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Related Concept Videos

Variance01:15

Variance

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The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.
The standard deviation measures the spread in the same units as the data....
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Estimating Population Mean with Unknown Standard Deviation01:22

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In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the...
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Estimating Population Mean with Known Standard Deviation01:16

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To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
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Linearization and Approximation01:26

Linearization and Approximation

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Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
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Estimating Population Standard Deviation01:26

Estimating Population Standard Deviation

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When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...
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Variability: Analysis01:11

Variability: Analysis

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Measures of variability are statistical metrics that reveal the dispersion pattern within a dataset. They are pivotal in biostatistics, providing insights into the heterogeneity within health and biological data. Variability signifies the degree to which data points diverge from one another, helping researchers understand the potential range of values and associated uncertainty within the data.
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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Regularized Variance Estimation and Variance Stabilization of High Dimensional Data.

Jean-Eudes Dazard1, J Sunil Rao2

  • 1Division of Bioinformatics, Center for Proteomics and Bioinformatics, Case Western Reserve University. Cleveland, OH 44106, USA.

Proceedings. American Statistical Association. Annual Meeting
|January 30, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a novel non-parametric adaptive regularization method to jointly estimate local mean and variance in high-dimensional data. This approach enhances statistical power and stabilizes variance for improved data preprocessing.

Keywords:
BioinformaticsInadmissibilityNormalizationRegularizationShrinkage EstimatorsVariance Stabilization

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • High-dimensional datasets (p >> n) present challenges like unreliable variance estimators and low statistical power due to limited degrees of freedom.
  • Variance in such datasets is often dependent on the mean, a relationship typically ignored by standard methods.

Purpose of the Study:

  • To introduce a non-parametric adaptive regularization procedure for jointly estimating local mean and variance.
  • To leverage the mean-variance relationship for improved statistical inference in high-dimensional settings.

Main Methods:

  • Developed a non-parametric adaptive regularization technique.
  • The method jointly generates local shrinkage estimators for both mean and variance.
  • Utilizes information from the sample mean to inform the regularization process.

Main Results:

  • Regularized t-like statistics demonstrated significantly greater statistical power compared to standard methods.
  • The proposed estimators outperformed common-value shrinkage estimators and methods ignoring the mean-variance relationship.
  • Achieved variance stabilization and normalization, beneficial for preprocessing high-dimensional multivariate data.

Conclusions:

  • The adaptive regularization procedure effectively addresses limitations in high-dimensional data analysis.
  • The joint estimation of mean and variance enhances statistical power and data preprocessing capabilities.
  • The method offers a robust approach for analyzing complex datasets where variance is mean-dependent.