大学数学基础课教学中融入数学建模思想的实践

数学建模不仅限于解决实际问题,也可以是数学发展过程中提出的理论问题。基于近年来在大学数学基础课教学中融入数学建模思想的教学实践,论述了其主要思路与做法。介绍了将数学建模思想融入数学基础课教学的两个教学案例,即微积分方法的本质和从方程组出发引入线性相关与线性无关概念。     

The Frisch–Waugh–Lovell theorem for the lasso and the ridge regression

The Frisch–Waugh–Lovell (FWL) (partitioned regression) theorem is essential in regression analysis. This is partly because it is quite useful to derive theoretical results. The lasso regression and the ridge regression, both of which are penalized least-squares regressions, have become popular statistical techniques. This article describes that the FWL theorem remains valid for these penalized least-squares regressions. More precisely, we demonstrate that the covariates corresponding to unpenalized regression parameters in these penalized least-squares regression can be projected out. Some other results related to the FWL theorem in such penalized least-squares regressions are also presented.     

Minimum variance unbiased estimation of stress–strength reliability under bivariate normal and its comparisons

In many industrial and natural phenomena, we need the probability that a component is smaller than the other component. Under a stress–strength model, this is reliability of an item. Under independent setup, there are different approaches for the estimation of such reliability. Here, estimation is considered under the dependent case. Under bi-variate setup uniformly minimum variance unbiased estimator is obtained. Also comparison with available estimator based on Maximum Likelihood Estimate (MLE) is done through Mean Square Error (MSE) and bias. Also these are compared by computing L₁ distance between their distribution functions. From this idea and numerical computations, UMVUE appears to be good.     

A probabilistic approach to Wallis’ formula

We present an alternative proof of Wallis’ formula from the probabilistic point of view. Based on the classical central limit theorem, some discrete distributions with additive property, such as binomial, negative binomial, Poisson and multinomial distributions, are considered to derive π/2.     

A simple approximation for the doubly noncentral t-distribution

A simple approximation for the doubly noncentral t-distribution, based upon the Fieller-Geary Theorem (1930) and approximate normality of the square root of the noncentral chi-square variable observed by Patnaik (1949), is developed, This approximation and an Edgeworth series expansion associated with it are evaluated. The simple approximation is seen to be reasonably accurate for most practical purposes.     

A process capability index with reliable confidence intervals

A well-know process capability index is slightly modified in this article to provide a new measure of process capability which takes account of the process location and variability, and for which point estimator and confidence intervals do exist that are insensitive to departures from the assumption of normal variability. Two examples of applications based on real data are presented.     

Proportional Trapping-Removal Model with a Known Ratio

ABSTRACT

Population size estimator is derived for a proportional trapping-removal model with a known ratio between two sub-population sizes, and the corresponding asymptotic properties is obtained. The performance of the proposed estimator is checked via simulation studies and an example.