scholarly journals On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach

2011 ◽  
Vol 28 (2) ◽  
pp. 128-139 ◽  
Author(s):  
Ewan Cameron
Keyword(s):  
Confidence Intervals ◽  
Normal Approximation ◽  
Astronomical Surveys ◽  
Software Packages ◽  
Under Sampling ◽  
Astronomical Research ◽  
Binomial Statistics ◽  
Quantities Of Interest ◽  
Fraction Active ◽  
Hole Fraction

AbstractI present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small to intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, active galactic nucleus fraction, supermassive black hole fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals — the ‘normal approximation’ and the Clopper & Pearson approach — are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (I) to reveal the ease with which (Bayesian) binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g.r, matlab, mathematica, idl, python); and (ii) to demonstrate convincingly the major flaws of both the ‘normal approximation’ and the Clopper & Pearson approach for error estimation.

Combining kNN Imputation and Bootstrap Calibrated

2010 ◽  
Vol 6 (4) ◽  
pp. 61-73 ◽  
Author(s):  
Yongsong Qin ◽  
Shichao Zhang ◽  
Chengqi Zhang
Keyword(s):  
Confidence Intervals ◽  
Incomplete Data ◽  
Asymptotic Theory ◽  
Nearest Neighbor ◽  
Normal Approximation ◽  
Simple Procedure ◽  
Important Research ◽  
K Nearest Neighbor ◽  
Research Topics ◽  
Data Discovery

The k-nearest neighbor (kNN) imputation, as one of the most important research topics in incomplete data discovery, has been developed with great successes on industrial data. However, it is difficult to obtain a mathematical valid and simple procedure to construct confidence intervals for evaluating the imputed data. This paper studies a new estimation for missing (or incomplete) data that is a combination of the kNN imputation and bootstrap calibrated EL (Empirical Likelihood). The combination not only releases the burden of seeking a mathematical valid asymptotic theory for the kNN imputation, but also inherits the advantages of the EL method compared to the normal approximation method. Simulation results demonstrate that the bootstrap calibrated EL method performs quite well in estimating confidence intervals for the imputed data with kNN imputation method.

scholarly journals Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
David Shilane ◽  
Derek Bean
Keyword(s):  
Confidence Intervals ◽  
Negative Binomial ◽  
Normal Approximation ◽  
Tail Probability ◽  
Expected Value ◽  
Alternative Methods ◽  
High Dispersion ◽  
Chi Square ◽  
Sample Mean ◽  
Wide Range

The negative binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard normal approximation often does not provide adequate inferences about the data's expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's inequality. We now propose growth estimators of the negative binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a normal-style confidence interval by effectively removing a small, predetermined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the normal approximation.

A note on coverage error of bootstrap confidence intervals for quantiles

1993 ◽  
Vol 114 (3) ◽  
pp. 517-531 ◽  
Author(s):  
D. De Angelis ◽  
Peter Hall ◽  
G. A. Young
Keyword(s):  
Confidence Intervals ◽  
Present Note ◽  
Normal Approximation ◽  
Poor Performance ◽  
Density Estimator ◽  
Bootstrap Confidence Intervals ◽  
Coverage Error ◽  
Bootstrap Percentile ◽  
Coverage Accuracy ◽  
Percentile Bootstrap

AbstractAn interesting recent paper by Falk and Kaufmann[11] notes, with an element of surprise, that the percentile bootstrap applied to construct confidence intervals for quantiles produces two-sided intervals with coverage error of size n−½, where n denotes sample size. By way of contrast, the error would be O(n−1) for two-sided intervals in more classical problems, such as intervals for means or variances. In the present note we point out that the relatively poor performance in the case of quantiles is shared by a variety of related procedures. The coverage accuracy of two-sided bootstrap intervals may be improved to o(n−½) by smoothing the bootstrap. We show too that a normal approximation method, not involving the bootstrap but incorporating a density estimator as part of scale estimation, can have coverage error O(n−1+∈), for arbitrarily small ∈ > 0. Smoothed and unsmoothed versions of bootstrap percentile-t are also analysed.

scholarly journals Confidence Intervals in Official Statistics: The Case of Lithuania

2012 ◽  
Vol 51 (1) ◽  
pp. 17-21
Author(s):  
Andrius Čiginas
Keyword(s):  
Normal Distribution ◽  
Confidence Intervals ◽  
Edgeworth Expansion ◽  
Normal Approximation ◽  
Official Statistics ◽  
Statistical Surveys

In the paper, the application of confidence intervals in the surveys of official statistics is discussed. It is noticedthat there are situations where at the first sight natural normal distribution-based confidence intervals are not suitable. Wedemonstrate it by examples taken from Lithuanian statistical surveys. We also discuss an Edgeworth expansion and abootstrap method as an alternative to the normal approximation.

scholarly journals Statistical inference on the accelerated competing failure model from the inverse weibull distribution under progressively type-II censored data

2021 ◽  
pp. 97-97
Author(s):  
Ying Wang ◽  
Zai-Zai Yan
Keyword(s):  
Confidence Intervals ◽  
Normal Approximation ◽  
Likelihood Method ◽  
Failure Model ◽  
Unknown Parameters ◽  
Bootstrap Confidence Intervals ◽  
Inverse Weibull Distribution ◽  
Type Ii Censored Data ◽  
Censored Sample ◽  
Better Than

In this paper, the parameter estimation is discussed by using the maximum likelihood method when the available data have the form of progressively censored sample from a constant-stress accelerated competing failure model. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared numerically. The simulation results show that bootstrap confidence intervals perform better than normal approximation. A thermal stress example is discussed.

Introducing high-school and university students to practical astronomy via Virtual observatory tools

2021 ◽  
pp. 413-425
Author(s):  
Alexander Kurtenkov ◽  
Keyword(s):  
High School ◽  
University Students ◽  
Summer School ◽  
Large Scale ◽  
Theoretical Knowledge ◽  
Virtual Observatory ◽  
Astronomical Surveys ◽  
Virtual Observatory Tools ◽  
Astronomical Research ◽  
Astronomy And Astrophysics

Large-scale astronomical surveys from the last decades have turned the usage of catalogs and archival data into one of the primary skills of contemporary observational astronomers. Virtual observatory tools give high-school and university students the opportunity to conduct astronomical research by themselves, using freely available observational data. For this purpose, they need basic theoretical knowledge in astronomy. The current paper includes a review of this theoretical knowledge as well as a review of Virtual observatory tools suitable for students. Results obtained by students using VO tools at the Beli Brezi Summer School in Astronomy and Astrophysics are presented as well.

Avoiding Problems With Normal Approximation Confidence Intervals for Probabilities

Technometrics ◽  
2008 ◽  
Vol 50 (1) ◽  
pp. 64-68 ◽  
Author(s):  
Yili Hong ◽  
William Q Meeker ◽  
Luis A Escobar
Keyword(s):  
Confidence Intervals ◽  
Normal Approximation

Interval Estimation for True Raw and Scale Scores Under the Binomial Error Model

2006 ◽  
Vol 31 (3) ◽  
pp. 261-281 ◽  
Author(s):  
Won-Chan Lee ◽  
Robert L. Brennan ◽  
Michael J. Kolen
Keyword(s):  
Confidence Intervals ◽  
Scale Score ◽  
Normal Approximation ◽  
Interval Estimation ◽  
Correct Score ◽  
True Number ◽  
Coverage Probabilities ◽  
Estimation Procedures ◽  
Scale Scores ◽  
Practical Guidelines

Assuming errors of measurement are distributed binomially, this article reviews various procedures for constructing an interval for an individual’s true number-correct score; presents two general interval estimation procedures for an individual’s true scale score (i.e., normal approximation and endpoints conversion methods); compares various interval estimation procedures through a computer simulation study; and provides some practical guidelines for use of the interval estimation procedures. To examine the effects of different types of scale scores, three nonlinearly transformed scale scores are employed. The conditional confidence intervals using conditional standard errors of measurement are recommended over the traditional confidence intervals using the overall standard error of measurement. For raw scores, the score confidence intervals, in general, tend to provide actual coverage probabilities that are closest to the nominal level. Results for scale score intervals seem to favor the endpoints conversion method using the true-score conversions over the normal approximation approach.

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