On a Generalized Savage Statistic with Applications to Life Testing

Let there be two samples X₁, X₂, , Xₘ and Y₁, Y₂, , Yₙ (N = m + n) from two populations with continuous cdf's F (x) and G (y). Let the first i ordered observations (out of N combined observations) contain mᵢx's and nᵢy's (mᵢ + nᵢ = i) where mᵢ and nᵢ are random numbers. To test equation*0. 1H₀: F = Gequation* against alternative that they are different we propose the statistic equation*0. 2S^ (N) ᵣ = ʳ₈ = ₁ aᵢzᵢ + (m - mᵣ) (N - r) ^-1 (Nₑ+₁aᵢ) - 12 (m + n) equation* based on the first r ordered observations only where aNᵢ = aᵢ = N₉=₍-₈+₁ 1/j, and alignzᵢ &= 1, if the ith ordered observation is an xᵢ, \\ &= 0, otherwise. align The statistic is the asymptotically most powerful rank test for censored data under the Lehmann alternative and is equivalent to the Savage statistic 14 when r = N. It is also known to maximize the minimum power over IFRA (or IFR) distributions asymptotically. Exact and large sample properties of S^ (N) ᵣ are studied and a k-sample extension of it is also considered. Various tables are also provided to facilitate the use of the S^ (N) ᵣ statistic.