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Interpretation of Confidence Intervals

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A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
Confidence intervals have confidence coefficients that are crucial for their interpretation. The most common confidence coefficients are 0.90, 0.95, and 0.99, which can be written as percentages–90%, 95%, and 99%, respectively.
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An unbiased point estimate is often insufficient to predict a population estimate, such as population mean or population proportion. In this scenario, a confidence interval is used. A confidence interval is an estimate similar to a sample proportion. However, unlike the point estimate which is a single value, the confidence interval contains a range of values. These values have lower and upper limits, known as confidence limits, and can be designated as L1 and L2, respectively.
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The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...

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The confidence coefficient is also known as the confidence level or degree of confidence. It is the percent expression for the probability, 1-α, that the confidence interval contains the true population parameter assuming that the confidence interval is obtained after sufficient unbiased sampling; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter. Here α is the area under the curve, distributed equally under...

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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.
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An R-Based Landscape Validation of a Competing Risk Model

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Adaptive Confidence Bands for Nonparametric Regression Functions.

T Tony Cai¹, Mark Low¹, Zongming Ma¹

¹University of Pennsylvania.

Journal of the American Statistical Association

|August 14, 2015

Summary

This summary is machine-generated.

A novel adaptive confidence band formulation is introduced for non-parametric function estimation. This method ensures bands adapt to function smoothness, controlling errors and improving accuracy for various Lipschitz classes.

Keywords:

Adaptive confidence band average coverage coverage probability excess mass lower bounds noncovered points nonparametric regression wavelets white noise model

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Area of Science:

Statistics
Non-parametric statistics
Machine learning

Background:

Confidence bands are crucial for quantifying uncertainty in non-parametric function estimation.
Existing methods may lack adaptivity to varying function smoothness.
Adaptive methods aim to provide more efficient and reliable estimations.

Purpose of the Study:

To propose a new formulation for constructing adaptive confidence bands.
To ensure confidence bands adapt to function smoothness while controlling estimation errors.
To demonstrate the adaptability of the proposed bands across a range of Lipschitz classes.

Main Methods:

Development of a new formulation for adaptive confidence bands.
Theoretical analysis to guarantee control over relative excess mass and measure of deviation.
Implementation using standard statistical software with wavelet support.

Main Results:

The proposed confidence bands adapt to function smoothness.
Guarantees are provided for small relative excess mass and measure of deviation.
The bands demonstrate adaptability over a maximum range of Lipschitz classes.
Numerical performance validated with simulated and real datasets, showing good agreement with theory.

Conclusions:

The new formulation provides effective adaptive confidence bands for non-parametric function estimation.
The method is computationally feasible and easily implementable.
The approach can be extended to other non-parametric function estimation models.