Gerhard Tintner, The Distribution of Symmetric Quadratic Forms in Normal and Independent Variables (1939)

Gerhard Tintner’s research article derives a distribution for a particular quadratic form in independent standard normal variables. Its scope is narrower than the title might suggest: every squared variable has the same coefficient (a), and every distinct cross-product has the same coefficient (2b). This uniform structure makes the calculation tractable. The paper proceeds from a Gaussian integral to a factored characteristic function, then to an inverse Fourier transform, a hypergeometric expression, and finally a series intended for calculation with existing chi-square tables. Its central contribution is the reduction of a seemingly many-variable distribution problem to two characteristic factors determined by the symmetry of the coefficients.

We want to establish the distribution of the quadratic form Q by means of the method of characteristic functions.

The method matters because Tintner does not begin by transforming the probability density directly into a density for (Q). He instead integrates (e^{iyQ}) against the joint normal density. Although the opening description says only that the variables are normally and independently distributed, equation (1) specifies zero means and unit variances. The quadratic form is therefore (Q=a\sum_i x_i^2+2b\sum_{i<j}x_ix_j). Independence belongs to the underlying variables, not to the separate terms of this expression: cross-products share variables, so their distributions cannot simply be combined as though all terms were independent.

The decisive conceptual move is an orthogonal transformation of the quadratic expression inside the Gaussian integral. Its coefficient matrix has constant diagonal and constant off-diagonal entries. Tintner evaluates the corresponding determinant as the product of one factor involving (a+(N-1)b) and another involving (a-b), raised to the power (N-1). These are the two eigenvalues of the original quadratic form: the first belongs to the direction in which all coordinates move together, while the second belongs to the perpendicular space of contrasts. Thus the algebra distinguishes one collective component from (N-1) equivalent remaining components, without requiring a separate calculation for each variable.

Integration gives the article’s clearest compact result: [ g(y)={1-2iy[a+(N-1)b]}^{-1/2} [1-2iy(a-b)]^{-(N-1)/2}. ] In probabilistic terms, this factorization corresponds to a sum of two independently scaled chi-square variables, with one and (N-1) degrees of freedom respectively. That interpretation explains why the number of original variables enters mainly through a multiplicity. It also provides a useful distinction between the general characteristic-function result and the more restricted density manipulations that follow: the scales can be positive, negative, or zero, with consequences for the support and form of the distribution.

Tintner next applies Fourier inversion. He rewrites both characteristic factors around the expression containing (a+(N-1)b), then expands the remaining factor in a binomial series. Each resulting term has the same basic inverse-transform structure, but with a successively increased exponent. A stated residue calculation supplies the integral needed for term-by-term inversion. The resulting density is presented first as an infinite series and then through ({}_1F_1), which the article calls the generalized hypergeometric function. This progression makes the special-function representation an outcome of the expansion rather than an unexplained substitution: its parameters encode the multiplicities already exposed by the determinant.

The final step translates that formal result into a familiar statistical vocabulary. Tintner introduces the chi-square density and rewrites the expression using densities with degrees of freedom increasing by two.

Then the distribution of Q can be expressed in terms of these distribution.

The grammatical irregularity in this sentence does not obscure its purpose. The successive terms involve (F_N), (F_{N+2}), (F_{N+4}), and so on, with coefficients inherited from the binomial expansion. These alternating terms should not be read as a probability mixture with nonnegative weights. They are a computational series. A footnote credits this formula to S. S. Wilks, distinguishing the concluding representation from the preceding derivation while locating the paper within contemporary work on statistical inference.

This form is probably the most convenient one for calculations, since the $\chi^2$ distribution has been frequently tabulated.

The closing sentence identifies the practical rationale for the article’s structure. A hypergeometric expression offers formal compactness, but a representation using tabulated chi-square densities promises easier numerical evaluation. The references to work on second-order moment statistics, Fourier integrals, special functions, and statistical tables likewise reflect the paper’s movement between probability theory and usable calculation. Tintner’s contribution is methodological; the article supplies no dataset or worked empirical application.

The supplied text nevertheless requires caution at the density stage. The displayed residue formula and subsequent density expressions carry leading negative signs that would produce a negative density in ordinary positive-scale cases. The normalization also appears inconsistent with standard Fourier inversion. Moreover, the exposition begins with arbitrary real coefficients but does not distinguish positive-definite, indefinite, and degenerate cases, or establish the conditions justifying the series expansion and termwise integration. These are concrete limitations of the displayed derivation, not grounds for silently correcting it. The determinant and characteristic-function factorization remain the strongest results to carry forward; the printed density formulas need verification before numerical use. Read with that qualification, the article demonstrates how coefficient symmetry compresses a Gaussian quadratic-form problem and how special-function expressions can be recast toward the computational resources available in 1939.