Abstract Generated abstract
This paper defines fiducial distributions for the parameters of a multivariate normal population under three standard cases: known covariance with unknown mean, unknown covariance with known mean, and both mean and covariance unknown. The definitions are formulated through equalities with invariant confidence set systems, reflecting translation invariance, linear coordinate transformations, and changes of measurement units as appropriate. The paper then derives the corresponding fiducial laws for the unknown parameters and the induced fiducial distributions of a future sample variable, expressed in terms of normal, inverse Wishart type, and multivariate Student type distributions.
Full Text
UDC 519.21
MATHEMATICS
G. P. KLIMOV
DEFINITION OF THE FIDUCIAL DISTRIBUTION FOR A MULTIVARIATE NORMAL POPULATION
(Presented by Academician A. N. Tikhonov, 4 IX 1969)
Let \(x_1,\ldots,x_n\) be independent observations of a random variable (r.v.) from an \(r\)-dimensional normal population \(N(\mu,A)\).
Case 1. \(A\) is known, \(\mu\) is unknown. A sufficient statistic is
\[ \bar{x}=\frac{1}{n}(x_1+\cdots+x_n). \]
For every \(\bar{x}\), define the fiducial distribution (fid. d.) \(P_{\bar{x}}\) of the parameter \(\mu\) by the equality
\[ P_\mu\{\mu\in S(\bar{x})\}=P_{\bar{x}}\{\mu\in S(\bar{x})\} \]
for every system of confidence (measurable) sets \(S(\bar{x})\) satisfying the condition
\[ \mu\in S(\bar{x}) \Longleftrightarrow \mu+\mu_0\in S(\bar{x}+\mu_0) \]
for any \(r\)-dimensional vector \(\mu_0\).
Case 2. \(A\) is unknown, \(\mu\) is known and \(\mu=0\). A sufficient statistic is
\[ T=\sum_1^n x_k x_k'. \]
For almost every, with \(n\ge r\), positive definite matrix \(T\), define the fid. d. \(P_T\) of the parameter \(A\) by the equality
\[ P_A\{A\in S(T)\}=P_T\{A\in S(T)\}; \]
for any system of confidence (measurable) sets \(S(T)\) satisfying the condition
\[ A\in S(T)\Longleftrightarrow A_C\in S(T_C), \]
where \(A_C=CAC'\), \(T_C=CTC'\), and \(C\) is an arbitrary nonsingular matrix of dimension \(r\times r\). Since any nonsingular real matrix \(C\) can be represented as the product of a finite number of matrices, each of which is either diagonal with positive elements on the main diagonal or orthogonal, the last condition means that the confidence sets are invariant with respect to the choice of the units of measurement and of the orthogonal coordinate system in which the sample variable is measured.
Case 3. \(A\) is unknown, \(\mu\) is unknown. A sufficient statistic is the pair \((T,\bar{x})\), where
\[ T=\sum_1^n (x_k-\bar{x})(x_k-\bar{x})'. \]
For every pair \((T,\bar{x})\), define the fid. d. \(P_{(T,\bar{x})}\) of the parameter \((A,\mu)\) by the equality
\[ P_{(A,\mu)}\{(A,\mu)\in S(T,\bar{x})\} = P_{(T,\bar{x})}\{(A,\mu)\in S(T,\bar{x})\} \]
for any system of confidence (measurable) sets \(S(T,\bar{x})\) satisfying the condition:
\[ (A,\mu)\in S(T,\bar{x})\Longleftrightarrow (A_C,C\mu+\mu_0)\in S(T_C,C\bar{x}+\mu_0) \]
for any nonsingular matrix \(C\) and any \(r\)-dimensional vector \(\mu_0\). The last condition means that the confidence sets are invariant with respect to the choice of the coordinate system in which the sample variable is measured.
Now one may introduce the fiducial distribution of a sample variable as the distribution of the sample variable when the unobserved unknown parameter has the fiducial distribution corresponding to some value of a sufficient statistic.
Denote by \(W^{*}(r,n,B)\) the distribution concentrated on the set \(A_r\) of positive definite matrices of dimension \(r \times r\), with density
\[ p(A)=\gamma_0(r,n)\frac{|B|^{n/2}}{|A|^{(n+r+1)/2}} \exp\left\{-\frac{n}{2}\operatorname{tr}(A^{-1}B)\right\},\qquad n\geqslant r, \]
where \(\gamma_0(r,n)=n^{nr/2}\gamma(r,n)\), and \(\gamma(r,n)\) is the normalizing factor in the Wishart distribution. For every matrix \(A\in A_r\), let \(K(r,n,A)\) denote the distribution concentrated on the Euclidean space \(E_r\), with density
\[ p(x)=\gamma_1(r,n)\frac{1}{|A|^{1/2}} \left[1+\frac{(A^{-1}x,x)}{n}\right]^{-(n+1)/2}; \qquad \gamma_1(r,n)=(\pi n)^{-r/2}\times \]
\[ \times \frac{\Gamma((n+1)/2)}{\Gamma((n-r+1)/2)}. \]
We shall mark the fiducial unobserved parameter and the fiducial sample variable with an asterisk as a superscript. 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 symbol \(\in\) means that, for example, \(\mu^{*}\) has distribution \(N\left(\bar{x},\frac{1}{n}A\right)\)).
In case 2,
\[ A^{*}\in W^{*}(r,n,\hat{A});\qquad x^{*}-\mu\in K(r,n,\hat{A});\qquad \hat{A}=\frac{1}{n}\sum_{1}^{n}(x_k-\mu)(x_k-\mu)'. \]
In case 3,
\[ \sqrt{n}(\mu^{*}-\bar{x})\in K(r,n-1,S);\qquad A^{*}\in W^{*}(r,n-1,S); \]
\[ x^{*}-\bar{x}\in K\left(r,n-1,\frac{n+1}{n}S\right);\qquad S=\frac{1}{n-1}\sum_{1}^{n}(x_k-\bar{x})(x_k-\bar{x})'. \]
Moscow State University
named after M. V. Lomonosov
Received
1 IX 1969
CITED LITERATURE
- G. P. Klimov, DAN, 191, No. 4 (1970).