Rong-Xian Yue

The Mean Square Discrepancy of Scrambled ( t , s )-Sequences

The discrepancy arises in the worst-case error analysis for quasi-Monte Carlo quadrature rules. Low discrepancy sets yield good quadrature rules. This article considers the mean square discrepancies for scrambled

META-ANALYSIS AND SENSITIVITY ANALYSIS FOR MULTI-ARM TRIALS WITH SELECTION BIAS

(\lam,t,m,s)

Optimal quadrature for Haar wavelet spaces

-nets and (t,s)-sequences in base b. It is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple Monte Carlo sampling. If the reproducing kernel defining the discrepancy satisfies a Lipschitz condition with respect to one of its variables separately, then the asymptotic order of the root mean square discrepancy is O(n-1[log n](s-1)/2) for scrambled nets. If the reproducing kernel satisfies a Lipschitz condition with respect to both of its variables, then the asymptotic order of the root mean square discrepancy is O(n-3/2[log n](s-1)/2) for scrambled nets. For an arbitrary number of points taken from a (t,s)-sequence, the root mean square discrepancy appears to be no better than O(n-1[log n](s-1)/2), regardless of the smoothness of the reproducing kernel.

Model-robust designs in multiresponse situations

Multi-arm trials meta-analysis is a methodology used in combining evidence based on a synthesis of different types of comparisons from all possible similar studies and to draw inferences about the effectiveness of multiple compared-treatments. Studies with statistically significant results are potentially more likely to be submitted and selected than studies with non-significant results; this leads to false-positive results. In meta-analysis, combining only the identified selected studies uncritically may lead to an incorrect, usually over-optimistic conclusion. This problem is known asbiselection bias. In this paper, we first define a random-effect meta-analysis model for multi-arm trials by allowing for heterogeneity among studies. This general model is based on a normal approximation for empirical log-odds ratio. We then address the problem of publication bias by using a sensitivity analysis and by defining a selection model to the available data of a meta-analysis. This method allows for different amounts of selection bias and helps to investigate how sensitive the main interest parameter is when compared with the estimates of the standard model. Throughout the paper, we use binary data from Antiplatelet therapy in maintaining vascular patency of patients to illustrate the methods.

The Discrepancy and Gain Coefficients of Scrambled Digital Nets

This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, Hwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands Hwav.

Statistical modeling for the optimal deposition of sputtered piezoelectric films

The multiresponse model E(y[alpha]x)=[summation operator]l=1p[alpha] [theta][alpha]lf[alpha]l(x)+h[alpha](x), [alpha]=1,...,r, is considered, where h[alpha](x) is an unknown bias or contamination function from some class with a probability measure. Optimal designs are studied in terms of generalized least squares estimation and the average expected quadratic loss. The performance of the uniform design is also explored.

On the variance of quadrature over scrambled nets and sequences

Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (?, t, m, s)-net in base b with n=?bm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error.

Strong tractability of integration using scrambled Niederreiter points

The quality of sputtered-deposited piezoelectric films used for integrating bulk acoustic wave (BAW) and surface acoustic wave (SAW) devices with semiconductor circuitry depends on several deposition parameters, including substrate temperature, background pressure, gas composition, gas flow rate, and deposition rate. It is desirable to establish the fabrication process based on a selection of the controllable parameter values that optimizes the film quality. It is common practice to perform a number of deposition experiments by varying the controllable parameters to determine the optimal film growth conditions. The films are grown under a number of different conditions within this space and a response parameter related to film performance is measured. Then a multiple linear regression model is fit to the data. By optimizing the fitted response, the best growth conditions can be obtained. This approach is illustrated with data from recent work on the development of very high quality magnetron sputtered aluminum nitride (AlN) films whose acoustic characteristics are like those of epitaxial films grown at considerably higher substrate temperatures. Because the resource cost involved can be high, depending upon the number of deposition runs made, it is desirable to minimize the number of experiments and maximize the amount of information gained from them. A discussion is given on how the statistical theory of experimental design can be used to obtain this goal.

Integration and Approximation Based on Scramble Sampling in Arbitrary Dimensions

The randomization of a (t,m,s)-net or a (t,s)-sequence proposed by Owen (1995, Monte Carlo and quasi-Monte Carlo Methods in Scientific Computing (Berlin), Lecture Notes in Statistics, vol. 106, pp. 299-317) combines the strengths of equidistribution and Monte Carlo integration. This paper shows that the order of variance found in Owen (1997b, Scrambled net variance for integrals of smooth functions. Ann. Statist. 25, 1541-1562) still holds for weaker smooth integrands. The main results in this paper are as follows: For the univariate Lipschitz integrands on [0,1), the variance over scrambled ([lambda],t,m,1)-nets is of order O(n-3); In the s-dimensional case, if the integrand satisfies a generalized Lipschitz condition on [0,1)s, then the variance over scrambled ([lambda],t,m,s)-nets is of order O(n-3(log n)s-1); Also shown in this paper is that the variance over a scrambled (t,s)-sequence is of order O(n-2(log n)s-1) for the integrands described above. These results seem to be sharper than any similar results in the literature.

A comparison of random and quasirandom points for nonparametric response surface design

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The e-exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.