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MR4488251 62G10 62G07 62G20
Ghosh, Santu [Ghosh, Santu2 ] (1-AUGU) ; Polansky, Alan M. (1-NIL-NDM)
Large-scale simultaneous testing using kernel density estimation. (Englishsummary)
Sankhya A 84 (2022), no. 2, 808–843.

The main contribution of this article is to demonstrate that when suitably implemented,
the two-sample t-test statistic using kernel density estimators (KDEs) can improve theerror rate of a normal approximation to its p-value by an order of magnitude. Letd{Xj }nj=1 ∈ Rd and {Yj }m
j=1 ∈ R be i.i.d. random samples drawn independently fromtwo populations X and Y. For notational simplicity, the i-th component of W ∈ {X, Y}
is denoted as W(i) hereinafter. To identify different component-wise features, consider
d sets of null and alternative hypotheses
H0i : µX(i) = µY(i)vs H1i : µX(i) 6= µY(i) ,i = 1, . . . , d,where µW(i) = E(W(i)).
To construct the KDE-based two-sample t-statistic, for i = 1, . . . , d, letnm1 X
Xj (i) − ·Yj (i) − ·1 X
fbX(i) (·) :=KKand fbY(i) (·) :=nhi j=1hi
mhi j=1hi√
2
be the KDEs of X(i) and Y(i), where K(u) = e−u /2 / 2π is the univariate Gaussian
kernel, and hi > 0 is the common bandwidth that shrinks toward zero at a certain rate.
bW(i) be the cumulative distribution function (CDF) implied by the KDE
Also, let F
fbW(i) . The test statistic for the testing of H0i against H1i is defined as
ei := q µ̃X(i) − µ̃Y(i)T,e 2 /n + S
e 2 /mSX(i)Y (i)2e2
where µ̃W(i) and SW(i) are plug-in estimators of µW(i) and σW(i) = Var(W(i)) basedbW(i) , respectively.
on FA benefit of employing the Gaussian kernel is that only the second-order cumulant ofbW(i) differs from that of the corresponding empirical CDF Gb W(i) (by a margin of h2 ).
Fiei can be rewritten asIt follows that Tµ̄X(i) − µ̄Y(i)ei = qT,
222SX(i) /n + SY(i) /m + hi (1/n + 1/m)
2where µ̄W(i) and SW(i) are the sample mean and variance of W(i), respectively. Observe
ei collapses to the conventional two-sample t-statisticthat when hi = 0, T

Ti := qµ̄X(i) − µ̄Y(i)22SX(i) /n + SY (i) /m.

ei takes some value t̃i . Then, an Edgeworth expansion of the p-value
Suppose that TeP(|Ti | ≥ |t̃i |) yields
ei | ≥ |t̃i | = P(|Z| ≥ |t̃i |) − C1 h2i − C2 N −1 + O(N −τ ), N = n + m → ∞,P |Twhere Z is a standard normal random variable, C1 > 0 and C2 are constants that dependon population moments of W(i), the standard normal density φ(·) and CDF Φ(·), andτ = min{2k + 1/2, 3/2} for some k > 0 satisfying h2i = O(N −k ). It follows that if C2 < 0,then putting h2i = −(C2 /C1 )N −1 leads toei | ≥ |t̃i | = 2 1 − Φ(|t̃i |) + O(N −3/2 ), N → ∞.
P |TOn the other hand, by an Edgeworth expansion of the p-value for the conventional
two-sample t-statistic,P |Ti | ≥ |t̃i |) = 2 1 − Φ(|t̃i |) + O(N −1 ), N → ∞.
Therefore, the two-sample t-statistic constructed by a smoothed version of the empirical
CDF can reduce the normal approximation error by an order of magnitude.
This article establishes that the O(N −3/2 ) approximation error can be attained evenafter all population moments are replaced by their sample counterparts. It is also
demonstrated that when C2 > 0, the KDE-based t-statistic using properly rescaled
observations can retrieve the O(N −3/2 ) rate. Furthermore, it is shown that as long as
the total number of tests d diverges but d = o(N 3/2 ), the KDE-based test can control
the false discovery rate.
Masayuki Hirukawa