Simplified Estimates for the Exponential Distribution

Sarhan, A. E.; Greenberg, Bernard; Ogawa, Junjiro

doi:10.1214/aoms/1177704246

Cited by 35 publications

(18 citation statements)

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“…In Section 4.5, we take just a specified number of k points and apply weighted least squares to obtain "quantile" estimators, as considered by Quandt (1966) for k ϭ 2 and Sarhan, Greenberg, and Ogawa (1963), Saleh and Ali (1966), and Koutrouvelis (1981) for arbitrary k Ն 2.…”

Section: Robust and Efficient Estimation Of The Tail Index Of A Singlmentioning

confidence: 99%

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Brazauskas¹,

Serfling²

2000

North American Actuarial Journal

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Estimation of the tail index parameter of a single-parameter Pareto model has wide application in actuarial and other sciences. Here we examine various estimators from the standpoint of two competing criteria: efficiency and robustness against upper outliers. With the maximum likelihood estimator (MLE) being efficient but nonrobust, we desire alternative estimators that retain a relatively high degree of efficiency while also being adequately robust. A new generalized median type estimator is introduced and compared with the MLE and several well-established estimators associated with the methods of moments, trimming, least squares, quantiles, and percentile matching. The method of moments and least squares estimators are found to be relatively deficient with respect to both criteria and should become disfavored, while the trimmed mean and generalized median estimators tend to dominate the other competitors. The generalized median type performs best overall. These findings provide a basis for revision and updating of prevailing viewpoints. Other topics discussed are applications to robust estimation of upper quantiles, tail probabilities, and actuarial quantities, such as stop-loss and excess-of-loss reinsurance premiums that arise concerning solvency of portfolios. Robust parametric methods are compared with empirical nonparametric methods, which are typically nonrobust.

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Section: Robust and Efficient Estimation Of The Tail Index Of A Singlmentioning

confidence: 99%

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Brazauskas¹,

Serfling²

2000

North American Actuarial Journal

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“…Such estimators were introduced and studied for the Pareto problem by Quandt (1966) for k = 2 and by Koutrouvelis (1981) for general k, and for the equivalent exponential problem by Sarhan, Greenberg, and Ogawa (1963) with ,u known and by Saleh and Ali (1966) with ,u unknown. As argued in Saleh and Ali (1966), the optimal choice of pl is 10…”

Section: Quantile Type Estimatorsmentioning

confidence: 99%

“…. ,pk then to be found from Sarhan, Greenberg and Ogawa (1963). (These values minimize the generalized variance, i.e., the determinant of the asymptotic covariance matrix, of the estimators of u and a , subject to (4).j Denoting the corresponding estimator of a by &$lk, we have in particular: n 0~4 2 0 For k = 2, i.e., for a , the optimal pis are pl = py and p2 = 30.…”

Section: Brazauskas and R Serflingmentioning

confidence: 99%

Small sample performance of robust estimators of tail parameters for pareto and exponential models

Brazauskas

Serfling

2001

Journal of Statistical Computation and Simulation

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Robust estimation of tail index parameters is treated for (equivalent) two-parameter Pareto and exponential models. These distributions arise as parametric models in actuarial science, economics, telecommunications, and reliability, for example, as well as in semiparametric modeling of upper observations in samples from distributions which are regularly varying or in the domain of attraction of extreme value distributions. In a recent previous paper, new estimators of "generalized median" (GM) type were introduced and shown to provide more favorable trade-offs between efficiency and robustness than several well-established estimators, including those corresponding to methods of maximum likelihood, trimming, and quantiles. Here we establish-via simulation-that the superiority of the GM type estimators remains valid even for small sample sizes n = 10 and 25. To bridge between "small" and "large" sample sizes, we also include the cases n = 50 and 100. Further, we arrive at guidelines for selection of a particular GM estimator in practice, depending upon the sample size, upon whether protection is desired against upper outliers only, or against both upper and lower outliers, and upon whether the level of possible contamination by outliers is high or low. Comparisons of estimators are made on the basis of relative efficiency with respect to the maximum likelihood estimator, breakdown points, and premium-protection plots.

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“…However, this is not surprising since the same observation has been reported in the literature for the exponential distribution (cf. Sarhan et al 1963, Saleh and Ali 1966, and Ali and Umbach 1998. In these cases, the generalized variance is unbounded if q 1 tends to 0.…”

Section: Introductionmentioning

confidence: 99%

An asymptotic approach to progressive censoring

Hofmann¹,

Cramer²,

Kunert³

2005

Journal of Statistical Planning and Inference

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Progressive Type-II censoring was introduced by Cohen (1963) and has since been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2 × 2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.

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Simplified Estimates for the Exponential Distribution

Cited by 35 publications

References 0 publications

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Small sample performance of robust estimators of tail parameters for pareto and exponential models

An asymptotic approach to progressive censoring

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