Kronecker delta method for testing independence between two vectors in high-dimension

2020 ◽

Vol 6 (1) ◽

pp. 19-54

Author(s):

Ryan Ka Yau Lai ◽

Youngah Do

Keyword(s):

Maximum Likelihood ◽

Corpus Linguistics ◽

Delta Method ◽

Confidence Bounds ◽

Likelihood Estimator ◽

Information Theoretic ◽

Leibler Divergence ◽

Information Theoretic Measures ◽

Data Points ◽

Measure Of Uncertainty

This article explores a method of creating confidence bounds for information-theoretic measures in linguistics, such as entropy, Kullback-Leibler Divergence (KLD), and mutual information. We show that a useful measure of uncertainty can be derived from simple statistical principles, namely the asymptotic distribution of the maximum likelihood estimator (MLE) and the delta method. Three case studies from phonology and corpus linguistics are used to demonstrate how to apply it and examine its robustness against common violations of its assumptions in linguistics, such as insufficient sample size and non-independence of data points.

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An approximate point-based alternative for the estimation of variance under big BAF sampling

Forest Ecosystems ◽

10.1186/s40663-021-00304-0 ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Thomas B. Lynch ◽

Jeffrey H. Gove ◽

Timothy G. Gregoire ◽

Mark J. Ducey

Keyword(s):

Forest Inventory ◽

Variance Estimation ◽

Basal Area ◽

Volume Estimation ◽

Delta Method ◽

Sampling Point ◽

Variance Estimator ◽

Computer Simulation Studies ◽

New Formula ◽

Estimation Of Variance

Abstract Background A new variance estimator is derived and tested for big BAF (Basal Area Factor) sampling which is a forest inventory system that utilizes Bitterlich sampling (point sampling) with two BAF sizes, a small BAF for tree counts and a larger BAF on which tree measurements are made usually including DBHs and heights needed for volume estimation. Methods The new estimator is derived using the Delta method from an existing formulation of the big BAF estimator as consisting of three sample means. The new formula is compared to existing big BAF estimators including a popular estimator based on Bruce’s formula. Results Several computer simulation studies were conducted comparing the new variance estimator to all known variance estimators for big BAF currently in the forest inventory literature. In simulations the new estimator performed well and comparably to existing variance formulas. Conclusions A possible advantage of the new estimator is that it does not require the assumption of negligible correlation between basal area counts on the small BAF factor and volume-basal area ratios based on the large BAF factor selection trees, an assumption required by all previous big BAF variance estimation formulas. Although this correlation was negligible on the simulation stands used in this study, it is conceivable that the correlation could be significant in some forest types, such as those in which the DBH-height relationship can be affected substantially by density perhaps through competition. We derived a formula that can be used to estimate the covariance between estimates of mean basal area and the ratio of estimates of mean volume and mean basal area. We also mathematically derived expressions for bias in the big BAF estimator that can be used to show the bias approaches zero in large samples on the order of $\frac {1}{n}$ 1 n where n is the number of sample points.

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Dental Expenditure by Household Income in Korea over the Period 2008–2017: A Review of the National Dental Insurance Reform

International Journal of Environmental Research and Public Health ◽

10.3390/ijerph18083859 ◽

2021 ◽

Vol 18 (8) ◽

pp. 3859

Author(s):

Hosung Shin ◽

Han-A Cho ◽

Bo-Ra Kim

Keyword(s):

Health Insurance ◽

Concentration Index ◽

Dental Health ◽

Age Groups ◽

The Elderly ◽

Dental Service ◽

Delta Method ◽

Dental Services ◽

Insurance Reform ◽

Health Insurance Reform

Since 2009, the National Health Insurance in Korea (NHI) has been implementing a series of policies to expand the scope of dental benefits. This study reviewed the changes in co-payments and dental use patterns before (2008 to 2012) and after (2013 to 2017) the NHI’s dental health insurance reform. The study used Korea Health Panel data of 7681 households (16,493 household members) from a 10-year period (2008–2017). Dental expenditures and equivalent income using square root of household size were analyzed. Dental services were categorized into 13 types and a concentration index and 95% confidence interval using the delta method was calculated to identify income-related inequalities by a dental service. Dental expenditures and the number of dental services used increased significantly, while the proportion of out-of-pocket spending by the elderly decreased. The expenditure ratio for implant services to total dental expenditures increased substantially in all age groups, but the ratio of expenditures for dentures and fixed bridges decreased relatively. The concentration index of implant services was basically in favor of the rich, but there was no longer a significant bias favoring the better-off after the reforms. The dental health insurance reform in Korea appears to contribute not only to lowering the ratio of out-of-pocket to total dental expenses per episode in the elderly but also to improving the inequality of dental expenses.

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