On Asymptotic Standard Normality of the Two Sample Pivot

Abstract

The asymptotic solution to the problem of comparing the means of two heteroscedastic populations, based on two random samples from the populations, hinges on the pivot underpinning the construction of the confidence interval and the test statistic being asymptotically standard Normal, which is known to happen if the two samples are independent and the ratio of the sample sizes converges to a finite positive number. This restriction on the asymptotic behavior of the ratio of the sample sizes carries the risk of rendering the asymptotic justification of the finite sample approximation invalid. It turns out that neither the restriction on the asymptotic behavior of the ratio of the sample sizes nor the assumption of cross sample independence is necessary for the pivotal convergence in question to take place. If the joint distribution of the standardized sample means converges to a spherically symmetric distribution, then that distribution must be bivariate standard Normal (which can happen without the assumption of cross sample independence), and the aforesaid pivotal convergence holds.! " # ! . .MSC 2010 subject classifications: 62E20, 62G20.

A Pedagogical Note on Asymptotic Normality of a Two-Sample Approximate Pivot for Comparing Means

“…This concept of distributional convergence is abundantly clear when n 1 = O(n 2 ), a situation that was emphasized in Mukhopadhyay [1,p.544] and few other sources cited by Majumdar and Majumdar [2] . We add some simple details in Section 2.2 for completeness.…”

“…A two-sample pivot for comparing the means µ 1 , µ 2 from two independent populations 1 and 2 with variances σ 2 1 , σ 2 2 respectively is well known. Having recorded independent observations X i,1 , ..., X i,n i of size n i from population i, one customarily uses the sample mean X i,n i (≡ n −1 i n i j=1 X i,j ) to estimate the population mean µ i , i = 1, 2 respectively.…”

“…Part (2) looks similar in spirit to part (1) except that in part (2), the sample variance S 2 i,n i , that is (n i − 1) −1 n i j=1 (X i,j − X i,n i ) 2 , replaces the population variance σ 2 i , i = 1, 2 respectively. In all fairness, however, neither W n 1 ,n 2 nor W n 1 ,n 2 may be a true pivot because their exact distributions may involve µ 1 , µ 2 , but we informally continue to refer to them as pivots.…”

“…Recently, Majumdar and Majumdar [2] briefly reviewed the literature on possible interpretation(s) of the phrase large-sample property (1.2). They nailed down precise statements of the asymptotic nature of the distributional convergences of the pivots from (1.1) via mathematical concepts of net and subnet.…”

“…The issue hinges upon sorting out the process of how n 1 , n 2 may converge to infinity -would this be a double limit or an iterated limit? This led Majumdar and Majumdar [2] to lay down an elegant treatment for precise understanding of both the double limit as well as the iterated limit of F n 1 ,n 2 (w) to (w) pointwise. However, a practicing statistician who routinely employs approximate pivots from (1.1) may feel genuinely frustrated to walk through a maze of such a treatment given that (1.2) has been taken for granted by nearly all users through close to eternity.…”