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Related Concept Videos

What are Estimates?01:06

What are Estimates?

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It isn't easy to measure a parameter such as the mean height or the mean weight of a population. So, we draw samples from the population and calculate the mean height or mean weight of the individuals in the sample. This sample data acts as a representative measure of the population parameter. These sample statistics are known as estimates. 
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Central Limit Theorem01:14

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The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
The sample size, n, that...
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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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Confidence Interval for Estimating Population Mean01:25

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A point estimate of the population mean is obtained from a single sample. Such a point estimate does not represent a population well because it needs to account for variability in the population. Single point estimate can also be biased despite the sample being selected randomly. Thus, a point estimate is often unreliable. A confidence interval is needed to reduce this unreliability.
A confidence interval for the mean is a range of values that provides an estimate of the population mean. As 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:
(point estimate - error bound, point estimate +...
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Testing a Claim about Mean: Unknown Population SD01:21

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A complete procedure of testing a hypothesis about a population mean when the population standard deviation is unknown is explained here.
Estimating a population mean requires the samples to be approximately normally distributed. The data should be collected from the randomly selected samples having no sampling bias. There is no specific requirement for sample size. But if the sample size is less than 30, and we don't know the population standard deviation, a different approach is used;...
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Three-Dimensional Shape Modeling and Analysis of Brain Structures
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STATISTICAL INFERENCE FOR MEAN FUNCTIONS OF COMPLEX 3D OBJECTS.

Yueying Wang1, Guannan Wang2, Brandon Klinedinst3

  • 1Amazon.com, Inc., Bellevue, WA 98170, USA.

Statistica Sinica
|September 11, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a new nonparametric method for analyzing complex 3D objects, improving signal estimation and effect detection. The approach enhances decision-making from 3D data by accurately identifying significant features in irregular shapes.

Keywords:
Complex object analysisFunctional principal component analysisLocalizationSimultaneous confidence corridorsTriangulationTrivariate splines

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Area of Science:

  • Computational Geometry
  • Statistical Learning
  • Medical Imaging Analysis

Background:

  • Increasing use of complex three-dimensional (3D) objects in data collection necessitates advanced analytical methods.
  • Identifying significant effects within 3D objects is crucial for informed decision-making.
  • Existing methods may struggle with the analysis of irregular-shaped 3D objects.

Purpose of the Study:

  • To present an advanced nonparametric method for learning and inferring complex 3D objects.
  • To enable accurate estimation of underlying signals and efficient detection/localization of significant effects in 3D data.
  • To provide methods for quantifying estimation uncertainty and comparing independent samples.

Main Methods:

  • Modeling irregular-shaped 3D objects as functional data.
  • Utilizing trivariate spline smoothing based on triangulations for signal estimation.
  • Developing procedures for estimating mean/covariance functions, eigenvalues/eigenfunctions, and constructing confidence corridors.

Main Results:

  • Accurate estimation of mean and covariance functions, eigenvalues, and eigenfunctions for 3D functional data.
  • Rigorous establishment of asymptotic properties for the proposed estimators.
  • Development of simultaneous confidence corridors for uncertainty quantification and extension for two-sample comparisons.

Conclusions:

  • The proposed nonparametric method effectively analyzes complex 3D objects, providing accurate signal estimation and effect localization.
  • The method offers robust statistical properties and practical tools for uncertainty quantification and comparative analysis.
  • Demonstrated utility through numerical experiments and application to Alzheimer's Disease Neuroimaging Initiative data.