The so-called Kumaraswamy distribution is a special probability developed to model doubled bounded random processes for which the mode do not necessarily have be within bounds. In this article, generalization of called T-Kumaraswamy family defined using T-R {Y} distributions framework. resulting obtained quantile functions some standardized distributions. Some general mathematical properties ne...

In this article, we introduce a new generalized family of Esscher transformed Laplace distribution, namely the Kumaraswamy distribution. We study various properties distribution including survival function, hazard rate cumulative function and reverse function. The parameters are estimated using maximum likelihood method estimation. A real application on breaking stress carbon fibres is also con...

It has been observed that return distributions in general and interest rates in particular exhibit skewness and kurtosis that cannot be explained by the lognormal distribution commonly used as an assumption in many option pricing models. We have replaced the lognormal assumption in the Black (1976) model with the g-and-h distribution and derived a simple, closed-form option pricing formula unde...

Although Zipf's law is widespread in natural and social data, one often encounters situations where one or both ends of the ranked data deviate from the power-law function. Previously we proposed the Beta rank function to improve the fitting of data which does not follow a perfect Zipf's law. Here we show that when the two parameters in the Beta rank function have the same value, the Lavalette ...

We show that, over Q, if an n-variate polynomial of degree d = nO(1) is computable by an arithmetic circuit of size s (respectively by an algebraic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O( √ d log d log n log s)) (respectively of size exp(O( √ d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O( √ ...

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For example, 77 = 21 · 55/(3 · 5) is a Fibonacci integer. Using some results about the structure of this multiplicative group, we determine a near-asymptotic formula for the counting function of the Fibonacci integers, showing that up to x the number of them is between exp(c(log x) − (log x)) and e...

In this note, we explain how f(a) = log(1− e−a) = log(1− exp(−a)) can be computed accurately, in a simple and optimal manner, building on the two related auxiliary functions log1p(x) (= log(1 + x)) and expm1(x) (= exp(x)− 1 = ex − 1). The cutoff, a0, in use in R since 2004, is shown to be optimal both theoretically and empirically, using Rmpfr high precision arithmetic. As an aside, we also sho...