pub. Inst. Stat. Univ., XXXII, fasc. 1-2, p. 93- 111, 1987. 169 [Ney.] J. Neyman, E.S. Pearson. On the testing of statistical hypotheses in relation to prob ability a priori. Proc. Cambridge Phil. Soc., 29, p. 492-510, 1933. [Nev.] J. Neveu. Calc...

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On a result of L.N. Bol’shev from the theory of the statist ical testing of hypotheses. Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 153, p. 129-137, 1986

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Estimation of the eﬃciency of Bol’shev’s decision rule i n the problem of distinguishing of two hypotheses. J. Soviet Math., 52, p.2955-2964, 1990. [Pfa.] J. Pfanzagl. A technical lemma for monotone likelihood ratio families. Ann. Math. Statist., 38, p. 611-613, 1967. [R´ en.] A. R ´enyi. Calcul des probabilit´ es.Dunod, Paris, 1966. [Reu.] M. Reuchlin. E...

View on Pith 1990

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Nous devons montrer que K et K ′ sont ´ egalesP0 et P1 presque sˆ urement. C’est-` a-dire que les ´ ev´ enementsB ={K >K ′} et C ={K <K ′} sont de probabilit´ es nulles pourP0 et P1. D´ emontrons queP1(B) est nulle. Il est ´ equivalent d’avoir P1(B′) = 0 avec B′ = B∩{p1 > 0}. Sur B′, la statistique K est ﬁnie, on a K ′<K < +∞ et donc p0 =K.p 1, p′ 0 =K ′....

View on Pith

[2] [2]

d´ epend deX2. Cette remarque reste vraie pour toutes les hypoth` eses qui font intervenir des param` etres expertisables ` a partir de statistiques ind´ ependantes. 142 ANNEXE I. Soient P0 et P1, deux probabilit´ es d´ eﬁnies sur (Ω,A). Elles admettent toujours des densit´ esp0 et p1 par rapport ` a une mesure µ sur (Ω ,A) (cf. [Leh.] p. 74). En 2.2 nous...

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[3] [3]

vaut +∞. Lemme 2 Soit (Ω,A, (pθ.µ )θ∈Θ 0∪Θ 1 ) un probl` eme de d´ ecision ` a hypoth` eses stables. On note ∆ s l’ensemble des fonctions de test simples d´ eﬁnies ` a partird’une famille {h(θ0,θ 1)}(θ0,θ 1)∈Θ 0×Θ 1 normalis´ ee. Pour tout θ1∈ Θ 1 ; on a inf{φ (θ0,θ 1) (0, 1) }θ0∈Θ 0 =f(t0,u 0) =inf ∆ s D´ emonstration La d´ emonstration est semblable ` a...

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[4] [4]

vaut 0. 147 ANNEXE III. Soit (Ω ,A, (Pθ =pθ.µ )θ∈Θ 0∪Θ 1=Θ ), un probl` eme de d´ ecision ` a hypoth` eses stables par rapport ` a la statistique r´ eelle T . D’apr` es la d´ eﬁnition 4.1.2, ceci implique l’existence, pour tout couple (θ0,θ 1)∈ Θ 0× Θ 1, d’une fonction croissante h(θ0,θ 1) : IR→IR+ v´ eriﬁantpθ0/p θ1 =h(θ0,θ 1)(T ) sur le domaine de d´ eﬁ...

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[5] [5]

Il existe une fonction normalis´ ee diﬀ´ erente de h(θ0,θ 1) en t si et seulement si : t∈IR−(Di∪Ds), t est ind´ etermin´ e pour(θ0,θ 1) et h(θ0,θ 1)(I − t )<h (θ0,θ 1)(I + t ) (les limites h(θ0,θ 1)(I − t ) eth(θ0,θ 1)(I + t ) ne d´ ependent pas de la fonction normalis´ ee choisie)

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[6] [6]

Si t∈IR−(Di∪Ds) est ind´ etermin´ e pour(θ0,θ 1), il est totalement ind´ etermin´ e :∀ω ∈ T −1(t) ∀θ ∈ Θ pθ (ω ) = 0 , lorsque h(θ0,θ 1)(I − t ) > 0 ou h(θ0,θ 1)(I + t )< +∞. D´ emonstration I — Condition n´ ecessaire de 1). La condition t∈IR−(Di∪Ds) est n´ ecessaire puisque pour t∈ Ds (resp. t∈ Di−Ds) toutes les fonctions normalis´ ees valent +∞ (resp. 0...

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[7] [7]

il existe ω ′∈T −1(t) et θ′∈ Θ 0 tels que pθ ′ (ω ′)> 0

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[8] [8]

il existe ω ′′∈T −1(t) et θ′′∈ Θ 1 tels que pθ ′′ (ω ′′)> 0. Nous n’en avons suppos´ e qu’une seule vraie au d´ epart, maisnous allons montrer que l’on ne peut pas avoir l’une sans l’autre. Si on avait la premi` ere condition sans la seconde, on aurait pour tout θ1 de Θ 1 : pθ1 (ω ′) = 0 et pθ ′ (ω ′) > 0, donc h(θ ′,θ 1)(t) = + ∞ ; ce qui implique : ∀θ1 ...

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[9] [9]

tel que : e(θ0,θ 1) t p.s. ≤ ϕ (θ ′ 0,θ ′ 1) t et ϕ (θ0,θ 1) t p.s. ≤ e(θ ′ 0,θ ′ 1) t . Ceci est en particulier r´ ealis´ e lorsquee(θ0,θ 1) t p.s. = ϕ (θ0,θ 1) t . Nous allons distinguer deux cas. 154 1er cas : les densit´ espθ0 et pθ1 ne sont pas s´ epar´ ees. Les densit´ espθ0 et pθ1 sont dites non s´ epar´ ees lorsque : IR−(Dθ0 i ∪Dθ1 s )⁄=∅ ; Dθ0 i ...

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[10] [10]

= (θ0,θ 1) convient. ii) t∈B. D’apr` es a),e(θ0,θ 1) t (resp. ϕ (θ0,θ 1) t ) est ´ egale ` a1IA (T ) ou f(−∞, 1). Nous avons vu en a-ii) que t appartient ` a IR−(Di∪ Ds) et qu’il est ind´ etermin´ e pour (θ0,θ 1) ; mais t n’est pas totalement ind´ etermin´ e puisque t /∈ N ; d’apr` es la ﬁn de la partie III de la d´ emonstration du lemme pr´ ec´ edent, il...

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[11] [11]

tel que : g(θ0,θ 1) t p.s. ≥ ψ (θ ′ 0,θ ′ 1) t et ψ (θ0,θ 1) t p.s. ≥ g(θ ′ 0,θ ′ 1) t . Ceci est en particulier r´ ealis´ e lorsqueg(θ0,θ 1) t p.s. = ψ (θ0,θ 1) t . Nous allons encore distinguer deux cas. 1er cas : les densit´ espθ0 et pθ1 ne sont pas s´ epar´ ees. On va d´ emontrer :g(θ0,θ 1) t p.s. = ψ (θ0,θ 1) t . La partie a) du 1 er cas de I reste v...

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[12] [12]

tel que : g(θ0,θ 1) t p.s. ≥ ψ (θ ′ 0,θ ′ 1) t et ψ (θ0,θ 1) t p.s. ≥ g(θ ′ 0,θ ′ 1) t . Soit t∈IR−(Di∪Ds∪N ). On consid` ere les trois possibilit´ es de la partition IR=A +B +C d´ eﬁnie, ` a partir de deux densit´ es s´ epar´ ees, dans le 2 ` eme cas de la partie I. i) t∈A. g(θ0,θ 1) t (resp. ψ (θ0,θ 1) t ) est ´ egale ` a1IA (T ) ou 1− 1IC (T ). Pour ob...

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[13] [13]

= (θ0,θ 1) convient. 160 ANNEXE IV. Soit (Ω ,A, (pθ.µ )θ∈Θ ) un mod` ele statistique ` a rapport de vraisemblance monotone par rapport ` a la statistique r´ eelleT , Θ ´ etant muni de la relation d’ordre totale : ≼ . Ceci suppose l’existence, pour θ′ ≺ θ′′, d’une fonction croissante h(θ ′′,θ ′) :IR→ IR+ v´ eriﬁantpθ ′′/p θ ′ = h(θ ′′,θ ′)(T ) sur le domai...

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[14] [14]

S’il existe ω t∈T −1(t) tel que : pθa(ω t)> 0 et pθb(ω t)> 0 (resp. pθb(ω t)> 0 et pθc(ω t)> 0), alors k2 =k3.k 1

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[15] [15]

Dans le premier cas : pθa (ω t)> 0 et pθb(ω t)> 0, on a φ (θc,θ b) (k3, 0) p.s. ≤ sup{φ (θb,θ a) (k1, 0) ,φ (θc,θ a) (k2, 0) , 1ID{θ ≻θ a} i (T )} et φ (θc,θ b) (k3, 1) p.s. ≥ inf{φ (θb,θ a) (k1, 1) ,φ (θc,θ a) (k2, 1) , 1− 1ID {θ ≺θ b} s (T )}

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[16] [16]

Dans le second cas : pθb (ω t)> 0 et pθc (ω t)> 0, on a φ (θb,θ a) (k1, 0) p.s. ≤ sup{φ (θc,θ a) (k2, 0) ,φ (θc,θ b) (k3, 0) , 1ID {θ ≻θ b} i (T )} et φ (θb,θ a) (k1, 1) p.s. ≥ inf{φ (θc,θ a) (k2, 1) ,φ (θc,θ b) (k3, 1) , 1− 1ID{θ ≺θ c} s (T )}. Ces ´ ecritures peuvent contenir des fonctions de test de la f orme φ (θ ′′,θ ′) (0, 0) (resp. φ (θ ′′,θ ′) (∞,...

View on Pith 1971

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Cours de probabilit´ es.Economica, Paris, 1980

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Cours de statistique math´ ematique. Economica, Paris, 1982. [Mor.] G. Morel

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Proc´ edures statistiques pour espace de d´ ecisions totalement ordonn´ e et famille de lois ` a vraisemblance monotone (th` ese). Universit´ e de Rouen, France, 1987

View on Pith 1987

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D´ ecisions li´ ees aux intervalles d’une partition : le probl` eme du choix dans la pratique de la recherche. Pub. Inst. Stat. Univ., XXXII, fasc. 1-2, p. 93- 111, 1987. 169 [Ney.] J. Neyman, E.S. Pearson. On the testing of statistical hypotheses in relation to prob ability a priori. Proc. Cambridge Phil. Soc., 29, p. 492-510, 1933. [Nev.] J. Neveu. Calc...

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On a result of L.N. Bol’shev from the theory of the statist ical testing of hypotheses. Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 153, p. 129-137, 1986

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Estimation of the eﬃciency of Bol’shev’s decision rule i n the problem of distinguishing of two hypotheses. J. Soviet Math., 52, p.2955-2964, 1990. [Pfa.] J. Pfanzagl. A technical lemma for monotone likelihood ratio families. Ann. Math. Statist., 38, p. 611-613, 1967. [R´ en.] A. R ´enyi. Calcul des probabilit´ es.Dunod, Paris, 1966. [Reu.] M. Reuchlin. E...

View on Pith 1990