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a-test approximation beta distribution binomial bution called characteristic function Consider the test continuous distribution F(x converges in probability decision function degrees of freedom denote the number density or probability distri distributed random variables distribution with mean distribution with parameters estimator of 9 exponential family F-distribution finite given H0 is rejected hence hypothesis H0 identically distributed random independent identically distributed independent of 9 independent random variables inequality large values least-squares estimate Let Xu likelihood ratio limiting N(0 Manoukian 1986 matrix median noncentral chi-square distribution noncentrality parameter Nonparametric null hypothesis One-Sample order statistics parameter 9 Pitman asymptotic efficiency Poisson population probability density function probability mass function rank rejecting H0 respectively Roussas SC(x sequence Serfling standard normal distribution Student distribution sufficient statistic symmetric test of hypothesis Theorem Two-Sample unbiased estimator v2 degrees variance a2 Xn be independent Yj=i