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Related Concept Videos

Prediction Intervals01:03

Prediction Intervals

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The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y. 
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Confidence Intervals01:21

Confidence Intervals

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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.
A...
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Interpretation of Confidence Intervals01:19

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.
Suppose a person calculates a confidence interval with a confidence coefficient of 0.95. In that case, they can...
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Uncertainty: Confidence Intervals00:54

Uncertainty: Confidence Intervals

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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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Confidence Interval for Estimating Population Mean01:25

Confidence Interval for Estimating Population Mean

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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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Confidence Coefficient01:24

Confidence Coefficient

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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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Confidence intervals and prediction intervals for two-parameter negative binomial distributions.

Md Mahadi Hasan1, K Krishnamoorthy1

  • 1Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA.

Journal of Applied Statistics
|September 13, 2024
PubMed
Summary

This study introduces simple confidence intervals (CIs) and prediction intervals (PIs) for the two-parameter negative binomial distribution. These new methods offer improved accuracy for moderate sample sizes compared to existing likelihood CIs.

Keywords:
Joint sampling approachmaximum likelihood estimatesover-dispersionpoisson distributionprediction intervalsscore method

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Area of Science:

  • Statistics
  • Probability Theory
  • Statistical Distributions

Background:

  • The two-parameter negative binomial distribution is frequently used in various fields, including biology and quality control.
  • Accurate confidence intervals (CIs) and prediction intervals (PIs) are crucial for reliable statistical inference with this distribution.
  • Existing methods for constructing CIs and PIs can be computationally intensive or less accurate for moderate sample sizes.

Purpose of the Study:

  • To develop and evaluate simple, accurate confidence intervals (CIs) for the mean of a two-parameter negative binomial distribution.
  • To propose and assess the performance of prediction intervals (PIs) for the mean of future samples from this distribution.
  • To compare the proposed methods with existing likelihood-based approaches.

Main Methods:

  • Development of large-sample based methods for constructing CIs for the mean.
  • Comparison of proposed CIs with traditional likelihood-based CIs.
  • Formulation and evaluation of prediction intervals (PIs) for future sample means.
  • Application of methods to real-world datasets for practical illustration.

Main Results:

  • Proposed CIs are computationally simpler than likelihood CIs.
  • The new CIs demonstrate superior performance over likelihood CIs for moderate sample sizes.
  • The proposed prediction intervals offer reliable accuracy for future observations.
  • Illustrative examples confirm the practical utility and effectiveness of the developed methods.

Conclusions:

  • The proposed simple confidence intervals provide a practical and accurate alternative to existing methods for the two-parameter negative binomial distribution.
  • The developed prediction intervals are effective for forecasting future sample means.
  • These methods enhance the statistical analysis capabilities for data following a negative binomial distribution.