Computation of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Non-central Wishart Matrix

We give an approximate formula for the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for the general dimension. The formula is expressed in terms of a definite integral with parameters. We derive a differential equation satisfied by the integral for the $2 \times 2$ matrix case and perform a numerical analysis of it.

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