Asymptotic Efficiency of Certain Locally Most Powerful Rank Tests

We are given independent random samples X₁, ⋯, Xₘ and Y₁, ⋯, Yₙ from populations with unknown cumulative distribution functions (cdf's) FX and FY, respectively. It is desired to test Hₒ:FX = FY against H₁ : FX = G_θ, FY = G_φ,θ, φ ε R, where G_θ is a specified family of cdf's (one for each θ), R is an interval on the real line, θ and φ are specified and very close to some specified value φₒ, and θ ≠ φ. A theorem of Hoeffding is used to show that the locally most powerful rank test (L.M.P.R.T.) of Hₒ against H₁ is based on a linear rank statistic TN = (m)⁻¹ ∑Ni = 1 aNiZNi, where ZNi = 1 when the ith smallest of $N = m + n$ observations is an X, and ZNi = 0, otherwise, and the aNi are given numbers. In a recent paper, Chernoff and Savage established the asymptotic normality of the test statistic TN, subject to some weak restrictions. The concept of asymptotic relative efficiency (A.R.E.) was introduced by Pitman to compare sequences of tests. It was pointed out by Chernoff and Savage that the asymptotic efficiency of a sequence of tests can be established by means of a likelihood ratio test. Using this method, in conjunction with the theorem of Chernoff and Savage on asymptotic normality, it is shown that the L.M.P.R.T. of Hₒ against H₁ is asymptotically efficient. Several applications to Cauchy, exponential, and normal populations are given.