Two-level orthogonal arrays of strength 3 permit the study of the main effects and the two-factor interactions of the experimental factors. These arrays are classified into regular and nonregular ...designs. Good regular designs are available in the literature for large run sizes that are a power of 2. In contrast, good nonregular designs, which have run sizes that are multiples of 8 and are more flexible alternatives to regular designs, are not available for large numbers of runs because their construction is challenging. In this article, we introduce a collection of strength-3 nonregular designs with large run sizes that, to the best of our knowledge, have not been explored before in the design literature. Using theoretical results and algorithmic approaches, we construct nonregular designs with up to 1280 runs. Our designs fill the gaps between the available strength-3 designs with large run sizes and outperform many comparably sized benchmark designs in terms of the aliasing among the two-factor interactions. We show the applicability of our collection of strength-3 designs using an infrared sensor experiment. Supplementary materials for this article are available online.
The maximum-length sequences, also called m-sequences, have received a lot of attention since the late 1960s. In terms of linear-feedback shift register (LFSR) synthesis they are usually generated by ...certain power polynomials over a finite field and in addition are characterized by a low cross correlation and high nonlinearity. We say that such a sequence is generated by a semi-bent function. Some new families of such function, represented by f(x)=/spl Sigma//sub i=1//sup (n-1)/2/c/sub i/Tr(x(2/sup i/)+1), n odd and c/sub i//spl isin/F/sub 2/, have recently (2002) been introduced by Khoo et al. We first generalize their results to even n. We further investigate the conditions on the choice of c/sub i/ for explicit definitions of new infinite families having three and four trace terms. Also, a class of nonpermutation polynomials whose composition with a quadratic function yields again a quadratic semi-bent function is specified. The treatment of semi-bent functions is then presented in a much wider framework. We show how bent and semi-bent functions are interlinked, that is, the concatenation of two suitably chosen semi-bent functions will yield a bent function and vice versa. Finally, this approach is generalized so that the construction of both bent and semi-bent functions of any degree in certain range for any n/spl ges/7 is presented, n being the number of input variables.
Kai Zhou (2008)
8 gave an explicit representation of the class of linear permutation polynomials and computed the number of them. In this paper, we give a simple proof of the above results.
We study permutation polynomials of the shape
G
(
X
)
+
γ
Tr
(
H
(
X
)
)
in
F
p
n
X
. Using a link with functions having a linear structure, we introduce an effective method to construct many such ...permutations, as well as
p-to-1 mappings.
Tests based on the permutation of observations are a common and attractive method of comparing two groups of outcomes, in part because they retain proper test size with minimal assumptions and can ...have high efficiency toward specific alternatives of interest. In addition, permutation tests may be used with discrete or categorical outcomes, for which linear rank tests are not designed. Permutation tests are now increasingly used to analyze discrete or continuous responses that themselves are functions of several statistics. Examples of such summary statistics include the area under the curve generated by repeated measures of a laboratory marker or an overall composite score from a quality of life study. Here even simple structures for the joint distribution of the component statistics can lead to complex differences between the distributions of summary statistics of the comparison groups. Despite their attractive features, surprisingly little is known about the behavior of linear permutation tests when the two groups differ even in simple ways. This lack of knowledge limits an assessment of the relative efficiency of different tests or the planning of the size of a study based on a permutation test. To address these issues, we derive the asymptotic distribution of permutation tests under a general contiguous alternative, and then investigate the implications for test selection and study design for several diverse areas of application. For discrete outcomes, areas of application include permutation tests for ordinal responses and for count data. For continuous outcomes, we explore several applications, including general results for location-scale families, a comparison of different data transformations, and a comparison to linear rank tests.
Probability
Statistics for Scientists and Engineers,
07/2015
Book Chapter
This chapter introduces the concepts, tools, and techniques of probability in an intuitive way. The study of probability became an essential part of statistics owing to the obvious reason that ...probability is deep rooted in a great majority of statistical models and procedures. Popular ways to express a probability are fractional form, decimal form, scientific form, percentage form, literal form, pictorial form, and as tail areas under empirical curves or functions. The chapter focuses mostly on probability mechanisms involved in random experiments. Events can be combined using set theoretic operations union, intersections, complements, and differences. The chapter discusses sample space, permutations and combinations, and also about the principle of inclusion and exclusion (PIE). This is followed by a discussion on recurrence relations, urn models and partitions. Finally, the chapter talks about conditional probability, including Bayes theorem.
Interpoint-distance-based graphs can be used to define measures of association that extend Kendall's notion of a generalized correlation coefficient. We present particular statistics that provide ...distribution-free tests of independence sensitive to alternatives involving non-monotonic relationships. Moreover, since ordering plays no essential role, the ideas are fully applicable in a multivariate setting. We also define an asymmetric coefficient measuring the extent to which (a vector) X can be used to make single-valued predictions of (a vector) Y. We discuss various techniques for proving that such statistics are asymptotically normal. As an example of the effectiveness of our approach, we present an application to the examination of residuals from multiple regression.
Let Ω denote the set of permutations performable by a pass through the Omega network (an n-stage shuffle-exchange network), and let π denote an arbitrary permutation of N = 2n elements. In this paper ...we characterize the sets {π|πΩ = Ω} (called left invariants) and {π|Ωπ = Ω} (called right invariants). This characterization provides a useful criterion for determining whether or not the composition of two permutations belongs to Ω; it also allows a number of known results to be generalized and their derivations simplified.
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