Multinomial distribution

Summary

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided dice rolled n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories. When k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial distribution. When k is bigger than 2 and n is 1, it is the categorical distribution. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so n determines the suffix, and k the prefix). The Bernoulli distribution models the outcome of a single Bernoulli trial. In other words, it models whether flipping a (poss

Official source

https://en.wikipedia.org/wiki/Multinomial_distribution

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Statistical methods for insect choice experiments

Ingrid Ricard

Olfactometer experiments are used to determine the effect of odours on the behaviour of organisms such as insects or nematodes, and typically result in data comprising many groups of small counts, overdispersed relative to the multinomial distribution. Overdispersion reflects a lack of independence or heterogeneity among individuals and can lead to statistics having larger variances than expected and possible losses of efficiency. In this thesis, some distributions which consist of generalisations of the multinomial distribution have been developed. These models are based on non-homogeneous Markov chain theory, take the overdispersion into account, and potentially provide a physical interpretation of the overdispersion seen in olfactometer data. Some inference aspects are considered, including comparison of the asymptotic relative efficiencies of three different sampling schemes. The fact that the empirical distributions well approximate the corresponding asymptotic distributions is checked. Observable differences in parameter estimates between data generated under different hypotheses are also studied. Finally, different models intended to shed light on various aspects of the data and/or the experiment procedure, are applied to three real olfactometer datasets.

EPFL2008

Consider a two-class classification problem where the number of features is much larger than the sample size. The features are masked by Gaussian noise with mean zero and covariance matrix Sigma, where the precision matrix Omega = Sigma(-1) is unknown but is presumably sparse. The useful features, also unknown, are sparse and each contributes weakly (i.e., rare and weak) to the classification decision. By obtaining a reasonably good estimate of Omega, we formulate the setting as a linear regression model. We propose a two-stage classification method where we first select features by the method of Innovated Thresholding (IT), and then use the retained features and Fisher's LDA for classification. In this approach, a crucial problem is how to set the threshold of IT. We approach this problem by adapting the recent innovation of Higher Criticism Thresholding (HCT). We find that when useful features are rare and weak, the limiting behavior of HCT is essentially just as good as the limiting behavior of ideal threshold, the threshold one would choose if the underlying distribution of the signals is known (if only). Somewhat surprisingly, when Omega is sufficiently sparse, its off-diagonal coordinates usually do not have a major influence over the classification decision. Compared to recent work in the case where Omega is the identity matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current setting is much more general, which needs a new approach and much more sophisticated analysis. One key component of the analysis is the intimate relationship between HCT and Fisher's separation. Another key component is the tight large-deviation bounds for empirical processes for data with unconventional correlation structures, where graph theory on vertex coloring plays an important role.

Institute of Mathematical Statistics2013

Correcting Jackknife Confidence Intervals

This paper presents a method for constructing confidence intervals using the jackknife. Asymptotic expansions are used to assess and reduce the skewness. It is proved that, although the error in the coverage rate remains of order O(1/n) for two-sided intervals, the main term due to the skewness in the distribution is eliminated, which constitute an improvement to the normal approximation that is important in the context of small samples. Several simulations are presented, with different distributions and functionals, and compared to the empirical accuracy obtained with a normal approximation. It is observed that our method produces in many cases a better empirical accuracy than the intervals due to a normal approximation, without requiring any knowledge on the underlying distribution.

1998