A distribution-free test of independence based on a modified mean variance index

Cui and Zhong(2019),(Computational Statistics&Data Analysis,139,117–133)proposed a test based on the mean variance(MV)index to test independence between a categorical random variable Y with R categories and a continuous random variable X.They ingeniously proved the asymptotic normality of the MV test statistic when R diverges to infinity,which brings many merits to the MV test,including making it more convenient for independence testing when R is large.This paper considers a new test called the integral Pearson chi-square(IPC)test,whose test statistic can be viewed as a modified MV test statistic.A central limit theorem of the martin-gale difference is used to show that the asymptotic null distribution of the standardized IPC test statistic when R is diverging is also a normal distribution,rendering the IPC test sharing many merits with the MV test.As an application of such a theoretical finding,the IPC test is extended to test independence between continuous random variables.The finite sample performance of the proposed test is assessed by Monte Carlo simulations,and a real data example is presented for illustration.

Extraction and separation of yttrium from other rare earths in chloride medium by phosphorylcarboxylic acids

Two phosphorylcarboxylic acids,3-((bis(2-ethylhexyloxy))phosphoryl)propanoic acid(PPA) and 3-((bis(2-ethylhexyloxy))phosphoryl)-3-phenylpropanoic acid(PPPA),were synthesized for separating yttrium from other rare earths in the chloride feed of ion-adsorption type rare earth concentrate.The effect of the factors such as pH_(1/2),temperature,saponification degree and phase modifiers was investigated.The separation efficiencies of PPA and PPPA are obviously better than the typical extractants such as sec-octylphenoxy acetic acid(CA-12) and naphthenic acid(NA).The extraction process of rare earths by PPA and PPPA is a cation exchanging reaction,which is similar to those of CA-12 and NA.The loaded rare earths in both PPA and PPPA systems can be effectively back-extracted by 0.5 mol/L HCl or higher concentration.A cascade extraction process for separating yttrium from other rare earths was developed using PPPA as the extractant.The yttrium product with the purity of 97.20 wt% was obtained by 35 stages of extraction and 12 stages of scrubbing.