On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher’s “Nile” example

Mukhopadhyay, Nitis; Zhuang, Yan

doi:10.1080/07474946.2016.1238264

Cited by 17 publications

(9 citation statements)

References 24 publications

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“…Obviously, J n;1 is a subset of R + , and thus takes into account the positivity of the parameter q. One may refer to Mukhopadhyay and Banerjee (2014), Banerjee and Mukhopadhyay (2016), Mukhopadhyay and Zhuang (2016), Bapat (2018) and other sources for additional background information on fixed-accuracy confidence interval estimation.…”

Section: Sequential Fixed-accuracy Confidence Intervals With Known α α αmentioning

confidence: 99%

Fixed-accuracy confidence interval estimation of P(X > c) for a two-parameter gamma population

Zhuang

Jun

Zou

2020

CSAM

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The gamma distribution is a flexible right-skewed distribution widely used in many areas, and it is of great interest to estimate the probability of a random variable exceeding a specified value in survival and reliability analysis. Therefore, the study develops a fixed-accuracy confidence interval for P(X > c) when X follows a gamma distribution, Γ(α, β), and c is a preassigned positive constant through: 1) a purely sequential procedure with known shape parameter α and unknown rate parameter β; and 2) a nonparametric purely sequential procedure with both shape and rate parameters unknown. Both procedures enjoy appealing asymptotic first-order efficiency and asymptotic consistency properties. Extensive simulations validate the theoretical findings. Three real-life data examples from health studies and steel manufacturing study are discussed to illustrate the practical applicability of both procedures.

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Section: Sequential Fixed-accuracy Confidence Intervals With Known α α αmentioning

confidence: 99%

Fixed-accuracy confidence interval estimation of P(X > c) for a two-parameter gamma population

Zhuang

Jun

Zou

2020

CSAM

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“…condition holds for T n ≡ U 2 n , which is an MLE from i.i.d. observations (more specifically, T n is the MLE of θ 2 by the invariance property of the MLE), Mukhopadhyay and Zhuang [32] obtained:…”

Section: A Fixed-width Confidence Interval In Fisher's "Nile" Examplementioning

confidence: 99%

Asymptotic Normality of Sequential Stopping Times with Applications: Confidence Intervals for an Exponential Mean

Mukhopadhyay

2020

Calcutta Statistical Association Bulletin

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In sequential methodologies, finally accrued data customarily look like [Formula: see text] where [Formula: see text] is the total number of observations collected through termination. Under mild regulatory conditions, a standardized version of [Formula: see text] follows an asymptotic normal distribution (Ghosh–Mukhopadhyay theorem) which we highlight with a number of illustrations from the recent literature for completeness. Then, we emphasize the role of such asymptotic normality results along with second-order approximations for stopping times in the construction of sequential fixed-width confidence intervals for the mean in an exponential distribution. Two kinds of confidence intervals are developed: (a) one centred at the randomly stopped sample mean [Formula: see text] and (b) the two other centred at appropriate constructs using the stopping variable [Formula: see text] alone. Ample comparisons among all three proposed methodologies are summarized via simulations. We emphasize our finding that the two fixed-width confidence intervals centred at appropriate constructs using the stopping variable [Formula: see text] alone perform as well or better than the customary one centred at the randomly stopped sample mean.

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“…Bandyopadhyay et al (2003) considered a nonparametric fixed-width confidence interval estimation for the parameter R by adopting a partial sequential sampling scheme. Recently, Mukhopadhyay and Zhuang (2016) have considered a fixed-accuracy interval for the reliability parameter under a bivariate exponential model, namely, the Fisher's Nile example. Bapat (2018) has constructed a purely sequential estimation methodology to find a fixed-accuracy confidence interval for the unknown parameter R in two separate cases, where (X, Y) have a bivariate exponential distribution due to Marshall and Olkin (1967) and Freund (1961).…”

Section: Introductionmentioning

confidence: 99%

Sequential-Based Approach for Estimating the Stress-Strength Reliability Parameter for Exponential Distribution

Khalifeh

Mahmoudi

Dolati

2020

JIRSS

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In this paper, two-stage and purely sequential estimation procedures are considered to construct fixed-width confidence intervals for the reliability parameter under the stress-strength model when the stress and strength are independent exponential random variables with different scale parameters. The exact distribution of the stopping rule under the purely sequential procedure is approximated using the law of large numbers and Monte Carlo integration. For the two-stage sequential procedure, explicit formulas for the distribution of the total sample size, the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress-strength model are provided. Moreover, it is shown that both proposed sequential procedures are finite, and in exceptional cases, the exact distribution of stopping times is degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally using real data, the procedures are clearly illustrated.

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On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher’s “Nile” example

Cited by 17 publications

References 24 publications

Fixed-accuracy confidence interval estimation of P(X > c) for a two-parameter gamma population

Fixed-accuracy confidence interval estimation of P(X > c) for a two-parameter gamma population

Asymptotic Normality of Sequential Stopping Times with Applications: Confidence Intervals for an Exponential Mean

Sequential-Based Approach for Estimating the Stress-Strength Reliability Parameter for Exponential Distribution

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