Asymptotically uniformly distributed priors and Bayes estimators in mixed models

H. Herrmann, Martin-Luther-Universität Halle-Wittenberg F. Liese, Universität Rostock

Abstract

For a general class of mixed models which includes the (Γ, γ)-model introduced by Shiryaev and Spokoiny (1993) we prove the minimaxity of a Pitman type estimator. This minimaxity is nearly related to the asymptotic minimaxity of a sequence of Bayes estimators which is a consequence of the asymptotic shift invariance of the priors. Such priors are called asymptotically uniformly distributed.

1 Introduction

Shiryaev and Spokoiny (1993) introduced a class of distributions, called (Γ, γ)-model which includes the mixed normal distribution as special case. The (Γ, γ)-model is a family of distributions P_{α}, α ∈ ℝ_{d}, on (Ω, 𝔉) which are absolutely continuous with respect to some σ-finite measure ν and the density is given by q(γ + Γα) where q is a Lebesgue density in ℝ_{d}. γ is a random d-dimensional vector and Γ a random d × d -matrix. The (Γ, γ)-model has been investigated also in Spokoiny (1994). For the investigation of Bayes estimates and the proof of the minimaxity of some estimator a key role is played by the question whether for a sequence {π_{n}} of priors for α the sample is approximately independent of the statistic T = γ + Γα. Sufficient conditions on {π_{n}} to guarantee this approximate independence have been formulated in Shiryaev, Spokoiny (1993). This condition turns out to be stronger than the asymptotic uniform distribution of {π_{n}} introduced in Kerstan, Matthes (1968). In this paper it will be shown that this weaker property is necessary and sufficient for the above formulated approximate independence in the restricted class of mixed (Γ, γ)-models. Furthermore we discuss in details the relation of different concepts for the asymptotic uniform distribution and compare them with the conditions formulated in Shiryaev, Spokoiny (1993). The results on asymptotically uniformly distributed priors are applied to show the minimaxity of the estimator −Γ^{−1}γ for α in the mixed (Γ, γ)-model which was already established in Shiryaev, Spokoiny (1993). The (Γ, γ)-model is used as a limit experiment in Shiryaev, Spokoiny (1993). Using the general way in the asymptotic decision theory one has to derive optimal inference in the limit experiment which is an inference for a sample of size one. But the examples in Shiryaev, Spokoiny (1993) and Spokoiny (1994) show that the (Γ, γ)-models form a wide and flexible class of distributions. If we take independent observations from a (Γ, γ)-model then the distribution of the sample belongs to a larger class of models called (Γ, γ, m, d)- models which we introduce in this paper. For (Γ, γ, m, d)-models we construct a Pitman type estimator for the unknown parameter α and prove the minimax property. For m = d the models (Γ, γ, d, d) and (Γ, γ) are identical and the Pitman estimator is the estimator −Γ^{−1}γ for which the minimax property was established in Shiryaev, Spokoiny (1993).

AMS subject classification: 62 F 12, 60 F 99.

Key words: (Γ, γ)-model, Pitman estimator, asymptotically uniformly distributed priors.

2 Asymptotically uniformly distributed priors

In this section we give for completeness of presentation a simplified proof of a characterization of asymptotically uniformly distributed sequences which was already established in Herrmann, Kerstan, Liese (1979). For this aim we summarize some well-known inequalities for the variational distance. Recall that for two distributions μ, ν on (ℝ_{d}, 𝔅_{d})

∥μ − ν∥ = 2 sup_{A∈𝔅_{d}} |μ(A) − ν(A)| .

Hence for a further distribution ϱ

∥ϱ ∗ μ − ϱ ∗ ν∥ ≤ ∫ 2 sup_{A∈𝔅_{d}} |μ(A − t) − ν(A − t)| ϱ(dt) = ∥μ − ν∥ . (1)

Analogously

∥μ − σ ∗ ν∥ ≤ ∫ ∥μ − δ_{t} ∗ ν∥ σ(dt) (2)

where δ_{t} is the Dirac measure concentrated at point t. Let X and Y be independent random vectors with distributions μ and ν, respectively. Then K(s, B) = ν(B − s) is the conditional distribution of X + Y given X. Furthermore

L(X, X + Y)(C) = ∫∫ I_{C}(s, s + t) μ(ds) ν(dt) =: (μ ⊕ ν)(C) .

Given two stochastic kernels K_{1}, K_{2} and a distribution μ we denote by μ×K_{i} the distribution defined by (μ × K_{i})(ds, dt) = μ(ds) K_{i}(s, dt) . The function s → ∥K_{1}(s, .) − K_{2}(s, .)∥ is measurable and it holds (see Herrmann, Kerstan, Liese (1979))

∥μ × K_{1} − μ × K_{2}∥ = ∫ ∥K_{1}(s, .) − K_{2}(s, .)∥ μ(ds) . (3)

Hence

∥L(X, X + Y) − L(X) × L(X + Y)∥ = ∥μ ⊕ ν − μ × (μ ∗ ν)∥
= ∫ ∥δ_{s} ∗ ν − μ ∗ ν∥ μ(ds)
≤ ∫∫ ∥δ_{s} ∗ ν − δ_{t} ∗ ν∥ μ(dt) μ(ds)
= ∫∫ ∥ν − δ_{t−s} ∗ ν∥ μ(dt) μ(ds)
= ∫ ∥ν − δ_{τ} ∗ ν∥ (μ ∗ μ̄)(dτ)

where we used the notation μ̄(A) = μ(−A) and inequality (2). By inequality (2) again we obtain

\begin{aligned}∥ν − (μ̄ ∗ μ) ∗ ν∥ & ≤ ∫ ∥ν − δ_{−s} ∗ μ ∗ ν∥ μ(ds) \\ & = ∫ ∥δ_{s} ∗ ν − μ ∗ ν∥ μ(ds) \\ & = ∥μ ⊕ ν − μ × (μ ∗ ν)∥ \\ & ≤ ∫ ∥ν − δ_{τ} ∗ ν∥ (μ ∗ μ̄)(dτ) . \end{aligned} (4)

For any measure μ on the Borel sets 𝔅_{d} of ℝ_{d} and any non-negative measurable function f we introduce the distribution μ_{f} by μ_{f}(A) = (∫_{A} f dμ)(∫ f dμ)^{−1} if 0 < ∫ f dμ < ∞ and μ_{f} = δ_{0} else. For f, g ∈ L_{1}^{+} and a distribution ν we set

(f ∗ g)(t) = ∫ f(t − s) g(s) ds ,
(f ∗ ν)(t) = ∫ f(t − s) ν(ds) .

Let 𝔹 be the set of all measurable functions h : ℝ_{d} → ℝ_{1} with |h(t)| ≤ 1 . Assume g ∈ L_{1}^{+} with ∫ g(t) dt = 1 and set ḡ(t) = g(−t) . Then by the definition of λ_{g}, g is the Lebesguedensity of λ_{g} and ḡ ∗ ν the Lebesgue-density of λḡ ∗ ν. Hence (λḡ ∗ ν)(0 < ḡ ∗ ν < ∞) = 1 and

∫ ∥(δ_{−s} ∗ ν)_{g} − λ_{g}∥ (λḡ ∗ ν) (ds)

= ∫ sup_{h∈𝔹}|\frac{1}{(ḡ ∗ ν)(s)} ∫ h(t − s) g(t − s) ν(dt) − ∫ h(t) g(t) dt| (ḡ ∗ ν)(s) ds

= ∫ sup_{h∈𝔹}| ∫ h(t − s) g(t − s) ν(dt) − ∫ h(t − s) g(t − s) dt (ḡ ∗ ν)(s)| ds

= ∫ sup_{φ∈𝔹}| ∫ φ(t) g(t − s) ν(dt) − ∫ φ(t) g(t − s) (ḡ ∗ ν)(s) dt| ds

= ∫ g(v) sup_{φ∈𝔹}| ∫ φ(t) ν(dt) − ∫ φ(t) (ḡ ∗ ν)(t − v) dt| dv

= ∫ g(v) sup_{φ∈𝔹}| ∫ φ(t) ν(dt) − ∫ φ(t) (δ_{v} ∗ λḡ ∗ ν)(dt)| dv

= ∫ ∥ν − δ_{v} ∗ λḡ ∗ ν∥ λ_{g}(dv) .

Hence

∫ ∥δ_{−s} ∗ ν)_{g} − λ_{g}∥ (λḡ ∗ ν) (ds) = ∫ ∥ν − δ_{v} ∗ λḡ ∗ ν∥ λ_{g}(dv) . (5)

Furthermore with g_{(u)}(t) = g(t + u)

∫ ∥(δ_{−s} ∗ ν)_{g} − λ_{g}∥ (λḡ ∗ ν) (ds) = ∫∫ ∥(δ_{−(u+v)} ∗ ν)_{g} − λ_{g}∥ λḡ(du) ν(dv)

= ∫∫ sup_{h∈𝔹}|\frac{1}{(ḡ ∗ ν)(u + v)} ∫ h(t − (u + v)) g(t − (u + v)) ν(dt)

− ∫ h(t) g(t) dt | ν(dv) λḡ(du)

= ∫∫ sup_{h∈𝔹}| ∫ \frac{1}{((ḡ)_{(−u)} ∗ ν)(v)}h_{(−u)}(t − v) g_{(−u)}(t − v) ν(dt)

− ∫ h_{(−u)}(t − v) g_{(−u)}(t − v) dt| ν(dv) λḡ(du)

= ∫∫ ∥(δ_{−v} ∗ ν)_{g_{(−u)}} − λ_{g_{(−u)}}∥ ν(dv) λḡ(du) .

Hence

∫ ∥(δ_{−s} ∗ ν)_{g} − λ_{g}∥ (λḡ ∗ ν) (ds) = ∫∫ ∥(δ_{−v} ∗ ν)_{g_{(u)}} − λ_{g_{(u)}}∥ ν(dv) λ_{g}(du) . (6)

We recall that by Kerstan, Matthes (1968) any sequence {ν_{n}} of distributions is called asymptotically uniformly distributed if

lim_{n→∞} ∥π_{n} − δ_{x} ∗ π_{n}∥ = 0 (7)

for every x ∈ ℝ_{d}. The next theorem gives equivalent conditions for the asymptotic uniform distribution of a sequence {ν_{n}}.

Theorem 1. Let {ν_{n}} be a sequence of on (ℝ_{d}, 𝔅_{d}). Then the following statements are equivalent.

1) {ν_{n}} is asymptotically uniformly distributed.

2a) ∥ν_{n} − ϱ ∗ ν_{n}∥ ⟶_{n→∞} 0 for every distribution ϱ.

2b) ∥ν_{n} − ϱ_{0} ∗ ν_{n}∥ ⟶_{n→∞} 0 for some ϱ_{0} ≪ λ.

3a) ∥μ ⊕ ν_{n} − μ × (μ ∗ ν_{n})∥ ⟶_{n→∞} 0 for every distribution μ.

3b) ∥μ_{0} ⊕ ν_{n} − μ_{0} × (μ_{0} ∗ ν_{n})∥ ⟶_{n→∞} 0 for some μ_{0} ≪ λ.

4) ν_{n}({s : ∥(δ_{−s} ∗ ν_{n})_{g} − λ_{g}∥ > ε}) ⟶_{n→∞} 0 for every ε > 0 and every g ∈ L_{1}^{+}.

Proof. The statements 2a) → 2b), 3a) → 3b) are trivial. The statement 1) → 2a) follows from inequality (2). Suppose now that 2b) is fulfilled. Then by inequality (1)

∥ν_{n} − δ_{t} ∗ ν_{n}∥ ≤ ∥ν_{n} − ϱ_{0} ∗ ν_{n}∥ + ⋯ + ∥ϱ_{0}^{(m−1)∗} ∗ ν_{n} − ϱ_{0}^{m∗} ∗ ν_{n}∥
+ ∥ϱ_{0}^{m∗} ∗ ν_{n} − δ_{t} ∗ ϱ_{0}^{m∗} ∗ ν_{n}∥ + ∥δ_{t} ∗ ϱ_{0}^{(m−1)∗} ∗ ν_{n} − δ_{t} ∗ ϱ_{0}^{m∗} ∗ ν_{n}∥ + …
+ ∥δ_{t} ∗ ϱ_{0} ∗ ν_{n} − δ_{t} ∗ ν_{n}∥
≤ 2m ∥ν_{n} − ϱ_{0} ∗ ν_{n}∥ + ∥ϱ_{0}^{m∗} − δ_{t} ∗ ϱ_{0}^{m∗}∥ .

As ϱ_{0} ≪ λ the sequence {ϱ_{0}^{m∗}} is asymptotically uniformly distributed by Kerstan, Matthes (1968). Hence

lim sup_{n→∞} ∥ν_{n} − δ_{t} ∗ ν_{n}∥ ≤ lim sup_{m→∞}(lim sup_{n→∞}(2m∥ν_{n} − ϱ_{0} ∗ ν_{n}∥ + ∥ϱ_{0}^{m∗} − δ_{t} ∗ ϱ_{0}^{m∗}∥)) = 0

and 1) is established. The statement 3a) follows from 1) by application of inequality (4). Suppose 3b) is fulfilled then by inequality (4) again we get 2b) with ϱ_{0} = μ_{0} ∗ μ̄_{0} by the first part of the proof and consequently statement 1) is established. Suppose now 1) is fulfilled and choose g ∈ L_{1}^{+}. Assume w.l.g. that ∫ g(t) dt = 1. To prove 4) it is enough to show that for every ε > 0

λḡ ∗ ν_{n}({s : ∥(δ_{−s} ∗ ν_{n})_{g} − λ_{g}∥ > ε}) ⟶_{n→∞} 0 (8)

as ∥ν_{n} − λḡ ∗ ν_{n}∥ ⟶_{n→∞} 0 by 2a). The variational distance is not larger than 2 and consequently by (5) the relation (8) is equivalent to

∫ ∥(δ_{−s} ∗ ν_{n})_{g} − λ_{g}∥ (λḡ ∗ ν_{n}) (ds) = ∫ ∥ν_{n} − δ_{v} ∗ λḡ ∗ ν_{n}∥ λ_{g}(dv) ⟶_{n→∞} 0 .

The application of inequality (2) to the right hand term yields

∫ ∥(δ_{−s} ∗ ν_{n})_{g} − λ_{g}∥ (λḡ ∗ ν_{n}) (ds) ≤ ∫∫ ∥ν_{n} − δ_{u} ∗ δ_{v} ∗ ν_{u}∥ λḡ(du) λ_{g}(dv)

= ∫ ∥ν_{n} − δ_{s} ∗ ν_{n}∥ (λ_{g} ∗ λḡ) (ds) ⟶_{n→∞} 0

by assumption 1) and Lebesgue’s dominated convergence theorem. Hence statement 4) is established. Assume now statement 4) is fulfilled. Fix g ∈ L_{1}^{+} then g_{(−u)} ∈ L_{1}^{+} for every u ∈ ℝ_{d}. Hence by statement 4)

∫ ∥(δ_{−v} ∗ ν_{n})_{g_{(u)}} − λ_{g_{(u)}}∥ ν_{n}(dv) ⟶_{n→∞} 0

for every u. Using (5) and (6) we get

∫ ∥ν_{n} − δ_{v} ∗ λḡ ∗ ν_{n}∥ λ_{g}(dv) ⟶_{n→∞} 0 .

Inequality (2) implies

∥ν_{n} − (λ_{g} ∗ λḡ) ∗ ν_{n}∥ ⟶_{n→∞} 0 .

Consequently 2b) is established for ϱ_{0} = λ_{g}∗λḡ und the proof of Theorem 1 is complete.

It is an interesting and remarkable fact that for every asymptotically uniformly distributed sequence {ν_{n}} the convergence in (7) is uniformly w.r.t. x belonging to a compact set K and w.r.t. ρ, μ belonging to tight families of distributions.

Proposition 1. Let K ⊆ ℝ_{d} be a compact set and 𝔓_{1}, 𝔓_{2} tight families of distributions on (ℝ_{d}, 𝔅_{d}). If {ν_{n}} is asymptotically uniformly distributed then the following equivalent conditions are fulfilled.

sup_{t∈K} ∥ν_{n} − δ_{t} ∗ ν_{n}∥ ⟶_{n→∞} 0, (9)

sup_{ρ∈𝔓_{1}} ∥ν_{n} − ρ ∗ ν_{n}∥ ⟶_{n→∞} 0, (10)

sup_{μ∈𝔓_{2}} ∥μ ⊕ ν_{n} − μ × μ ∗ ν_{n}∥ ⟶_{n→∞} 0. (11)

Proof. Let ρ_{0} be any distribution which is absolutely continuous w.r.t. the Lebesgue measure λ and denote by p_{0} the corresponding density. Then by the L_{1}(λ)-continuity of p_{0}

∥ ρ_{0} − δ_{h_{n}} ∗ ρ_{0} ∥ = ∫ | p_{n}(t) − p_{n}(t − h_{n})| dt ⟶_{n→∞} 0

if h_{n} → 0. Consequently for every ε > 0 there exists δ(ε) such that

sup_{|h|≤δ(ε)} ∥ ρ_{0} − δ_{h} ∗ ρ_{0} ∥ ≤ ε.

Let t_{1}, …, t_{N} be a δ(ε)-net of K. Then

sup_{t∈K} ∥ ν_{n} − δ_{t} ∗ ν_{n} ∥ ≤ ∑_{i=1}^{N} ∥ ν_{n} − δ_{t_{i}} ∗ ν_{n} ∥ + ∑_{i=1}^{N} sup_{∥t−t_{i}∥≤ε} ∥ δ_{t} ∗ ν_{n} − δ_{t_{i}} ∗ ν_{n} ∥

and

lim sup_{n→∞} sup_{t∈K} ∥ ν_{n} − δ_{t} ∗ ν_{n} ∥ ≤ N lim sup_{n→∞} sup_{∥h∥≤δ(ε)} ∥ ν_{n} − δ_{h} ∗ ν_{n} ∥

≤ 2N lim sup_{n→∞} ∥ ν_{n} − ρ_{0} ∗ ν_{n} ∥ + sup_{∥h∥≤δ(ε)} ∥ ρ_{0} − δ_{h} ∗ ρ_{0} ∥ ≤ ε.

Hence (9) is established. Statement (10) follows from (9) and inequality (7). To see that (10) implies (9) we have only to set 𝔓_{1} = {δ_{t}, t ∈ K}. The equivalence of (10) and (11) follows from (4).

Now we investigate conditions under which the ν_{n}-stochastic convergence in statement 4) of Theorem 1 holds uniformly w.r.t. g providing g belongs to some subset G ⊆ L_{1}^{+}. If a sequence {π_{n}} is asymptotically uniformly distributed then by Kerstan, Matthes (1968) the weight of the part of π_{n} being singular w.r.t. λ tends to zero as n → ∞. Hence we assume that π_{n} ≪ λ and denote by p_{n} the density \frac{dπ_{n}}{dλ}. We will need the following technical assumption. Set

sup_{g∈G} ∫ |p_{n}(t + α) − p_{n}(α)| g(t) dt =: ψ_{n}(α) and sup_{g∈G} ∥(π_{n} ∗ δ_{−α})_{g} − λ_{g}∥ =: φ_{n}(α) .

To avoid difficulties with the measurability of ψ_{n} and φ_{n}, respectively, we suppose that there is some at most countable subset G_{0} of G such that

ψ_{n}(α) = sup_{g∈G_{0}} ∫ |p_{n}(t + α) − p_{n}(α)| g(t) dt, φ(α) = sup_{g∈G_{0}} ∥(π_{n} ∗ δ_{−α})_{g} − λ_{g}∥. (12)

Theorem 2. Assume π_{n} ≪ λ with density p_{n}, n = 1, 2, …, and G ⊆ L_{1}^{+} is a family of probability densities such that the condition (12) is fulfilled. If

∫ ( sup_{g∈G} ∫ |p_{n}(t + α) − p_{n}(α)| g(t) dt)dα ⟶_{n→∞} 0 (13)

then

(14) π_{n}({α : sup_{g∈G} ∥(π_{n} ∗ δ_{−α})_{g} − λ_{g}∥ > ε}) ⟶_{n→∞} 0 for every ε > 0.

Proof. Denote by γ the distribution with the Lebesgue density g and set

\overline{g}(x) = g(−x) , x ∈ ℝ_{d}; \overline{γ}(A) = γ(−A) , A ∈ 𝔅_{d}; φ_{n}(α, g) = ∥(π_{n} ∗ δ_{−α})_{g} − λ_{g}∥.

Because of 0 ≤ φ_{n}(α, g) ≤ 2 and consequently 0 ≤ φ_{n}(α) ≤ 2 the condition (14) is equivalent to

D_{1,n} := ∫ φ_{n}(α) π_{n}(dα) ⟶_{n→∞} 0.

\overline{g} ∗ p_{n} is the Lebesgue density of \overline{γ} ∗ π_{n} and consequently \overline{γ} ∗ π_{n}-a.s. positive. Hence

φ_{n}(α, g) = ∫ |\frac{g(t) p_{n}(t + α)}{(\overline{g} ∗ p_{n})(α)} − g(t)| dt , \overline{γ} ∗ π_{n}-a.s. and

φ_{n}(α, g) (\overline{g} ∗ p_{n})(α) = ∫ |p_{n}(t + α) − (\overline{g} ∗ p_{n})(α)| g(t) dt , λ-a.e.

Therefore

\begin{aligned}D_{2,n} : & = ∫ sup_{g∈G_{0}} (φ_{n}(α, g) (\overline{g} ∗ p_{n})(α)) dα \\ & = ∫ ( sup_{g∈G_{0}} ∫ |p_{n}(t + α) − (\overline{g} ∗ p_{n})(α)| g(t) dt) dα \end{aligned}

and consequently

|D_{1,n} − D_{2,n}|

= | ∫ sup_{g∈G_{0}} φ_{n}(α, g) p_{n}(α) dα − ∫ sup_{g∈G_{0}}(φ_{n}(α, g) (\overline{g} ∗ p_{n})(α)) dα |

≤ ∫ sup_{g∈G_{0}}(φ_{n}(α, g)|p_{n}(α) − (\overline{g} ∗ p_{n})(α)|) dα

≤ 2 ∫ sup_{g∈G_{0}} ∫ |p_{n}(α) − p_{n}(α + t)| g(t) dt dα ,

D_{2,n} = ∫ ( sup_{g∈G_{0}} ∫∫ |p_{n}(t + α) − p_{n}(s + α)| g(s) g(t) ds dt) dα

≤ ∫ ( sup_{g∈G_{0}} ∫∫ {|p_{n}(t + α) − p_{n}(α)| + |p_{n}(α) − p_{n}(s + α)|} g(s) g(t) ds dt) dα

≤ 2 ∫ ( sup_{g∈G_{0}} ∫ |p_{n}(t + α) − p_{n}(α)| g(t) dt) dα .

Now it follows from (13) that

|D_{1,n} − D_{2,n}| ⟶_{n→∞} 0, D_{2,n} ⟶_{n→∞} 0,

which completes the proof.

It is clear that the separability condition (12) is fulfilled if the expressions under the suprema depend continuously on g.

Proposition 2. If for n = 1, 2, …, the density p_{n} is bounded or the set G of densities is uniformly bounded then the mapping g ↦ ∫ p_{n}(t + α) g(t) dt is continuous in the norm L_{1}(λ) and (12) is fulfilled for any countable in G dense subset of G.

Proof. For a fixed α ∈ ℝ_{d} set

h_{n}(g) = ∫ p_{n}(t + α) g(t) dt, g ∈ G.

Let p_{n} be bounded, say by the constant c_{1}. Then for g_{1}, g_{2} ∈ G

| h_{n}(g_{1}) − h_{n}(g_{2})| ≤ ∫ p_{n}(t + α) | g_{1}(t) − g_{2}(t)| dt ≤ c_{1} ∫ | g_{1}(t) − g_{2}(t)| dt.

Suppose now g(t) ≤ c_{2}, ∀t ∈ ℝ_{d}, ∀g ∈ G, for some c_{2}. For every ε > 0 there exists a constant C(ε) > 0 so that

∫_{{t: p_{n}(t+α)>C(ε)}} p_{n}(t + α) dt ≤ ε.

Therefore

| h_{n}(g_{1}) − h_{n}(g_{2})| ≤
C(ε) ∫_{{t: p_{n}(t+α)≤C(ε)}} | g_{1}(t) − g_{2}(t)| dt + 2c_{2} ∫_{{t: p_{n}(t+α)>C(ε)}} p_{n}(t + α) dt ≤
C(ε) ∫ | g_{1}(t) − g_{2}(t)| dt + 2c_{2} ε.

Hence for g_{m} ∈ G, m = 0, 1, 2, …, with

∫ | g_{0}(t) − g_{m}(t)| dt ⟶_{m→∞} 0 we get in both cases lim_{m→∞} h_{n}(g_{m}) = h_{n}(g_{0}),

that is h_{n} : G → ℝ_{1} is continuous w.r.t. the L_{1}-distance. Consequently, the suprema in ψ_{n}(α) and φ_{n}(α) may be substituted by suprema over a countable in G dense subset of G, both suprema are measurable functions of α.

Remark 1. It follows from Theorem 1 that condition (14) implies that {π_{n}} is asymptotically uniformly distributed. It is clear that the converse statement will hold only if the set G is not too large. One way is to require that the family Γ of distributions γ with densities g ∈ G is tight. But the following example will show that such sets are still too large in order to guarantee that (14) follows from (7).

Example 1. We restrict ourselves to the case d = 1. For general d the construction is similar. Denote by π_{n} the uniform distribution on

[−n, n] \ ⋃_{k=−n}^{n}[k − \frac{1}{n^{2}} , k + \frac{1}{n^{2}} ].

π_{n} has the Lebesgue density

\frac{1}{2n − \frac{4}{n}} I_{[−n,n]\⋃_{k=−n}^{n}[k− \frac{1}{n^{2}} ,k+ \frac{1}{n^{2}} ]}.

Let σ_{n} be the measure with the Lebesgue density

\frac{1}{2n − \frac{4}{n}} I_{[−n+ \frac{1}{n^{2}} ,n− \frac{1}{n^{2}} ]}.

Then for every α ∈ ℝ_{1} we have

∥π_{n} − δ_{α} ∗ π_{n}∥ ≤ ∥π_{n} − σ_{n}∥ + ∥σ_{n} − δ_{α} ∗ σ_{n}∥ + ∥δ_{α} ∗ σ_{n} − δ_{α} ∗ π_{n}∥ = ∥σ_{n} − δ_{α} ∗ σ_{n}∥ + 2∥σ_{n} − π_{n}∥ ≤

2 |α| \frac{1}{2n − \frac{4}{n}} + 2 (2n − 1) \frac{1}{2n − \frac{4}{n}} \frac{2}{n^{2}} ⟶_{n→∞} 0.

For every α, n ≥ 2 we introduce the function f_{α,n} by

f_{α,n} = \frac{n^{2}}{4} I_{( [α]−α− \frac{2}{n^{2}} , [α]−α+ \frac{2}{n^{2}} )} .

Note that −1 < [α] − α ≤ 0 and the subset

([α] − α − \frac{1}{n^{2}} , [α] − α + \frac{1}{n^{2}} ) of ([α] − α − \frac{2}{n^{2}} , [α] − α + \frac{2}{n^{2}} )

does not belong to the support of the distribution π_{n} ∗ δ_{−α}. Hence

∥(π_{n} ∗ δ_{−α})_{f_{α,n}} − λ_{f_{α,n}}∥ > \frac{1}{2}

for every α, n ≥ 2. To complete the example we have only to remark that the family of distributions belonging to the densities f_{α,n} is obviously tight.

Suppose the condition (12) is fulfilled and there is some h ∈ L_{1}^{+} such that g ≤ h for every g ∈ G. Then

∫ ( sup_{g∈G} ∫ |p_{n}(t + α) − p_{n}(α)| g(t) dt)dα ≤ ∫ ∥π_{n} − δ_{t} ∗ π_{n}∥ h(t) dt ⟶_{n→∞} 0 (15)

provided the sequence {π_{n}} is asymptotically uniformly distributed. Neverthless for special asymptotically uniformly distributed sequences less restrictive conditions on G are sufficient to imply the statement (13).

Example 2. If G ⊆ L_{1}^{+} is a family of probability densities with the property that for some h ∈ L_{1}^{+} and some r > 0

g(x) ≤ h(x), ∀g ∈ G, ∀x ∈ ℝ_{d} : |x| ≥ r,

and π_{n}, n = 1, 2, …, is the uniform distribution on the ball

K_{n} := {x ∈ ℝ_{d} : |x| ≤ r_{n}}, 0 < r_{n} ⟶_{n→∞} ∞.

Then by Proposition 2 the condition (12) is fulfilled and it holds:

lim sup_{n→∞} ∫ ( sup_{g∈G} ∫ |p_{n}(t + α) − p_{n}(α)| g(t) dt)dα ≤

lim sup_{n→∞} ∫ ( sup_{g∈G} ∫_{|t|<r} |p_{n}(t + α) − p_{n}(α)| g(t) dt)dα +

lim sup_{n→∞} ∫ ∫_{|t|≥r} |p_{n}(t + α) − p_{n}(α)| h(t) dt dα ≤

lim sup_{n→∞} (∫_{|α| ≥ r_{n}+\sqrt{r_{n}}} sup_{g∈G} ∫_{|t|<r} … +

∫_{r_{n}−\sqrt{r_{n}} ≤ |α| < r_{n}+\sqrt{r_{n}}} sup_{g∈G} ∫_{|t|<r} … + ∫_{|α| < r_{n}−\sqrt{r_{n}}} sup_{g∈G} ∫_{|t|<r} …),

the first and the third integral vanish for sufficiently large n and the second integral is not larger than

\frac{1}{λ(K_{n})} ∫_{r_{n}−\sqrt{r_{n}} ≤ |α| < r_{n}+\sqrt{r_{n}}} sup_{g∈G} ∫_{|t|<r} g(t) dt dα ≤

\frac{1}{λ(K_{n})} λ(\{α : r_{n} − \sqrt{r_{n}} ≤ |α| < r_{n} + \sqrt{r_{n}} \}) ⟶_{n→∞} 0 .

Consequently the condition (13) is fulfilled and Theorem 2 yields statement (14).

Now we study a family of densities generated by the multivariate scale and location model. We suppose that the generating density is bounded by a spherically symmetric function which depends monoton on the distance to the zero point. Let f ∈ L_{1}^{+} be a Legesgue density with the following property.

(16) There exists a bounded function h ∈ L_{1}^{+} with

h(x_{1}) = h(x_{2}) if |x_{1}| = |x_{2}|, h(x_{1}) ≥ h(x_{2}) if |x_{1}| < |x_{2}|, ∀x_{1}, x_{2} ∈ ℝ_{d},

such that f(x) ≤ h(x), ∀x ∈ ℝ_{d}.

For f and constants c_{1}, c_{2}, c_{3} > 0, c_{2}c_{3} > 1 set

(17) F (c_{1}, c_{2}, c_{3}, f) = {g : g(x) = |det(M)| f(m + Mx),

m ∈ ℝ_{d}, |m| ≤ c_{1}, M ∈ M_{d,d}, det(M) ≠ 0, ∥M^{−1}∥ ≤ c_{2}, ∥M∥ ≤ c_{3}}.

Proposition 3. If π_{n} ≪ λ, n = 1, 2, …, and {π_{n}} is asymptotially uniformly distributed then for every Lebesgue density f which satisfies (16) and any constants c_{1}, c_{2}, c_{3} > 0, c_{2}c_{3} > 1, it holds

π_{n}({α : sup_{g∈F (c_{1},c_{2},c_{3},f)} ∥(π_{n} ∗ δ_{−α})_{g} − λ_{g}∥ > ε}) ⟶_{n→∞} 0 (18)

for every ε > 0.

Proof. The condition ∥M∥ ≤ c_{3} implies | det(M)| ≤ c̃_{3} with some c̃_{3}. It holds

|m + Mt| ≥ |Mt| − |m| ≥ \frac{|t|}{∥M^{−1}∥} − |m| ≥ \frac{|t|}{c_{2}} − c_{1}.

Therefore by (16)

|det(M)| f(m + Mt) ≤ |det(M)| h(m + Mt) ≤ h̃(t) := \{ \begin{matrix}c̃_{3} h(0) , & if |t| ≤ c_{1}c_{2}, \\ c̃_{3} h(\frac{|t|}{c_{2}} − c_{1}) , & if |t| > c_{1}c_{2} \end{matrix}

for every |m| ≤ c_{1}, ∥M^{−1}∥ ≤ c_{2}, ∥M∥ ≤ c_{3} with a bounded function h̃ ∈ L_{1}^{+}. Proposition 2, relation (15) and Theorem 2 yield (18).

3 Asymptotically uniformly distributed priors in (Γ, γ, m, d)-models

Gaussian experiments appear as typical limit experiments for i.i.d. observations under weak regularity conditions. But in the statistic of stochastic processes so-called mixed normal distributions appear for non-ergodic processes, see e.g. Basawa, Scott (1983). A general class of models which includes the mixed normal distributions as special case have been introduced in Shiryaev and Spokoiny (1993). We generalize this approach. To be more precise we suppose that (Ω, 𝔉, ν) is a measurable space with σ-finite ν. Let 𝔅_{m} be the σ-algebra of Borel sets of ℝ_{m} and M_{m,d} the set of m × d matrices. 𝔐 is the σ-algebra of Borel sets of M_{m,d}. Assume we are given γ : Ω → ℝ_{m}, Γ : Ω → M_{m,d}, measurable w.r.t. the corresponding σ-algebras and a Lebesgue density q on ℝ_{m}. Suppose that

∫ q(γ + Γα) dν = 1 (19)

for every α ∈ ℝ_{d} and introduce the distribution P_{α} on (Ω, 𝔉) by dP_{α} = q(γ + Γα) dν. The experiment (Ω, 𝔉, P_{α}, α ∈ ℝ_{d}) is called a (Γ, γ, m, d)-model. If q(γ) > 0 ν - a.e. then P_{α} ≪ P_{0} and

\frac{dP_{α}}{dP_{0}} = \frac{q(γ + Γα)}{q(γ)} .

Now we turn to a special (Γ, γ, m, d)-model. Assume (Ω, 𝔉, P_{0}) is a given probability space and the r.v. γ : Ω → ℝ_{m}, Γ : Ω → M_{m,d} are independent w.r.t. P_{0}. Put P_{0}^{γ} = L(γ|P_{0}), P_{0}^{Γ} = L(Γ|P_{0}) and assume P_{0}^{γ} ≪ λ where λ is the Lebesgue measure. Put q = \frac{dP_{0}^{γ}}{dλ}. Suppose q > 0 λ-a.e. Then for every B ∈ 𝔉 the integral

P_{α}(B) = ∫_{B} \frac{q(γ + Γα)}{q(γ)} dP_{0}

is well-defined and we get by the independence of γ and Γ

P_{α}(Ω) = ∫∫ q(t + Mα) dt P_{0}^{Γ}(dM) = 1 ,

so that the condition (19) is fulfilled with dν = \frac{1}{q(γ)} dP_{0}. In general every (Γ, γ, m, d)-model is called mixed (Γ, γ, m, d)-model if γ and Γ are independent w.r.t. P_{0}, P_{0}^{γ} ≪ λ and the corresponding density q = \frac{dP_{0}^{γ}}{dλ} fulfils q(γ) > 0 ν-a.e. Denote by ∥P_{α} − P_{β}∥ the variational distance between P_{α} and P_{β}. Then for a mixed (Γ, γ, m, d)-model

∥P_{α} − P_{β}∥ = ∫ |\frac{q(γ + Γα)}{q(γ)} − \frac{q(γ + Γβ)}{q(γ)}|dP_{0}
= ∫∫ |q(t + Mα) − q(t + Mβ)| dtP_{0}^{Γ}(dM)

As for every function f : ℝ_{d} → ℝ_{1} which is integrable w.r.t. the Lebesgue-measure

lim_{h→0} ∫ |f(t + h) − f(t)|dt = 0

we get

lim_{α→β} ∥P_{α} − P_{β}∥ = 0. (20)

For every (Γ, γ, m, d)-model, P_{α}, α ∈ ℝ_{d}, the mapping α → P_{α}(B) is measurable for every B ∈ 𝔉 and therefore the family P_{α} defines a stochastic kernel ℙ. For every distribution π on ℝ_{d} we introduce the distribution ℙ ⊗ π on (Ω × ℝ_{d}, 𝔉 ⊗ 𝔅_{d}) by P_{α}(dω)π(dα). The distribution ℙ ⊗ π has the marginal distributions ℙπ(dω) = ∫ P_{α}(dω) π(dα) and π, respectively. The model (Γ, γ, d, d) has been introduced and studied in Shiryaev, Spokoiny (1993). The more general class of (Γ, γ, m, d)-models has the advantage to be closed w.r.t. the product of experiments.

Proposition 4. If P_{α,i}, i = 1, 2, are (Γ_{i}, γ_{i}, m_{i}, d)-models on the measure spaces (Ω_{i}, 𝔉_{i}, ν_{i}) then P_{α} = P_{α,1} × P_{α,2} is a (Γ, γ, m, d)-model on (Ω_{1} × Ω_{2}, 𝔉_{1} ⊗ 𝔉_{2}, ν_{1} × ν_{2}) with Γ = \binom{Γ_{1}}{Γ_{2}}, γ = \binom{γ_{1}}{γ_{2}}, q(x_{1}, x_{2}) = q_{1}(x_{1})q_{2}(x_{2}), x_{i} ∈ ℝ_{m_{i}}.

Proof.

\frac{d(P_{α,1} × P_{α,2})}{d(ν_{1} × ν_{2})} (ω_{1}, ω_{2}) = \frac{dP_{α,1}}{dν_{1}} (ω_{1}) \frac{dP_{α,2}}{dν_{2}} (ω_{2})

= q_{1}(γ(ω_{1})) + Γ_{1}(ω_{1})α) q_{2}(γ_{2}(ω_{2})) + Γ_{2}(ω_{2})α) (ν_{1} × ν_{2}) -a.e.

Example 3. Put Ω_{i} = ℝ_{1}, 𝔉_{i} = 𝔅_{1}, ν_{i} = Lebesgue measure on ℝ_{1}, Γ_{i}(t) = −1, γ_{i}(t) = t, t ∈ ℝ_{1}, d = 1, i = 1, …, m. Then P_{α} = P_{α,1} × ⋯ × P_{α,m} has Lebesgue density q(t_{1} − α) ⋯ q(t_{m} − α) and is the distribution of m independent observations from a shift model.

The following property of mixed (Γ, γ, m, d)-models is important for all further investigations.

Lemma 1. Let P_{α}, α ∈ ℝ_{d}, be a mixed (Γ, γ, m, d)-model. Then for T = γ + Γα holds

L((T, Γ)|P_{α}) = L((γ, Γ)|P_{0}) = P_{0}^{γ} × P_{0}^{Γ} .

Proof. Suppose B ∈ 𝔅_{m} ⊗ 𝔐_{d} and denote by I_{B} the indicator function of B. Then

∫ I_{B}(γ + Γα, Γ) dP_{α} = ∫ I_{B}(γ + Γα, Γ)\frac{q(γ + Γα)}{q(γ)} dP_{0}
= ∫ I_{B}(t + Mα, M) q(t + Mα) dt P_{0}^{Γ}(dM)
= ∫ I_{B}(s, M) q(s) ds P_{0}^{Γ}(dM)
= ∫ I_{B}(s, M) P_{0}^{γ}(ds) P_{0}^{Γ}(dM) .

Denote by X and A the projection of Ω×ℝ_{d} onto Ω and ℝ_{d}, respectively. Put T = γ+ΓA and consider this statistic to be defined on (Ω × ℝ_{d}, 𝔉 ⊗ 𝔅_{d}, ℙ ⊗ π). We calculate the distribution L(T |ℙ ⊗ π) in the mixed (Γ, γ, m, d)-model. It holds with Q = P_{0}^{γ} and B ∈ 𝔅_{m}

∫ I_{B}(T) d(ℙ ⊗ π) = ∫∫ I_{B}(γ(ω) + Γ(ω)α)\frac{q(γ(ω) + Γ(ω)α)}{q(γ(ω))} P_{0}(dω) π(dα)
= ∫∫∫ I_{B}(t + Mα) q(t + Mα) dt P_{0}^{Γ}(dM) π(dα) = Q(B) .

Hence

L(T |ℙ ⊗ π) = Q . (21)

Now we confine us to (Γ, γ, d, d)-models which are but not necessarily mixed. We replace the fixed distribution π by a sequence of priors {π_{n}}. Let Q be the distribution with the Lebesgue density q. It turns out that from the statistical point of view the situation becomes nice in some sense (see Shiryaev, Spokoiny (1993), chapter 3) if (21) (with π = π_{n}) is true or at least L(T |ℙ ⊗ π_{n}) converges to Q and the r.v. T and X, considered on (Ω×ℝ_{d}, 𝔉⊗𝔅_{d}, ℙ⊗π_{n}) are approximately independent, i.e. in view of L(X|ℙ⊗π_{n}) = ℙπ_{n}

∥L((X, T)|ℙ ⊗ π_{n}) − (ℙπ_{n}) × Q∥ ⟶_{n→∞} 0 , (22)

where ∥ ... ∥ is the variational distance. The condition (22) may be rewritten in the following way. Denote by K_{n} a regular conditional distribution of T under the condition X then for C ∈ 𝔉 ⊗ 𝔅_{m}

(ℙ ⊗ π_{n})((X, T) ∈ C) = ∫∫ I_{C}(ω, t) K_{n}(ω, dt)(ℙπ_{n})(dω)

and

∥L((X, T)|ℙ ⊗ π_{n}) − (ℙπ_{n}) × Q∥ = ∫ ∥K_{n}(ω, .) − Q( .)∥ (ℙπ_{n})(dω) .

The condition (22) therefore means that K_{n}(ω, .) tends in variational distance ℙπ_{n}-stocha-stically to the distribution Q. It turns out that some asymptotically uniformly distributional properties of the distributions π_{n} are responsible for the validity of (22).

Theorem 3. The condition (22) holds for every mixed (Γ, γ, d, d)-model with P_{0}(det(Γ) = 0) = 0 iff the sequence of priors {π_{n}} is asymptotically uniformly distributed.

Proof. 1.) Suppose P_{α}, α ∈ ℝ_{d}, is a mixed (Γ, γ, d, d)-model. Denote by 𝔹 the set of all measurable functions h : Ω × ℝ_{d} → ℝ_{1} with

sup_{ω,x} |h(ω, x)| ≤ 1 .

Then

∥L((X, T)|ℙ ⊗ π_{n}) − (ℙπ_{n}) × Q∥

= sup_{h∈𝔹}|∫∫ h(ω, γ(ω) + Γ(ω)α) P_{α}(dω) π_{n}(dα) − ∫∫∫ h(ω, t) P_{α}(dω) π_{n}(dα) Q(dt)|

= sup_{h∈𝔹}|∫∫∫ (h(ω, γ(ω) + Γ(ω)α) − h(ω, t))\frac{q(γ(ω) + Γ(ω)α)}{q(γ(ω))}P_{0}(dω) π_{n}(dα) q(t) dt| .

Set ν_{n}(ω, B) = π_{n}({t : Γ(ω)t ∈ B}), (δ_{s} ∗ ν_{n})(ω, B) = ν_{n}(ω, B − s), and v = γ(ω) + Γ(ω)α, then

(23)  ∥L((X, T)|ℙ ⊗ π_{n}) − (ℙπ_{n}) × Q∥

= sup_{h∈𝔹}|∫∫∫ h(ω, v)\frac{q(v)q(t)}{q(γ(ω))} δ_{−γ(ω)} ∗ ν_{n}(ω, dv) P_{0}(dω) dt

− ∫∫∫ h(ω, t)\frac{q(v)q(t)}{q(γ(ω))} δ_{−γ(ω)} ∗ ν_{n}(ω, dv) P_{0}(dω) dt|

= sup_{h∈𝔹}|∫∫∫ h(ω, t)\frac{q(t)q(v)}{q(γ(ω))}[δ_{−γ(ω)} ∗ ν_{n}(ω, dt) dv

− δ_{−γ(ω)} ∗ ν_{n}(ω, dv) dt]P_{0}(dω)| .

Let π_{n} = π_{n}^{a}+π_{n}^{s} be the decomposition of π_{n} into the part π_{n}^{a} ≪ λ and π_{n}^{s} which is singular w.r.t. λ. Now we suppose that the sequence {π_{n}} is asymptotically uniformly distributed.

It is well-known (see Kerstan, Matthes (1968)) that in this case π_{n}^{s}(ℝ_{d}) → 0, as n → ∞. It is easily seen from (23) that for n → ∞ the singular part π_{n}^{s} may be neglected so that we may assume without loss of generality that π_{n} ≪ λ and denote by φ_{n} the Lebesgue density of π_{n}. By assumption P_{0}(det(Γ) = 0) = 0. The distribution δ_{−γ(ω)} ∗ ν_{n}(ω, ·) has P_{0}-a.s. the Lebesgue density

ψ_{n}(ω, t) = \frac{1}{det(Γ(ω))} φ_{n}(Γ^{−1}(ω)(t − γ(ω))

and it holds by the independence of γ and Γ w.r.t. P_{0}

∫ |ψ_{n}(ω, t_{1}) − ψ_{n}(ω, t_{2})|\frac{1}{q(γ(ω))}P_{0}(dω)

= ∫∫ \frac{1}{det(M)} |φ_{n}(M^{−1}(t_{1} − v)) − φ_{n}(M^{−1}(t_{2} − v))| \frac{1}{q(v)} P_{0}^{γ}(dv) P_{0}^{Γ}(dM)

= ∫ [∫ \frac{1}{det(M)} |φ_{n}(M^{−1}(t_{1} − v) − φ_{n}(M^{−1}(t_{2} − v))| dv] P_{0}^{Γ}(dM)

= ∫ [∫ \frac{1}{det(M)} |φ_{n}(M^{−1}s) − φ_{n}(M^{−1}(t_{2} − t_{1} + s)| ds] P_{0}^{Γ}(dM) .

\frac{1}{det(M)} φ_{n}(M^{−1}s) is the Lebesgue density of the distribution π_{n} ∘ M^{−1}. For a linear oneto-one mapping A : ℝ_{d} → ℝ_{d} we have

∥π_{n} ∘ A^{−1} − δ_{x} ∗ (π_{n} ∘ A^{−1})∥ = 2 sup_{B∈𝔅_{d}} |π_{n}(A^{−1}(B)) − π_{n}(A^{−1}(B) − x)|

= 2 sup_{B∈𝔅_{d}} |π_{n}(A^{−1}(B)) − π_{n}(A^{−1}(B − A(x)))|

= ∥π_{n} − δ_{A(x)} ∗ π_{n}∥ .

Hence by the asymptotic uniform distribution of {π_{n}} and Lebesgue’s dominated convergence theorem

∫ |ψ_{n}(ω, t_{1}) − ψ_{n}(ω, t_{2})|\frac{1}{q(γ(ω))} P_{0}(dω)

= ∫ ∥π_{n} ∘ M^{−1} − δ_{M^{−1}(t_{1}−t_{2})} ∗ π_{n} ∘ M^{−1}∥ P_{0}^{Γ}(dM) ⟶_{n→∞} 0

for every t_{1}, t_{2} ∈ ℝ_{d}. Consequently by (23) and the Theorem of Lebesgue

lim sup_{n→∞} ∥L((X, T)|ℙ ⊗ π_{n}) − (ℙπ_{n}) × Q∥

= lim sup_{n→∞} sup_{h∈𝔹} |∫∫∫ h(ω, t)\frac{q(t)q(v)}{q(γ(ω))} [ψ_{n}(ω, t) − ψ_{n}(ω, v)] P_{0}(dω) dt dv|

≤ lim sup_{n→∞} ∫∫∫ ∥π_{n} ∘ M^{−1} − δ_{M^{−1}(t−v)} ∗ π_{n} ∘ M^{−1}∥ P_{0}^{Γ}(dM) Q(dt) Q(dv) ⟶_{n→∞} 0 .

2.) Now we construct a special mixed (Γ, γ, d, d)-model for which condition (22) is even equivalent to the asymptotically uniform distribution of the sequence {π_{n}}. To this end we set Ω = ℝ_{d}, 𝔉 = 𝔅_{d} and denote by Q a probability distribution absolutely continuous w.r.t. the Lebesgue-measure with strictly positive density q. Set P_{0} = Q, γ(ω) = ω, Γ = −1, ω ∈ ℝ_{1}. Then

(ℙ ⊗ π_{n})((T, X) ∈ C) = ∫∫ I_{C}(ω − α, ω) P_{α}(dω) π_{n}(dα)
= ∫∫ I_{C}(ω − α, ω) q(ω − α) dω π_{n}(dα)
= ∫∫ I_{C}(ω, ω + α) q(ω) dω π_{n}(dα) =: (Q ⊕ π_{n})(C) .

Note that the marginal distributions of Q ⊕ π_{n} are given by L(T |ℙ ⊗ π_{n}) = ℙπ_{n} = Q ∗ π_{n} and L(X|ℙ ⊗ π_{n}) = Q. Hence

∥L((X, T)|ℙ ⊗ π_{n}) − (ℙπ_{n}) × Q∥ = ∥L((T, X)|ℙ ⊗ π_{n}) − Q × (Q ∗ π_{n})∥
= ∥Q ⊕ π_{n} − Q × (Q ∗ π_{n})∥ .

But in view of Theorem 1 the convergence to zero of the right-hand term as n → ∞ is equivalent to ∥π_{n} − δ_{x} ∗ π_{n}∥ ⟶_{n→∞} 0 for every x ∈ ℝ_{d} which has to be proved.

For arbitrary (Γ, γ, d, d)-models the statement (22) have been proved in Shiryajev and Spokoiny (1993) under the following assumptions. Firstly, the sequence {π_{n}} fulfils (14) for every set G ⊆ L_{1}^{+} of probability densities for which the family of corresponding distributions is tight, and secondly, the sequences

{L(γ + Γα |ℙ ⊗ π_{n})} and {L(Γ^{−1}|ℙ ⊗ π_{n})}

are tight. Note that by Lemma 1 the second condition is fulfilled in mixed (Γ, γ, d, d)- models with P_{0}(det(Γ) = 0) = 0). However an analyse of the proof in Shiryaev and Spokoiny shows that one needs the condition (14) only for the sets

{g : g(x) = |det(M)| q(m + Mx),

m ∈ ℝ_{d}, |m| ≤ c_{1}, M ∈ M_{d,d}, det(M) ≠ 0, ∥M^{−1}∥ ≤ c_{2}}

for every c_{1}, c_{2} > 0. Moreover, using the arguments in Shiryaev and Spokoiny and the additional assumption that the sequence {L(Γ |ℙ ⊗ π_{n})} is tight one can see that it is enough to assume (14) for the sets

F (c_{1}, c_{2}, c_{3}, q) = {g : g(x) = |det(M)| q(m + Mx),

m ∈ ℝ_{d}, |m| ≤ c_{1}, M ∈ M_{d,d}, det(M) ≠ 0, ∥M^{−1}∥ ≤ c_{2}, ∥M∥ ≤ c_{3}}

for every c_{1}, c_{2}, c_{3} > 0, c_{2}c_{3} > 1. Under the additional assumption that the Lebesgue density q has the property (16) by Proposition 3 the condition (14) is fulfilled for every asymptotically uniformly distributed sequence {π_{n}} and every set F (c_{1}, c_{2}, c_{3}, q), c_{1}, c_{2}, c_{3} > 0, c_{2}c_{3} > 1.

Proposition 5. Assume (P_{α}, α ∈ ℝ_{d}) is a (Γ, γ, d, d)-model and {π_{n}} a sequence of distributions on ℝ_{d}, so that

{L(Γ |ℙ ⊗ π_{n})}, {L(γ + Γα |ℙ ⊗ π_{n})} and {L(Γ^{−1}|ℙ ⊗ π_{n})}

are tight. If q has the property (16) then for every asymptotically uniformly distributed sequence {π_{n}} the statement (22) holds.

4 Minimax-estimation in (Γ, γ, m, d)-models

Now we are going to demonstrate how the statement (22) can be used to prove a minimax property of some estimators in mixed (Γ, γ, m, d)-models and in general (Γ, γ, m, d)-models. Assume (P_{α}, α ∈ ℝ_{d}) is a mixed (Γ, γ, d, d)-model. Let κ : Ω → ℝ_{d} be an estimator for the unknown parameter α. Let L : ℝ_{d} → [0, ∞) be a loss function and introduce the risk of estimator κ by

R(α, κ) = ∫ L(κ(ω) − α) P_{α}(dω) .

Given a prior distribution π the Bayes risk is defined by

R_{π}(κ) = ∫ R(α, κ) π(dα) .

We now suppose P_{0}(det(Γ) = 0) = 0. Introduce the estimator κ_{0} = −Γ^{−1}γ. It holds

(24)  R(α, κ_{0}) = ∫ L(κ_{0}(ω) − α) P_{α}(dω)

= ∫ L(−Γ^{−1}γ − α) \frac{q(γ + Γα)}{q(γ)} dP_{0}

= ∫∫ L(−M^{−1}t − α) \frac{q(t + Mα)}{q(t)} q(t) dt P_{0}^{Γ}(dM)

= ∫∫ L(−M^{−1}s) q(s) ds P_{0}^{Γ}(dM)

= R(0, κ_{0}) .

Consequently, the estimator κ_{0} has constant risk in the mixed (Γ, γ, d, d)-models. We show that κ_{0} is optimal in some sense.

Theorem 4. Let (P_{α}, α ∈ ℝ_{d}) be a general (Γ, γ, d, d)-model. Suppose the loss function L is bounded and fulfils the condition

∫ L(y + Mx) Q(dx) ≥ ∫ L(Mx) Q(dx) (LQ)

for every y ∈ ℝ_{d} and every M ∈ M_{d,d}. If {π_{n}} is an asymptotically uniformly distributed sequence and at least one of the following conditions

(I) (P_{α}, α ∈ ℝ_{d}) is a mixed model with P_{0}(det(Γ) = 0) = 0

(II) q has the property (16) and the sequences

{L(Γ |ℙ ⊗ π_{n})}, {L(γ + Γα |ℙ ⊗ π_{n})} and {L(Γ^{−1}|ℙ ⊗ π_{n})}

are tight

holds then for every of estimator κ lim inf_{n→∞} R_{π_{n}}(κ) ≥ lim inf_{n→∞} R_{π_{n}}(κ_{0}).

Corollary. If condition (I) holds then

lim_{n→∞} R_{π_{n}}(κ_{0}) = ∫∫ L(−M^{−1}t) Q(dt) P_{0}^{Γ}(dM) = R(0, κ_{0})

and κ_{0} is minimax.

Proof. In view of Theorem 3 and Proposition 5 the statement (22) holds in the cases (I) and (II). Consequently

R_{π_{n}}(κ) = ∫∫ L(κ(ω) − α) (ℙ ⊗ π_{n})(dω, dα)
= ∫∫ L(κ + Γ^{−1}γ − Γ^{−1}T) d(ℙ ⊗ π_{n})
= ∫∫ L(κ(ω) + Γ^{−1}(ω)γ(ω) − Γ^{−1}(ω) t) Q(dt) (ℙπ_{n})(dω) + o(1)
≥ ∫∫ L(−Γ^{−1}(ω) t) Q(dt) (ℙπ_{n})(dω) + o(1)
= ∫∫∫ L(−Γ^{−1}(ω) t) (ℙ ⊗ π_{n})(dω, dt) + o(1)
= R_{π_{n}}(κ_{0}) + o(1)

and the statement of Theorem 4 is established. In order to prove the Corollary we note that by (24)

R_{π_{n}}(κ_{0}) = ∫ R(α, κ_{0})π_{n}(dα) = R(0, κ_{0})

and consequently in view of Theorem 4

sup_{α} R(α, κ) ≥ lim inf_{n→∞} R_{π_{n}}(κ) ≥ R(0, κ_{0}) = sup_{α} R(α, κ_{0})

that is the minimax property of κ_{0} = −Γ^{−1}γ.

Remark 2. A sufficient condition for assumption (LQ) is given by the Anderson-Lemma (see Ibragimov, Khasminski (1979), Lemma 2.11.1) which says that condition (LQ) is fulfilled if for every b > 0 the sets {L ≤ b}, {q ≥ b} are convex and centrally symmetric.

Remark 3. Under stronger assumptions the statement of Theorem 4 has been already established in Shiryaev, Spokoiny (1993).

Now we turn to the (Γ, γ, m, d)-model. If d = 1, Ω = ℝ_{m}, γ((t_{1}, …, t_{m})^{T}) = (t_{1}, …, t_{m})^{T}, Γ = −(1, …, 1)^{T}, ν = Lebesgue measure λ on ℝ_{m} then

\frac{dP_{α}}{dλ} = q(t_{1} − α, …, t_{m} − α) .

Consequently P_{α} is a shift model for which the Pitman estimator is minimax for the quadratic loss function (see Lehmann (1983) p. 282-284, Ferguson (1967), p. 186-190). We will construct an estimator for the mixed (Γ, γ, m, d)-model, m ≥ d, in analogy to the Pitman estimator in the shift model. Suppose that

\begin{aligned}∫ q(γ + Γα) dα & < ∞ P_{0} -a.s. , \\ ∫ |α| q(γ + Γα) dα & < ∞ P_{0} -a.s. \end{aligned} (25)

We show that the first integral in (25) is P_{0}-a.s. positive. Put

A = {ω : ∫ q(γ(ω) + Γ(ω)α) dα = 0} .

Then

0 = ∫∫ I_{A}(ω) q(γ(ω) + Γ(ω)α) dα ν(dω) = ∫∫ I_{A} dP_{α} dα = ∫ P_{α}(A) dα .

In view of (20) the mapping α → P_{α}(A) is continuous. Hence ∫ P_{α}(A) dα = 0 implies P_{β}(A) = 0 for every β ∈ ℝ_{d} or equivalent

∫ q(γ + Γα) dα > 0 P_{β} -a.s.

Therefore

(26)  θ̂ = \frac{∫ α q(γ + Γα) dα}{∫ q(γ + Γα) dα}, that is

θ̂ = (θ̂_{1}, …, θ̂_{d})^{T} with θ̂_{i} = \frac{∫ α_{i} q(γ + Γα) dα}{∫ q(γ + Γα) dα}, i = 1, …, d,

is P_{β}-a.s. defined for every β ∈ ℝ_{d}. θ̂ will be called the Pitman estimator in the (Γ, γ, m, d)-model. We have

θ̂ − β = \frac{∫ (α − β) q(γ + Γα) dα}{∫ q(γ + Γα) dα} = \frac{∫ s q(T + Γs) ds}{∫ q(T + Γs) ds}

where T = γ + Γβ. Note that the vector of the last fractions is some measurable mapping φ : (T, Γ) → ℝ_{d}. Hence by Lemma 1

L(φ(T, Γ)| P_{β}) = L(φ(γ, Γ)| P_{0})

which yields

L((θ̂ − β) | P_{β}) = L(θ̂| P_{0}) . (27)

Assume π is any prior distribution on ℝ_{d}. Then the distribution ℙ ⊗ π defined by P_{α}(dω) π(dα) is absolutely continous w.r.t. P_{0} × π with density

\frac{d(ℙ ⊗ π)}{d(P_{0} × π)} = \frac{q(γ + Γα)}{q(γ)} = f(ω, α) .

Then the conditional density given ω is

f(α | ω) = \frac{f(ω, α)}{∫ f(ω, α) π(dα)} = \frac{q(γ(ω) + Γ(ω)α)}{∫ q(γ(ω) + Γ(ω)α) π(dα)} .

For the loss function L(x) = |x|^{2} the Bayes estimate θ̂_{π} of α is then given by the conditional expectation of α given ω provided the expectation exists. Assume

∫ |α| d(ℙ ⊗ π) < ∞ ,

then

θ̂_{π} = ∫ β f(β|ω) π(dβ) = \frac{∫ β q(γ(ω) + Γ(ω)β) π(dβ)}{∫ q(γ(ω) + Γ(ω)α) π(dα)} .

Now we choose π as uniform distribution π_{A} on [−A, A]^{d} and write θ̂_{A} instead of θ̂_{π_{A}}. Then

θ̂_{A}(ω) = \frac{∫_{[−A,A]^{d}} β q(γ(ω) + Γ(ω)β) dβ}{∫_{[−A,A]^{d}} q(γ(ω) + Γ(ω)β) dβ} .

Consequently, under assumption (25) θ̂_{A} → θ̂ as A → ∞, P_{0} -a.s. Using this fact we are able to show that the Pitman-estimator is minimax. Denote by E_{β} the expectation w.r.t. P_{β}.

Theorem 5. If the condition (25) is fulfilled and E_{0} |θ̂|^{2} < ∞ then the Pitman estimator (26) is minimax w.r.t. the quadratic loss function in the mixed (Γ, γ, m, d)-model (m ≥ d).

Proof. For every estimator θ̃

sup_{β} E_{β}|θ̃ − β|^{2} ≥ ∫ E_{β}|θ̃ − β|^{2} π_{A}(dβ) ≥ \frac{1}{(2A)^{d}} ∫_{[−A,A]^{d}} E_{β}|θ̂_{A} − β|^{2} dβ .

In view of (27) it holds

E_{0} |θ̂|^{2} = sup_{β} E_{β}|θ̂ − β|^{2} .

Therefore it remains to prove that for some A_{n} → ∞

(28)  lim sup_{n→∞} \frac{1}{(2A_{n})^{d}} ∫_{[−A_{n},A_{n}]^{d}} E_{β}|θ̂_{A_{n}} − β|^{2} dβ ≥ E_{0} |θ̂|^{2} .

We have

\frac{1}{(2A_{n})^{d}} ∫_{[−A_{n},A_{n}]^{d}} E_{β} |θ̂_{A_{n}} − β|^{2} dβ

= \frac{1}{(2A_{n})^{d}} ∫ I_{[−A_{n},A_{n}]^{d}} (β) E_{β}[∫ I_{[−A_{n},A_{n}]^{d}} (α) q(γ + Γα) dα]^{−2}

|∫ I_{[−A_{n},A_{n}]^{d}} (α) (α − β) q(γ + Γα) dα |^{2} dβ

≥ \frac{1}{(2A_{n})^{d}} ∫ I_{[−A_{n},A_{n}]^{d}} (β) E_{β}[∫ q(γ + Γα) dα]^{−2}

|∫ I_{[−A_{n},A_{n}]^{d}} (s + β) s q(γ + Γβ + Γs) ds|^{2} dβ

= \frac{1}{(2A_{n})^{d}} ∫ I_{[−A_{n},A_{n}]^{d}} (β) E_{β}[∫ q(T + Γα) dα]^{−2}

[∫ I_{[−A_{n},A_{n}]^{d}} (s + β) s q(T + Γ s) ds]^{2} dβ

with T = γ +Γβ. The expression under the symbol E_{β} is as function of ω some measurable function of (T, Γ). Hence by Lemma 1 with

B_{N} = \{ω : ∫ \frac{|s| q(γ(ω) + Γ(ω) s}{∫ q(γ(ω) + Γ(ω) α) dα} ≤ N\}

and < ·, · > as scalar product

\frac{1}{(2A_{n})^{d}} ∫_{[−A_{n},A_{n}]^{d}} E_{β} |θ̂_{A_{n}} − β|^{2} dβ

≥ \frac{1}{(2A_{n})^{d}} ∫ I_{[−A_{n},A_{n}]^{d}} (β) E_{0}\{I_{B_{N}} [∫ q(γ + Γα) dα]^{−2}

∫∫ I_{[−A_{n},A_{n}]^{d}} (s_{1} + β) I_{[−A_{n},A_{n}]^{d}} (s_{2} + β) < s_{1}, s_{2} > q(γ + Γs_{1}) q(γ + Γs_{2}) ds_{1} ds_{2}} dβ ≥ E_{0} ∫∫ φ_{n}(s_{1}, s_{2}) I_{B_{N}} [∫ q(γ + Γα) dα]^{−2} < s_{1}, s_{2} > q(γ + Γs_{1}) q(γ + Γs_{2}) ds_{1} ds_{2}

where

φ_{n}(s_{1}, s_{2}) = \frac{1}{(2A_{n})^{d}} ∫ I_{[−A_{n},A_{n}]^{d}} (β) I_{[−A_{n},A_{n}]^{d}} (β + s_{1}) I_{[−A_{n},A_{n}]^{d}} (β + s_{2}) dβ .

It is easy to see that

0 ≤ φ_{n}(s_{1}, s_{2}) ≤ 1 and lim_{n→∞} φ_{n}(s_{1}, s_{2}) = 1 .

Note that

E_{0} |∫∫ φ_{n}(s_{1}, s_{2}) I_{B_{N}} [∫ q(γ + Γα) dα]^{−2} < s_{1}, s_{2} > q(γ + Γs_{1}) q(γ + Γs_{2}) ds_{1} ds_{2}|

≤ E_{0} I_{B_{N}} ([∫ q(γ + Γα) dα]^{−1} ∫ |s| q(γ + Γs) ds)^{2} ≤ N^{2} .

Hence by the Theorem of Lebesgue

lim inf_{n→∞} ∫ E_{β} |θ̂_{A_{n}} − β|^{2} dβ ≥ E_{0} I_{B_{N}} [∫ q(γ + Γα) dα]^{−2} |∫ s q(γ + Γs) ds|^{2} .

We now apply the monotone convergence theorem and let N → ∞ to get (28).

Because of the different assumptions the statements of Theorem 5 in the case m = d and Theorem 4 are not identical. But by a additional condition the estimator (26) passes into the estimator −Γ^{−1}γ.

Proposition 6. Let m = d. If the assumptions of Theorem 5 are fulfilled and the expectation vector of the distribution Q vanishes then

\frac{∫ α q(γ + Γα) dα}{∫ q(γ + Γα) dα} = −Γ^{−1}γ P_{0} -a.s.

∫ x Q(dx) = 0 is equivalent to the (LQ)-condition in Theorem 4.

Proof. Firstly we show for m ≥ d that from (25), more precise from

0 < ∫ q(γ + Γα) dα < ∞ P_{0}-a.s.,

it follows that the matrix Γ has the rank d P_{0}-a.s. Let r, r < d, be the rank of Γ. With a suitable matrix A ∈ M_{d,d} and β = Aα we obtain

q(γ + Γα) = q(γ + ΓA^{−1}α) = q̃(β_{1}, …, β_{r}) and

∫ q(γ + Γα) dα = | det(A)|^{−1} ∫ (∫ q̃(β_{1}, …, β_{r}) dβ_{1} ⋯ dβ_{r})dβ_{r+1} ⋯ dβ_{d}.

This integral is 0 or ∞. By (25) both cases have the P_{0}-probability 0. Therefore in the case m = d

\frac{∫ α q(γ + Γα) dα}{∫ q(γ + Γα) dα} = \frac{∫ Γ^{−1}(γ + Γα) q(γ + Γα) dα}{∫ q(γ + Γα) dα} − Γ^{−1}γ

= Γ^{−1}(∫ β q(β) dβ) − Γ^{−1}γ = −Γ^{−1}γ

if ∫ x Q(dx) = 0. For L(x) = |x|^{2} the condition (LQ) is equivalent to

|y|^{2} + 2 < y, M(∫ x Q(dx) ) > ≥ 0, ∀y ∈ ℝ_{d}, ∀M ∈ M_{d,d} .

Put y = − ∫ x Q(dx) and M = (δ_{i,j}). Then it is to see that ∫ x Q(dx) = 0 is the necessary and sufficient condition for the validity of this inequality.

Under a stronger additional assumption than ∫ x Q(dx) = 0 we obtain in the case m > d a result corresponding to Proposition 6.

Proposition 7. Let m > d. If the assumptions of Theorem 5 are fulfilled and the density q satisfies the condition q(x_{1}) = q(x_{2}) if |x_{1}| = |x_{2}|, ∀x_{1}, x_{2} ∈ ℝ_{m}, then

\frac{∫ α q(γ + Γα) dα}{∫ q(γ + Γα) dα} = −(Γ^{T} Γ)^{−1} Γ^{T} γ P_{0} -a.s.

Proof. The rank of the matrix Γ is d P_{0}-a.s. (see proof of Proposition 6). The function |γ + Γα|, α ∈ ℝ_{d}, is minimal for α_{0} = −(Γ^{T} Γ)^{−1} Γ^{T} γ. Hence the vector (γ + Γα_{0}) is orthogonal to the linear subspace {Γα, α ∈ ℝ_{d}}, that is

(γ + Γα_{0}) ⊥ (γ + Γα) , ∀α ∈ ℝ_{d}.

Consequently |γ + Γα_{0} + Γα|^{2} = |γ + Γα_{0}|^{2} + |Γα|^{2} = |γ + Γα_{0} − Γα|^{2} whence we deduce q(γ + Γ(α_{0} + α)) = q(γ + Γ(α_{0} − α)). Thus α_{0} is the expectation vector of the distribution with the Lebesgue density

\frac{q(γ + Γα)}{∫ q(γ + Γβ) dβ} , that is α_{0} = \frac{∫ α q(γ + Γα) dα}{∫ q(γ + Γα) dα} .

Finally we note that −(Γ^{T} Γ)^{−1} Γ^{T} γ is also the maximum likelihood estimator of α in a (Γ, γ, m, d)-model, m ≥ d, if

P_{0}(det(Γ^{T} Γ) = 0) = 0,

q(x_{1}) = q(x_{2}) if |x_{1}| = |x_{2}|, and q(x_{1}) ≥ q(x_{2}) if |x_{1}| < |x_{2}|, ∀x_{1}, x_{2} ∈ ℝ_{m}.

References

[1] Basawa, I.V., Scott, D.J. (1983): Asymptotic optimal inference for non-ergodic models. Lect. Notes Statistics, 17, Springer-Verlag.

[2] Ferguson, T. S. (1967): Mathematical Statistics. A Decision Theoretic Approach. Academic Press.

[3] Herrmann, H., Kerstan, J., Liese, F. (1979): Zweidimensionale und lokale Charakterisierung der asymptotischen Gleichverteilung. Math. Nachrichten 89, 87-126.

[4] Ibragimov, I. A., Khasminski, R. Z. (1979): Asymptotic Theory of Estimation (in Russian). Moscow.

[5] Kerstan, J., Matthes, K. (1968): Gleichverteilungseigenschaften von Faltungen von Verteilungsgesetzen auf lokalkompakten abelschen Gruppen I. Math. Nachr. 37, 267-312.

[6] Lehmann, E. L. (1983): Theory of Point Estimation. J. Wiley & Sons.

[7] Shiryaev, A., Spokoiny, V. (1993): Statistical Experiments and Decisions. Asymptotic Theory. VINITI, Moscow (in Russian). English transl.

[8] Spokoiny, V. (1994): Efficient Estimation and Experimental Design for Controlled Systems. Asymptotic Approach. Stochastics and Stochastics Reports. 50, 129-160.