Using zero-norm constraint for sparse probability density function estimation

A new sparse kernel probability density function (pdf) estimator based on zero-norm constraint is constructed using the classical Parzen window (PW) estimate as the target function. The so-called zero-norm of the parameters is used in order to achieve enhanced model sparsity, and it is suggested to minimize an approximate function of the zero-norm. It is shown that under certain condition, the kernel weights of the proposed pdf estimator based on the zero-norm approximation can be updated using the multiplicative nonnegative quadratic programming algorithm. Numerical examples are employed to demonstrate the efficacy of the proposed approach.

Asymptotic expansions for Taylor coefficients of the composition of two functions

We give an asymptotic expansion for the Taylor coe±cients of L(P(z)) where L(z) is analytic in the open unit disc whose Taylor coe±cients vary `smoothly' and P(z) is a probability generating function. We show how this result applies to a variety of problems, amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal sequences.

Curriculum theory, curriculum policy and the problem of ill-disciplined thinking

This paper examines the implications of policy fracture and arms length governance within the decision making processes currently shaping curriculum design within the English education system. In particular it argues that an unresolved ‘ideological fracture’ at government level has been passed down to school leaders whose response to the dilemma is distorted by the target-driven agenda of arms length agencies. Drawing upon the findings of a large scale on-line survey of history teaching in English secondary schools, this paper illustrates the problems that occur when policy making is divorced from curriculum theory, and in particular from any consideration of the nature of knowledge. Drawing on the social realist theory of knowledge elaborated by Young (2008), we argue that the rapid spread of alternative curricular arrangements, implemented in the absence of an understanding of curriculum theory, undermines the value of disciplined thinking to the detriment of many young people, particularly those in areas of social and economic deprivation.