In this work in progress, we study the asymptotic behaviour of the p-quantile of the Beta distribution, i.e. the quantity q defined implicitly by ş q 0 t a´1 p1´tq b´1 dt " pBpa, bq, as a function of the first parameter a. In particular, we derive asymptotic expansions of and q and its logarithm at 0 and 8. Moreover, we provide some relations between Bell and Nørlund Polynomials, a generalisation of Bernoulli numbers. Finally, we provide Maple and Sage algorithms for computing the terms of the asymptotic expansions.
When infants start mastering their first language, they may start to notice when words are used incorrectly. Around 14-months of age, infants detect incorrect labeling when they are presented with an object which is labeled while still visible. However, things that are referred to are often out of sight when we communicate about them. The present study examined infants’ detection of semantic mismatch when the object was occluded at the time of labeling. Specifically, we investigated whether mislabeling that referred to an occluded object could elicit a semantic mismatch. We showed 14-month-old Danish-speaking infants events where an onscreen agent showed an object and then hid it in a box. This was followed by another agent’s hand pointing at the box, and a concurrent auditory category label played, which either matched or did not match the hidden object. Our results indicate that there is an effect of semantic mismatch with a larger negativity in incongruent trials. Thus, infants detected a mismatch, as indicated by a larger n400, when occluded objects were mislabeled. This finding suggests that infants can sustain an object representation in memory and compare it to a semantic representation of an auditory category label.
The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as a univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.
The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.
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