Sankhya: The Indian Journal of Statistics

1968, Volume 30, Series A, Pt. 2, pp. 167--180

SOLUTIONS TO SOME FUNCTIONAL EQUATIONS AND THEIR APPLICATIONS TO CHARACTERIZATION OF PROBABILITY DISTRIBUTIONS

By

C. G. KHATRI and C. RADHAKRISHNA RAO, Indian Statistical Institute

SUMMARY. Three sets of results are contained in this paper. The first is on a new matrix product. If $A$ and $B$ are two matrices of orders $p\times r$ and $q\times r$ respectively, and if $\alpha_1, \cdots, \alpha_r$ are column vectors of $A$ and $\beta_1, \cdots, \beta_r$ are those of $B$ then the new product $A\odot B$ is the partitioned matrix $$(\alpha_1\otimes \beta_1 \vdots \alpha_2\otimes \beta_2 \vdots \cdots \vdots \alpha_r\otimes \beta_r)$$ where \otimes denotes the Kronecker product. Propositions involving the new product of matrices are stated. The second is on the solution of functional equations of two types. One is of the form $$\sum_{u-1}^{p}c_{ju}\psi_u({\bf e}^\prime_u{\bf t})+ \sum_{i=1}^{r}b_{ji}\phi_i({\bf \alpha}^\prime_i{\bf t})=g_j{\mbox{(constant)}}\,j=1,\cdots,q$$ involving a vector variable $t$ where $e_u$ are unit vectors of an identity matrix of order $p$, $\alpha_i$ are given column vectors and $\psi_u$, $\phi_i$ are unknown continuous functions. Another is of the form $$\sum_{j=1}^nd_{ij}\phi(b_j t)=g_i\, i=1, \cdots q$$ involving an unknown function $\phi$ of a single variable $t$. Conditions under which the unknown functions in these two types of equations are polynomials of an assigned degree are given. The third, on the characterization of normal and gamma distributions, extends the earlier work of the authors (Rao, 1967 and Khatri and Rao, 1968). We consider two sets of functions $L_1, \cdots, \L_q$ and $M_1, \cdots, M_p$ of the independent random variables $X_1, \codts, X_n$ with the condition $$E(L_i|M_1, \cdots, M_p)=g_i{\mbox{(constant)}}$$ for $i=1, \cdots, q$. When $L_i$ and $M_j$ are linear, the $X_i$ have normal distributions. When $L_i$ are linear in the reciprocals of the variables and $M_j$ are linear in the variables, the $X_i$ have gamma or conjugate gamma distributions. When the $X_i$ variables are non-negative, $L_i$ are linear in the variables, and $M_j$ are linear in the logarithms of the variables, the $X_i$ have gamma distributions. These results are proved under some conditions on the compounding coefficients for $p> 1$, and in the case of $p=1$ with the further condition that the $X_i$ are identically distributed.

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