Normal approximation and smoothness for sums of means of lattice-valued random variables.(English)Zbl 1291.62046

Let $$\hat{\theta}$$ be a given statistic for which a central limit theorem applies. To obtain estimates for the difference between the exact distribution of $$T=(\hat{\theta} - E[ \hat{\theta}])/\sqrt{\mathrm{Var} ( \hat{\theta})}$$ and the standard normal distribution $$\Phi (x)$$, several expansions for the distribution of $$T$$ are available. An important class of these expansions is given by Edgeworth expansions, which are expansions of the form $P (T \leq x) = \Phi (x) + \sum_{j=0}^{r} \frac{p_j(x)}{n^{j/2}} \, \phi (x) + O(n^{-(r+1)/2}), \qquad r \geq 0,$ where $$p_0 (x) \equiv 0$$, $$\phi (x)$$ is the derivative of $$\Phi (x)$$, and for $$j \geq 1$$, $$p_j(x)$$ are polynomials whose coefficients depend on the cumulants of $$\hat{\theta} - E[ \hat{\theta}]$$. For examples, see [P. Hall, The bootstrap and Edgeworth expansion. Springer Series in Statistics. New York etc.: Springer-Verlag. (1992; Zbl 0744.62026)], and [X. H. Zhou, C. M. Li and Z. Yang, “Improving interval estimation of binomial proportions”, Phil. Trans. Roy. Soc. Ser. A 366, 2405–2418 (2001)].
In this article the authors investigate the first order Edgeworth expansions of sums of independent means of independent lattice-valued random variables. Sums or differences of binomial proportions are special cases of the problem under investigation. Let $$\{X_{j1}, X_{j2}, \dots , X_{jn_j}\}$$, $$j=1,2,\dots , k$$, $$k\geq 2$$, be $$k$$ independent samples of independent lattice-valued random variables, with $$E[|X_{j1}|^3]< +\infty$$. Put $$\mu_j = E(X_{j1})$$, $$\sigma_j^2 = \mathrm{Var} (X_{j1})$$, $$\overline{X}_j = n_j^{-1} \sum_i X_{ji}$$, and $$S = \sum_{j=1}^k \overline{X}_j$$. Under these assumptions one would expect $$S$$ to have a first order Edgeworth expansion of the form $P\left(\frac{S - E(S)}{\sqrt{\mathrm{Var} (S)}}\leq x \right) = \Phi (x) + \frac{\beta (1-x^2) \phi (x) }{6 \sqrt{n}} + \frac{d_n(x) \phi (x)}{\sqrt{n}} + o(n^{-1/2}),$ where $$n = n_1 + \dots + n_k$$, $\beta = \beta (n) = \frac{\sqrt{n} \, E[(S - E(S))^3]}{{(\mathrm{Var} (S))^{3/2}}},$ and $$d_n$$ is a discontinuous term in general needed when dealing with lattice distributions see [C.-G. Esseen, Acta Math. 77, 1–125 (1945; Zbl 0060.28705)]. The terms $$d_n$$ are often referred to as continuity corrections.
The authors investigate the distribution of S, and describe a methodology and conditions under which continuity corrections are not needed for this multi-sample problem. Specifically, suppose that the sample sizes $$n_1, n_2, \dots , n_k$$ are changing in such a way that the correspondent sequence of values of $$n$$ is strictly increasing, and that $\min_{1 \leq j \leq k} \liminf_{n\rightarrow +\infty} \frac{n_j}{n} > 0.$ Let $$e_j$$ denote the span of the distribution of $$X_{j1}$$, and for every $$1 \leq j_1 < j_2 \leq k$$ put $$\rho_{j_1j_2} = (e_{j_2}n_{j_1})/ (e_{j_1}n_{j_2})$$. The authors prove that if for at least one of the $$\rho_{j_1j_2}$$, $\lim_{n\rightarrow +\infty} \sqrt{n} \, | \sin (l \rho_{j_1j_2} \pi)| = +\infty \qquad \text{ for every positive integer $$l$$, }$ then $P\left(\frac{S - E(S)}{\sqrt{\mathrm{Var} (S)}}\leq x \right) = \Phi (x) + \frac{\beta (1-x^2) \phi (x) }{6 \sqrt{n}} + o(n^{-1/2})$ holds uniformly in $$x$$. The authors also give conditions under which a continuity correction $$d_n$$ is needed, and for the case $$k=2$$, $$d_n$$ is derived. Extensions to problems where distributions are estimated using the bootstrap are also given.

MSC:

 62E17 Approximations to statistical distributions (nonasymptotic) 60F05 Central limit and other weak theorems 62F40 Bootstrap, jackknife and other resampling methods

Citations:

Zbl 0744.62026; Zbl 0060.28705
Full Text:

References:

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