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Journal articles on the topic 'Sampling with loaded probabilities'

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Published: 10 December 2022
Last updated: 28 January 2023

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1

Cumpston, J. Richard. "A more efficient sampling procedure, using loaded probabilities." International Journal of Microsimulation 5, no. 1 (2011): 21–30. http://dx.doi.org/10.34196/ijm.00065.

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2

Hultgren, Gustav, Leo Myrén, Zuheir Barsoum, and Rami Mansour. "Digital Scanning of Welds and Influence of Sampling Resolution on the Predicted Fatigue Performance: Modelling, Experiment and Simulation." Metals 11, no. 5 (May 18, 2021): 822. http://dx.doi.org/10.3390/met11050822.

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Digital weld quality assurance systems are increasingly used to capture local geometrical variations that can be detrimental for the fatigue strength of welded components. In this study, a method is proposed to determine the required scanning sampling resolution for proper fatigue assessment. Based on FE analysis of laser-scanned welded joints, fatigue failure probabilities are computed using a Weakest-link fatigue model with experimentally determined parameters. By down-sampling of the scanning data in the FE simulations, it is shown that the uncertainty and error in the fatigue failure probability prediction increases with decreased sampling resolution. The required sampling resolution is thereafter determined by setting an allowable error in the predicted failure probability. A sampling resolution of 200 to 250 μm has been shown to be adequate for the fatigue-loaded welded joints investigated in the current study. The resolution requirements can be directly incorporated in production for continuous quality assurance of welded structures. The proposed probabilistic model used to derive the resolution requirement accurately captures the experimental fatigue strength distribution, with a correlation coefficient of 0.9 between model and experimental failure probabilities. This work therefore brings novelty by deriving sampling resolution requirements based on the influence of stochastic topographical variations on the fatigue strength distribution.
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3

Pflug, G. Ch. "Sampling derivatives of probabilities." Computing 42, no. 4 (December 1989): 315–28. http://dx.doi.org/10.1007/bf02243227.

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4

Singh, M. P. "Sampling with unequal probabilities." Metrika 33, no. 1 (December 1986): 92. http://dx.doi.org/10.1007/bf01894732.

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5

Milbrodt, Hartmut. "Comparing inclusion probabilities and drawing probabilities for rejective sampling and successive sampling." Statistics & Probability Letters 14, no. 3 (June 1992): 243–46. http://dx.doi.org/10.1016/0167-7152(92)90029-5.

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6

Goulionis, J. E. "Strategies for sampling with varying probabilities." Journal of Statistical Computation and Simulation 81, no. 11 (November 2011): 1753. http://dx.doi.org/10.1080/00949655.2011.622434.

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7

Chaudhuri, Arijit, and Arun Kumar Adhikary. "Circular Systematic Sampling with Varying Probabilities." Calcutta Statistical Association Bulletin 36, no. 3-4 (September 1987): 193–96. http://dx.doi.org/10.1177/0008068319870310.

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Certain conditions connecting the population size, sample size and the sampling interval in circular systematic sampling with equal probabilities are known. We present here a simple “condition” connecting the sample size, size-measures and the sampling interval in pps circular systematic sampling. The condition is important in noting limitations on sample-sizes when a sampling interval is pre-assigned.
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8

Greco, Luigi, and Stefania Naddeo. "Inverse Sampling with Unequal Selection Probabilities." Communications in Statistics - Theory and Methods 36, no. 5 (April 3, 2007): 1039–48. http://dx.doi.org/10.1080/03610920601033926.

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9

Ng, Meei Pyng, and Martin Donadio. "Computing inclusion probabilities for order sampling." Journal of Statistical Planning and Inference 136, no. 11 (November 2006): 4026–42. http://dx.doi.org/10.1016/j.jspi.2005.03.010.

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10

Chauvet, G., D. Bonnéry, and J. C. Deville. "Optimal inclusion probabilities for balanced sampling." Journal of Statistical Planning and Inference 141, no. 2 (February 2011): 984–94. http://dx.doi.org/10.1016/j.jspi.2010.09.005.

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11

Letac, Gérard. "Comment: Lancaster Probabilities and Gibbs Sampling." Statistical Science 23, no. 2 (May 2008): 187–91. http://dx.doi.org/10.1214/08-sts252a.

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12

Hunt, P. J., and F. P. Kelly. "On critically loaded loss networks." Advances in Applied Probability 21, no. 4 (December 1989): 831–41. http://dx.doi.org/10.2307/1427769.

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This paper studies the behaviour of large loss networks, paying particular attention to links at a certain critical loading where the load offered very nearly matches capacity. We correct and extend an earlier central limit theorem for the stationary distribution of a loss network with critically loaded links. We then use this result to show that acceptance probabilities have a limiting product-form decomposition, despite marked dependencies between the occupancies of critically loaded links.
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13

Hunt, P. J., and F. P. Kelly. "On critically loaded loss networks." Advances in Applied Probability 21, no. 04 (December 1989): 831–41. http://dx.doi.org/10.1017/s0001867800019078.

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This paper studies the behaviour of large loss networks, paying particular attention to links at a certain critical loading where the load offered very nearly matches capacity. We correct and extend an earlier central limit theorem for the stationary distribution of a loss network with critically loaded links. We then use this result to show that acceptance probabilities have a limiting product-form decomposition, despite marked dependencies between the occupancies of critically loaded links.
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14

Hudson, Richard R. "Two-Locus Sampling Distributions and Their Application." Genetics 159, no. 4 (December 1, 2001): 1805–17. http://dx.doi.org/10.1093/genetics/159.4.1805.

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AbstractMethods of estimating two-locus sample probabilities under a neutral model are extended in several ways. Estimation of sample probabilities is described when the ancestral or derived status of each allele is specified. In addition, probabilities for two-locus diploid samples are provided. A method for using these two-locus probabilities to test whether an observed level of linkage disequilibrium is unusually large or small is described. In addition, properties of a maximum-likelihood estimator of the recombination parameter based on independent linked pairs of sites are obtained. A composite-likelihood estimator, for more than two linked sites, is also examined and found to work as well, or better, than other available ad hoc estimators. Linkage disequilibrium in the Xq28 and Xq25 region of humans is analyzed in a sample of Europeans (CEPH). The estimated recombination parameter is about five times smaller than one would expect under an equilibrium neutral model.
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15

Kemp, Adrienne W. "Absorption sampling and the absorption distribution." Journal of Applied Probability 35, no. 2 (June 1998): 489–94. http://dx.doi.org/10.1239/jap/1032192864.

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The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.
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16

Kemp, Adrienne W. "Absorption sampling and the absorption distribution." Journal of Applied Probability 35, no. 02 (June 1998): 489–94. http://dx.doi.org/10.1017/s0021900200015114.

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The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.
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17

Brádler, Kamil, Shmuel Friedland, Josh Izaac, Nathan Killoran, and Daiqin Su. "Graph isomorphism and Gaussian boson sampling." Special Matrices 9, no. 1 (January 1, 2021): 166–96. http://dx.doi.org/10.1515/spma-2020-0132.

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Abstract We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.
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18

Chaudhuri, Arijit. "Network and Adaptive Sampling with Unequal Probabilities." Calcutta Statistical Association Bulletin 50, no. 3-4 (September 2000): 237–54. http://dx.doi.org/10.1177/0008068320000310.

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19

Lehtonen, T., and H. Nyrhinen. "Simulating level-crossing probabilities by importance sampling." Advances in Applied Probability 24, no. 4 (December 1992): 858–74. http://dx.doi.org/10.2307/1427716.

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Let X1, X2, · ·· be independent and identically distributed random variables such that ΕΧ1 < 0 and P(X1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X1 + X2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P(T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P(T <∞) and . The paper ends with an example.
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20

AIRES, NIBIA, JOHAN JONASSON, and OLLE NERMAN. "Order Sampling Design with Prescribed Inclusion Probabilities." Scandinavian Journal of Statistics 29, no. 1 (March 2002): 183–87. http://dx.doi.org/10.1111/1467-9469.00120.

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21

Rosén, Bengt. "On inclusion probabilities for order πps sampling." Journal of Statistical Planning and Inference 90, no. 1 (September 2000): 117–43. http://dx.doi.org/10.1016/s0378-3758(00)00104-x.

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22

Lehtonen, T., and H. Nyrhinen. "Simulating level-crossing probabilities by importance sampling." Advances in Applied Probability 24, no. 04 (December 1992): 858–74. http://dx.doi.org/10.1017/s0001867800024988.

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Let X 1, X 2, · ·· be independent and identically distributed random variables such that ΕΧ 1 &lt; 0 and P (X 1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X 1 + X 2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P (T &lt;∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P (T &lt;∞) and . The paper ends with an example.
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23

Stuart, Alan. "Location-Shifts in Sampling with Unequal Probabilities." Journal of the Royal Statistical Society. Series A (General) 149, no. 4 (1986): 349. http://dx.doi.org/10.2307/2981721.

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24

Kluever, Bryan M., Eric M. Gese, and Steven J. Dempsey. "The influence of road characteristics and species on detection probabilities of carnivore faeces." Wildlife Research 42, no. 1 (2015): 75. http://dx.doi.org/10.1071/wr14244.

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Context Determining reliable estimates of carnivore population size and distributions are paramount for developing informed conservation and management plans. Traditionally, invasive sampling has been employed to monitor carnivores, but non-invasive sampling has the advantage of not needing to capture the animal and is generally less expensive. Faeces sampling is a common non-invasive sampling technique and future use is forecasted to increase due to the low costs and logistical ease of sampling, and more advanced techniques in landscape and conservation genetics. For many species, faeces sampling often occurs on or alongside roads. Despite the commonality of road-based faeces sampling, detectability issues are often not addressed. Aim We sought to test whether faeces detection probabilities varied by species – coyote (Canis latrans) versus kit fox (Vulpes macrotis) – and to test whether road characteristics influenced faeces detection probabilities. Methods We placed coyote and kit fox faeces along roads, quantified road characteristics, and then subsequently conducted ‘blind’ road-based faeces detection surveys in Utah during 2012 and 2013. Technicians that surveyed the faeces deposition transects had no knowledge of the locations of the placed faeces. Key results Faeces detection probabilities for kit foxes and coyotes were 45% and 74%, respectively; larger faeces originated from coyotes and were more readily detected. Misidentification of placed faeces was rare and did not differ by species. The width of survey roads and the composition of a road’s surface influenced detection probabilities. Conclusion We identified factors that can influence faeces detection probabilities. Not accounting for variable detection probabilities of different species or not accounting for or reducing road-based variables influencing faeces detection probabilities could hamper reliable counts of mammalian faeces, and could potentially reduce precision of population estimates derived from road-based faeces deposition surveys. Implications We recommend that wildlife researchers acknowledge and account for imperfect faeces detection probabilities during faecal sampling. Steps can be taken during study design to improve detection probabilities, and during the analysis phase to account for variable detection probabilities.
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25

Wichert, Andreas. "Quantum-Like Sampling." Mathematics 9, no. 17 (August 24, 2021): 2036. http://dx.doi.org/10.3390/math9172036.

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Probability theory is built around Kolmogorov’s axioms. To each event, a numerical degree of belief between 0 and 1 is assigned, which provides a way of summarizing the uncertainty. Kolmogorov’s probabilities of events are added, the sum of all possible events is one. The numerical degrees of belief can be estimated from a sample by its true fraction. The frequency of an event in a sample is counted and normalized resulting in a linear relation. We introduce quantum-like sampling. The resulting Kolmogorov’s probabilities are in a sigmoid relation. The sigmoid relation offers a better importability since it induces the bell-shaped distribution, it leads also to less uncertainty when computing the Shannon’s entropy. Additionally, we conducted 100 empirical experiments by quantum-like sampling 100 times a random training sets and validation sets out of the Titanic data set using the Naïve Bayes classifier. In the mean the accuracy increased from 78.84% to 79.46%.
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26

Oaksford, Mike, and Michelle Wakefield. "Data selection and natural sampling: Probabilities do matter." Memory & Cognition 31, no. 1 (January 2003): 143–54. http://dx.doi.org/10.3758/bf03196089.

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27

Salcedo, Lorenzo Luis. "Representation of complex probabilities and complex Gibbs sampling." EPJ Web of Conferences 175 (2018): 07037. http://dx.doi.org/10.1051/epjconf/201817507037.

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Complex weights appear in Physics which are beyond a straightforward importance sampling treatment, as required in Monte Carlo calculations. This is the wellknown sign problem. The complex Langevin approach amounts to effectively construct a positive distribution on the complexified manifold reproducing the expectation values of the observables through their analytical extension. Here we discuss the direct construction of such positive distributions paying attention to their localization on the complexified manifold. Explicit localized representations are obtained for complex probabilities defined on Abelian and non Abelian groups. The viability and performance of a complex version of the heat bath method, based on such representations, is analyzed.
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28

Barbu, A., and Song-Chun Zhu. "Generalizing Swendsen-Wang to sampling arbitrary posterior probabilities." IEEE Transactions on Pattern Analysis and Machine Intelligence 27, no. 8 (August 2005): 1239–53. http://dx.doi.org/10.1109/tpami.2005.161.

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29

Bondesson, Lennart, and Imbi Traat. "On Sampling Designs with Ordered Conditional Inclusion Probabilities." Scandinavian Journal of Statistics 40, no. 4 (September 11, 2013): 724–33. http://dx.doi.org/10.1111/sjos.12024.

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30

Kumar, Prenesh. "On sampling of three units using varying probabilities." Statistics 18, no. 3 (January 1987): 373–77. http://dx.doi.org/10.1080/02331888708802032.

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31

BONDESSON, LENNART. "On Sampling with Prescribed Second-order Inclusion Probabilities." Scandinavian Journal of Statistics 39, no. 4 (August 31, 2012): 813–29. http://dx.doi.org/10.1111/j.1467-9469.2012.00808.x.

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32

Sándor, Zsolt, and Péter András. "Alternative sampling methods for estimating multivariate normal probabilities." Journal of Econometrics 120, no. 2 (June 2004): 207–34. http://dx.doi.org/10.1016/s0304-4076(03)00212-4.

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33

Ozturk, Omer, and Mohammad Jafari Jozani. "Inclusion probabilities in partially rank ordered set sampling." Computational Statistics & Data Analysis 69 (January 2014): 122–32. http://dx.doi.org/10.1016/j.csda.2013.07.034.

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34

Peřina, J., and J. Křepelka. "Sampling of quasidistributions, nonclassical behavior and negative probabilities." Physics Letters A 380, no. 22-23 (May 2016): 1932–35. http://dx.doi.org/10.1016/j.physleta.2016.04.007.

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35

Szapudi, Istvan, and Alexander S. Szalay. "Effects of Sampling on Measuring Galaxy Count Probabilities." Astrophysical Journal 459 (March 1996): 504. http://dx.doi.org/10.1086/176913.

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36

Glasserman, Paul, and Yashan Wang. "Counterexamples in importance sampling for large deviations probabilities." Annals of Applied Probability 7, no. 3 (August 1997): 731–46. http://dx.doi.org/10.1214/aoap/1034801251.

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37

White, Gary C. "Correcting wildlife counts using detection probabilities." Wildlife Research 32, no. 3 (2005): 211. http://dx.doi.org/10.1071/wr03123.

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One of the most pervasive uses of indices of wildlife populations is uncorrected counts of animals. Two examples are the minimum number known alive from capture and release studies, and aerial surveys where the detection probability is not estimated from a sightability model, marked animals, or distance sampling. Both the mark–recapture and distance-sampling estimators are techniques to estimate the probability of detection of an individual animal (or cluster of animals), which is then used to correct a count of animals. However, often the number of animals in a survey is inadequate to compute an estimate of the detection probability and hence correct the count. Modern methods allow sophisticated modelling to estimate the detection probability, including incorporating covariates to provide additional information about the detection probability. Examples from both distance and mark–recapture sampling are presented to demonstrate the approach.
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38

Li, Min-Tang, Lee-Fang Chow, Fang Zhao, and Shi-Chiang Li. "Geographically Stratified Importance Sampling for the Calibration of Aggregated Destination Choice Models for Trip Distribution." Transportation Research Record: Journal of the Transportation Research Board 1935, no. 1 (January 2005): 85–92. http://dx.doi.org/10.1177/0361198105193500110.

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A key feature in estimating and applying destination choice models with aggregate alternatives is to sample a set of nonchosen traffic analysis zones (TAZs), plus the one a trip maker chose, to construct a destination choice set. Computational complexity is reduced because the choice set would be too large if all study area TAZs were included in the calibration. Commonly, two types of sampling strategies are applied to draw subsets of alternatives from the universal choice set. The first, and simplest, approach is to select randomly a subset of nonchosen alternatives with uniform selection probabilities and then add the chosen alternative if it is not otherwise included. The approach, however, is not an efficient sampling scheme because most alternatives for a given trip maker may have small choice probabilities. The second approach, stratified importance sampling, draws samples with unequal selection probabilities determined on the basis of preliminary estimates of choice probabilities for every alternative in the universal choice set. The stratified sampling method assigns different selection probabilities to alternatives in different strata. Simple random sampling is applied to draw alternatives in each stratum. However, it is unclear how to divide the study area so that destination TAZs may be sampled effectively. The process of and findings from implementing a stratified sampling strategy in selecting alternative TAZs for calibrating aggregate destination choice models in a geographic information system (GIS) environment are described. In this stratified sampling analysis, stratum regions varied by spatial location and employment size in the adjacent area were defined for each study area TAZ. The sampling strategy is more effective than simple random sampling in regard to maximum log likelihood and goodness-of-fit values.
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39

Gamrot, Wojciech. "A Stopping Rule for Simulation‑Based Estimation of Inclusion Probabilities." Acta Universitatis Lodziensis. Folia Oeconomica 4, no. 349 (September 22, 2020): 67–80. http://dx.doi.org/10.18778/0208-6018.349.04.

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Design‑based estimation of finite population parameters such as totals usually relies on the knowledge of inclusion probabilities characterising the sampling design. They are directly incorporated into sampling weights and estimators. However, for some useful sampling designs, these probabilities may remain unknown. In such a case, they may often be estimated in a simulation experiment which is carried out by repeatedly generating samples using the same sampling scheme and counting occurrences of individual units. By replacing unknown inclusion probabilities with such estimates, design‑based population total estimates may be computed. The calculation of required sample replication numbers remains an important challenge in such an approach. In this paper, a new procedure is proposed that might lead to the reduction in computational complexity of simulations.
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40

Zibdeh, H. S. "Reliability of Thermally Loaded Cylinders." Journal of Pressure Vessel Technology 112, no. 3 (August 1, 1990): 303–8. http://dx.doi.org/10.1115/1.2928630.

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Cylinders often contain fluids at various temperatures. Temperature gradients develop across the cylinder which in turn produce thermal stresses. Using a perturbation technique, expressions for means and standard deviations of thermal stresses in cylinders are presented in this paper. These expressions include the probabilistic effects of changes due to inner and outer radii, modulus of elasticity, Poisson’s ratio, coefficient of thermal expansion, and temperature. It is found out that the cylinder is more sensitive to the geometrical changes than the mechanical and thermal changes. It is also noticed that the effect of variation in the inner radius is more than the outer radius at the inner half of the cylinder. On the other hand, the effect of variation in the outer radius is more than the inner radius at the outer half of the cylinder. Probabilities of failure are calculated for normal and log-normal distributions.
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41

Aydogan, Ilke, and Yu Gao. "Experience and rationality under risk: re-examining the impact of sampling experience." Experimental Economics 23, no. 4 (January 16, 2020): 1100–1128. http://dx.doi.org/10.1007/s10683-019-09641-y.

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Abstract A recent strand of the literature on decision-making under uncertainty has pointed to an intriguing behavioral gap between decisions made from description and decisions made from experience. This study reinvestigates this description-experience gap to understand the impact that sampling experience has on decisions under risk. Our study adopts a complete sampling paradigm to address the lack of control over experienced probabilities by requiring complete sampling without replacement. We also address the roles of utilities and ambiguity, which are central in most current decision models in economics. Thus, our experiment identifies the deviations from expected utility due to over- (or under-) weighting of probabilities. Our results confirm the existence of the behavioral gap, but they provide no evidence for the underweighting of small probabilities within the complete sampling treatment. We find that sampling experience attenuates rather than reverses the inverse S-shaped probability weighting under risk.
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42

Metzler, Adam, and Alexandre Scott. "Importance Sampling in the Presence of PD-LGD Correlation." Risks 8, no. 1 (March 10, 2020): 25. http://dx.doi.org/10.3390/risks8010025.

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This paper seeks to identify computationally efficient importance sampling (IS) algorithms for estimating large deviation probabilities for the loss on a portfolio of loans. Related literature typically assumes that realised losses on defaulted loans can be predicted with certainty, i.e., that loss given default (LGD) is non-random. In practice, however, LGD is impossible to predict and tends to be positively correlated with the default rate and the latter phenomenon is typically referred to as PD-LGD correlation (here PD refers to probability of default, which is often used synonymously with default rate). There is a large literature on modelling stochastic LGD and PD-LGD correlation, but there is a dearth of literature on using importance sampling to estimate large deviation probabilities in those models. Numerical evidence indicates that the proposed algorithms are extremely effective at reducing the computational burden associated with obtaining accurate estimates of large deviation probabilities across a wide variety of PD-LGD correlation models that have been proposed in the literature.
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43

Smith, Peter J. "Underestimation of Rare Event Probabilities in Importance Sampling Simulations." SIMULATION 76, no. 3 (March 2001): 140–50. http://dx.doi.org/10.1177/003754970107600301.

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44

Mehta, Cyrus R., Nitin R. Patel, and Pralay Senchaudhuri. "Importance Sampling for Estimating Exact Probabilities in Permutational Inference." Journal of the American Statistical Association 83, no. 404 (December 1988): 999–1005. http://dx.doi.org/10.1080/01621459.1988.10478691.

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45

Moors, J. J. A., R. Smeets, and F. W. M. Boekema. "Sampling with probabilities proportional to the variable of interest." Statistica Neerlandica 52, no. 2 (June 1998): 129–40. http://dx.doi.org/10.1111/1467-9574.00073.

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46

Hedayat, A. S., and John Stufken. "Sampling designs to control selection probabilities of contiguous units." Journal of Statistical Planning and Inference 72, no. 1-2 (September 1998): 333–45. http://dx.doi.org/10.1016/s0378-3758(98)00041-x.

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47

Ross, Sheldon M. "Using the Importance Sampling Identity to Bound Tail Probabilities." Probability in the Engineering and Informational Sciences 12, no. 4 (October 1998): 445–52. http://dx.doi.org/10.1017/s0269964800005313.

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48

Frey, Jesse. "Recursive computation of inclusion probabilities in ranked-set sampling." Journal of Statistical Planning and Inference 141, no. 11 (November 2011): 3632–39. http://dx.doi.org/10.1016/j.jspi.2011.05.017.

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49

Metcalfe, A. V. "Ratio Estimators with Unequal Probabilities Sampling: A Practical Application." Statistician 38, no. 4 (1989): 275. http://dx.doi.org/10.2307/2349060.

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50

PALMGREN, JUNI. "Precision of double sampling estimators for comparing two probabilities." Biometrika 74, no. 4 (1987): 687–94. http://dx.doi.org/10.1093/biomet/74.4.687.

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