UDC 519.21

MATHEMATICS

G. P. KLIMOV

ON THE FIDUCIAL APPROACH IN STATISTICS

(Presented by Academician A. N. Tikhonov, 4 IX 1969)

Let (\mathscr P={P_\theta,\ \theta\in\Omega}) be a family of distributions defined on one and the same sample space ((X,B)) and absolutely continuous with respect to a fixed measure (\mu). We denote the density of (P_\theta) with respect to (\mu) by (p(x\mid\theta)). Consider the class (\mathscr P_\nu) of (prior) distributions (P), defined on the parameter space ((\Omega,F)) and absolutely continuous with respect to the measure (\nu). Suppose that

[
0< p(x)=\int p(x\mid\theta)\,d\nu(\theta)<\infty
]

for almost all (with respect to (\mu)) (x).

Definition. Let ({\Omega_n}{n\ge1}) be an arbitrary monotonically increasing sequence of subsets of (\Omega) such that (\bigcup \Omega_n=\Omega), (\nu(\Omega_n)<\infty). Denote by ({P_n}) a sequence of distributions from (\mathscr P_\nu) such that the distribution (P_n) is concentrated on (\Omega_n) and has maximal entropy (-\int p_n(\theta)\ln p_n(\theta)\,d\nu(\theta)) among such distributions. Here (p_n(\theta)) is the density of the distribution (P_n) with respect to the measure (\nu). Let (p_n(\theta\mid x)) be the density of the posterior distribution (with respect to (\nu)), computed by Bayes’ formula starting from the prior density (p_n(\theta)). Then the distribution (P_x) on ((\Omega,F)), defined for almost all (x) and having density with respect to (\nu) equal to (p(\theta\mid x)=\lim p_n(\theta\mid x)), will be called the fiducial distribution (corresponding to the observation (x)). The density (p(\theta\mid x)) is computed uniquely and is equal to (p(x\mid\theta)/p(x)). Thus the fiducial distribution (fid. d.) (P_x) is defined uniquely up to the measure (\nu). We shall now present a statistical model for which the requirements of invariance of statistical conclusions uniquely single out the measure (\nu), and hence also the fid. d. (P_x).

Let (G) be a group of measurable transformations of (X) onto (X). We make the following assumptions.

P.1. (\theta_1=\theta_2 \Longleftrightarrow P_{\theta_1}(E)=P_{\theta_2}(E)) for every (E\in B).

P.2. The family (\mathscr P) is closed with respect to transformations from (G). Thus, if the random variable (r.v.) (x) has distribution (P_\theta) and (g\in G), then there exists (\theta^\in\Omega) such that the r.v. (gx) has distribution (P_{\theta^}). By P.1 such an element (\theta^) is unique. Thus to each (g\in G) there corresponds a mapping (g^) of the set (\Omega) into itself. The set of such mappings (g^) will be denoted by (G^).

P.3. Every element (g^\in G^) is a mapping of (\Omega) onto itself. Now the set (G^*) becomes a group.

P.4. The groups (G) and (G^*) are isomorphic.

P.5. The group (G) of transformations of the set (X) is strictly transitive, i.e., for any points (x,y\in X) there exists a unique transformation (g\in G) taking (x) into (y): (y=gx).

P.6. The group (G^*) of transformations of the set (\Omega) is strictly transitive.

The elements of the group (G) can now be used to describe points (x\in X) and points (\theta\in\Omega). To this end choose arbitrary points (x_0) and (\theta_0) from (X) and (\Omega), respectively. These points will play the role of reference points (scale elements), relative to which the remaining points of (X) and (\Omega) will be “measured.” Since for each (x\in X) there exists a unique transformation (g_x\in G) taking (x_0) into (x=g_xx_0), it follows thereby ...

the elements (x \in X) are parametrized by the elements (g_x \in G). Similarly, since for each (\theta \in \Omega) there exists a unique transformation (g_\theta^ \in G^) taking (\theta_0) into (\theta = g_\theta^ \theta_0), the elements (\theta \in \Omega) are described by the elements (g_\theta^ \in G^). By virtue of the isomorphism of the groups (G) and (G^), to the element (g_\theta^ \in G^) there corresponds an element (g_\theta \in G). We have indicated a one-to-one correspondence between the elements of the sets (X,\Omega,G,G^*). In particular, this correspondence generates a (\sigma)-algebra (F) of subsets of the set (\Omega), starting from the (\sigma)-algebra (B) of subsets of the set (X). Moreover, using such a correspondence, we shall denote by the same letter (x) both an element of (X) and the corresponding element (g_x) of (G). We shall do the same for elements of (\Omega). For example, (\theta^{-1}x) is understood to mean (g_\theta^{-1}g_x \in G).

Suppose that the following two principles are fulfilled: the invariance principle of the fiducial distribution and the invariance principle of the entropy of the fiducial distribution with respect to the choice of the “scale elements” (x_0) and (\theta_0). It turns out that if the left Haar measure on the group (G) is chosen as (\mu), then from the first principle it follows that the measure (\nu) is relatively invariant (see ((^1)), p. 257, problem 6). If the second principle is also fulfilled, then (\nu) becomes a right-invariant measure, i.e. a right Haar measure, which is determined uniquely up to a constant positive factor. Thus the fiducial distribution (P_x) is determined uniquely. In this case
[
p(\theta\mid x)=\Delta(x)\cdot q(\theta^{-1}x),
]
where (q) is the density of the distribution (P_{\theta_0}\in\mathcal P), and (\Delta(x)) is the modular function (see ((^1)), p. 256, problem 5).

Remark. For the fiducial distribution so defined, the equality (of fiducial and confidence probabilities)
[
P_\theta{\theta\in S(x)}=P_x{\theta\in S(x)}
]
holds for every class ({S(x), x\in X}) of measurable (confidence) sets (S(x)\subseteq\Omega), invariant with respect to the group (G), i.e. such that (g^*S(x)=S(gx)) for any (g\in G). This equality may be taken as the basis for the definition of the fiducial distribution (P_x): it uniquely determines (P_x), if one does not require the fulfillment of the two principles indicated above. For the definition of the fiducial distribution on the basis of the so-called central function and the frequency interpretation of the fiducial distribution, see ((^2)).

Now one may introduce the fiducial distribution of a sample variable as the distribution of the sample variable when the unobserved parameter (\theta) has the fiducial distribution corresponding to some observation (x).

Example. Let (x_1,\ldots,x_n) be independent observations on a random variable from an (r)-dimensional normal population (N(\mu,A)).

Case 1. (A) is known, (\mu) is unknown. (\mathcal P) is the family of distributions of sufficient statistics
[
\bar{x}=\frac1n(x_1+\ldots+x_n),
]
corresponding to different values of (\mu); (X) is the Euclidean space (E_r); (G) is the group of translations in (E_r).

Case 2. (A) is unknown, (\mu) is known and (\mu=0). Denote by (G^{-}) the group of lower triangular matrices of dimension (r\times r) with positive elements on the main diagonal. Let (\mathcal P) be the family of distributions of sufficient statistics (t), corresponding to different values (a\in G^{-}); here (t) and (a) are determined uniquely (almost everywhere) by the requirement:
[
aa'=A,\quad tt'=T=\sum_1^n x_k x_k';\qquad a,t\in G^{-};\quad n\ge r.\qquad X=\Omega=G=G^{-}.
]

Case 3. (A) is unknown, (\mu) is unknown; (\mathcal P) is the family of distributions of sufficient statistics ((s,\bar{x})), corresponding to different ((a,\mu)\in G^{-}\times E_r); here (a) and (s) are determined uniquely (almost everywhere) by the requirement:
[
aa'=A,\quad ss'=S=\frac1{n-1}\sum_1^n(x_k-\bar{x})(x_k-\bar{x})';\qquad a,\ s\in G^{-},
]

(X=\Omega=G^{-}\times E_r.\quad G={[a,\mu]\mid a\in G^{-},\ \mu\in E_r}) is a group with group operation ([a,\mu][s,x]=[as, ax+\mu]).

Let (W^{-}(r,n,B)) be a distribution concentrated on the set of positive definite matrices of dimension (r\times r), with density

[
p(A)=\gamma_0(r,n)\frac{|B|^{(n-r-1)/2}d_{-}(B)}{|A|^{n/2}d_{-}(A)}
\exp\left{-\frac{n}{2}\operatorname{tr}(A^{-1}B)\right},
]

where (d_{-}(A)) is the product of the principal minors. Denote by (S^{-}(r,n)) the distribution on (E_r) with density

[
p(t)=\gamma_1(r,n)\left[1+\frac{(t,t)}{n}\right]^{-(n+1)/2}
\frac{[1+(t,t)/n]^{(r-1)/2}}{\displaystyle\prod_{1}^{r-1}[1+(t,t)_k/n]},
]

where (t=(t_1,\ldots,t_r)); ((t,t)_k=t_1^2+\cdots+t_k^2). If the random variable (t) has distribution (S^{-}(r,n)) and (a\in G^{-}), then the distribution of the random variable (at) will be denoted by (K^{-}(r,n,A)), where (A=aa'). A fiducial random unobservable parameter and a fiducial sample variable will be marked by an asterisk above. Then, in case 1,

[
\mu^{}\in N\left(\bar{x},\frac{1}{n}A\right);\qquad
x^{}\in N\left(\bar{x},\frac{n+1}{n}A\right)
]

(the sign (\in) means that, for example, (x^{*}) has distribution (N\left(\bar{x},\frac{n+1}{n}A\right))).

In case 2

[
A^{}\in W^{-}(r,n,\hat{A});\quad
x^{}-\mu\in K^{-}(r,n,A);\quad
\hat{A}=\frac{1}{n}\sum_{1}^{n}(x_k-\mu)(x_k-\mu)'.
]

In case 3

[
\sqrt{n}(\mu^{}-\bar{x})\in K^{-}(r,n-1,S);\qquad
A^{}\in W^{-}(r,n-1,S);
]

[
x^{*}-\bar{x}\in K^{-}\left(r,n-1,\frac{n+1}{n}S\right).
]

Moscow State University
named after M. V. Lomonosov

Received
1 IX 1969

CITED LITERATURE

(^{1}) P. Halmos, Measure Theory, Moscow, 1953.
(^{2}) D. A. S. Fraser, Biometrika, 48, 261 (1961).