Two-stage Sequential Sampling Research Articles

Estimation of fixed-accuracy confidence interval of the stress–strength reliability for inverse Pareto distribution using two-stage sampling technique

In recent years, several probability distributions have been introduced in the literature to analyze the data exhibiting an upside-down bathtub–shaped failure rate; an inverse Pareto distribution (IPD) is an appropriate choice among them. For stress–strength reliability models, estimation of parameters is an interesting area of research. In this article, we estimate the stress–strength reliability parameter R = P ( X > Y ) (where X and Y are strength and stress variables, respectively) of the IPD, whereby we focus on the problem of fixed-accuracy confidence interval estimation of R. It is established that the proposed interval estimation problem cannot be solved with the help of any fixed sample technique. As a result, we propose a two-stage sequential sampling strategy (which reduces the sample size significantly) to solve the given estimation problem. We obtain the expressions of several exact operating characteristics associated with our two-stage sampling technique. We also establish that the proposed two-stage procedure enjoys interesting first-order asymptotic properties. The detailed simulation analyses support our theoretical findings, and two real data sets based on insurance claims reinforce the practical utility of the proposed technique.

Two-stage estimation of the combination of location and scale parameter of the exponential distribution under the constraint of bounded risk per unit cost index

We consider the problem of bounded risk point estimation for the linear combination of the form where and are the location and scale parameters of a exponential distribution and and are constant. We aim to estimate under the modified squared error loss function using the constraint that the risk per unit cost is bounded above with fixed preassigned, . The two-stage sequential sampling is proposed for estimating The performances of the proposed methodologies are investigated with the help of simulations. Finally, using an actual dataset, the procedure is clearly illustrated.

Bounded risk per unit cost index constraint for sequential estimation of the mean in a two-parameter exponential distribution

In this article, the two-stage sequential sampling procedure is proposed to estimate the mean of an exponential distribution under the modified square error loss function. The main aim of the article is to consider the associated risk per unit cost function by bounding it from above with a fixed preassigned positive number, ω. We provide the exact distribution of the total sample size, explicit formulas for the mathematical expectation of the rth power of the total sample size, the expected value and mean squared error of the maximum likelihood estimate of the scale, and mean parameters of the exponential distribution under the two-stage sequential procedure. The performances of the proposed methodologies are investigated with the help of simulations. In the end, we supplement with extensive sets of data analysis via computer simulations, validating that the two-stage sequential method performs very well. Finally, using a real data set, the procedure is clearly illustrated.

A Comparison of Two-Stage Scheme and Fully Sequential Sampling Scheme for Estimating Software Reliability

In this paper, we compare two-stage sequential sampling scheme with fully sequential sampling scheme to test software and estimate reliability. In two-stage sampling scheme, test cases can be allocated among partitions in two phases. Our goal of this scheme is to obtain the near-optimal choices for distributing of test cases among sub-domains by minimizing the variance of the overall software reliability estimator. The two-stage sampling scheme is expected to be more convenient than a fully sequential sampling scheme because it requires fewer computations than the fully sequential sampling scheme. Also, the two-stage sampling scheme is expected to perform better than a balanced sampling scheme by virtue of lower the variance incurred by the overall estimated software reliability

Regression-type estimators for adaptive two-stage sequential sampling

Adaptive two-stage sequential sampling (ATSSS) design was developed to observe more rare units and gain higher efficiency, in the sense of having a smaller variance estimator, than conventional sampling designs with equal effort for rare and spatially cluster populations. For certain rare populations, incorporating auxiliary variables into a sampling design can further improve the observation of rare units and increase efficiency. In this article, we develop regression-type estimators for ATSSS so that auxiliary variables can be incorporated into the ATSSS design when warranted. Simulation studies on two populations show that the regression-type estimators can significantly increase the efficiency of ATSSS and the detection of more rare units as compared to conventional sampling counterparts. Simulation of sampling of desert shrubs in Inner Mongolia (one of the two populations studied) showed that by incorporating a GIS auxiliary variable into ATSSS with the regression estimators resulted in a gain in efficiency over ATSSS without the auxiliary variable. Further, we found that the use of the GIS auxiliary variable in a conventional two-stage design with a regression estimator did not show a gain in efficiency.

Adaptive survey designs for sampling rare and clustered populations

Designing an efficient large-area survey is a challenge, especially in environmental science when many populations are rare and clustered. Adaptive and unequal probability sampling designs are appealing when populations are rare and clustered because survey effort can be targeted to subareas of high interest. For example, higher density subareas are usually of more interest than lower density areas. Adaptive and unequal probability sampling offer flexibility for designing a long-term survey because they can accommodate changes in survey objectives, changes in underlying environmental habitat, and changes in species-habitat models. There are many different adaptive sampling designs including adaptive cluster sampling, two-phase stratified sampling, two-stage sequential sampling, and complete allocation stratified sampling. Sample efficiency of these designs can be very high compared with simple random sampling. Large gains in efficiency can be made when survey effort is targeted to the subareas of the study site where there are clusters of individuals from the underlying population. These survey methods work by partitioning the study area in some way, into strata, or primary sample units, or in the case of adaptive cluster sampling, into networks. Survey effort is then adaptively allocated to the strata or primary unit where there is some indication of higher species counts. Having smaller, and more numerous, strata improves efficiency because it allows more effective targeting of the adaptive, second-phase survey effort.

Two-stage sequential sampling for two rare species in western Inner Mongolia of China

To evaluate the efficiencies of different sampling methods for a rare and clustered population, we investigated the sampling effects for the two species Tamarix chinensis (Salt cedar) and Elaeagnus angustifolia (Russian olive) in western Inner Mongolia with two-stage sequential sampling, which is a new sampling method, traditional simple random sampling and two-stage sampling. Based on two-stage sequential sampling and two-stage sampling, each population was partitioned into four primary sampling units, and then two of them were randomly selected. Sampling designs were simulated based on the conditions of five secondary sampling unit areas, two criterion values, five initial secondary sampling units and two sequential secondary sampling units in 1000 repetitions. To evaluate the performance of the sampling designs for each method, the variance and relative error of the density estimates were used. The relative sampling efficiencies of the three sampling methods were compared using the same final sampling sizes. We analyzed the sampling efficiency generated by two-stage sequential sampling and found that it yielded smaller variances than those of simple random sampling and two-stage sampling in all sampling designs, and that two-stage sampling was more efficient than simple random sampling. Density estimates from the two-stage sequential sampling were very close to the true values. We also determined the optimum secondary sampling unit areas for the two species in the two-stage sequential sampling. It was best for Tamarix chinensis and Elaeagnus angustifolia when the secondary sampling unit areas were 200 and 100 m2, respectively.

Performance comparison of traditional sampling designs and adaptive sampling designs for fishery-independent surveys: A simulation study

We compared the performance of two traditional sampling designs with three adaptive sampling designs using simulated data based on fishery-independent surveys for yellow perch in Lake Erie. Traditionally, the fishery-independent survey has been conducted with a stratified random sampling design based on basin and depth strata; however, adaptive sampling designs are thought to be more suitable for surveying heterogeneous populations. A simulation study was conducted to compare these designs by examining the accuracy and precision of the estimators. Initially in the simulation study, we used bias, variance of the mean, and mean squared error (MSE) of the estimators to compare simple random sampling (SRS), stratified random sampling (StRS), and adaptive two-phase sampling (ATS). ATS was the best design according to these measurements. We then compared ATS, adaptive cluster sampling (ACS), adaptive two-stage sequential sampling (ATSS), and the currently used stratified random sampling design. ATS performed better than the other two approaches and the current stratified random sampling design. We concluded that ATS is preferable for yellow perch fishery-independent surveys in Lake Erie. Simulation study is a preferred approach when we seek an appropriate sampling design or evaluate the current sampling design.

Efficient estimators for adaptive stratified sequential sampling

In stratified sampling, methods for the allocation of effort among strata usually rely on some measure of within-stratum variance. If we do not have enough information about these variances, adaptive allocation can be used. In adaptive allocation designs, surveys are conducted in two phases. Information from the first phase is used to allocate the remaining units among the strata in the second phase. Brown et al. [Adaptive two-stage sequential sampling, Popul. Ecol. 50 (2008), pp. 239–245] introduced an adaptive allocation sampling design – where the final sample size was random – and an unbiased estimator. Here, we derive an unbiased variance estimator for the design, and consider a related design where the final sample size is fixed. Having a fixed final sample size can make survey-planning easier. We introduce a biased Horvitz–Thompson type estimator and a biased sample mean type estimator for the sampling designs. We conduct two simulation studies on honey producers in Kurdistan and synthetic zirconium distribution in a region on the moon. Results show that the introduced estimators are more efficient than the available estimators for both variable and fixed sample size designs, and the conventional unbiased estimator of stratified simple random sampling design. In order to evaluate efficiencies of the introduced designs and their estimator furthermore, we first review some well-known adaptive allocation designs and compare their estimator with the introduced estimators. Simulation results show that the introduced estimators are more efficient than available estimators of these well-known adaptive allocation designs.

Two-stage sequential sampling: A neighborhood-free adaptive sampling procedure

Designing an efficient sampling scheme for a rare and clustered population is a challenging area of research. Adaptive cluster sampling, which has been shown to be viable for such a population, is based on sampling a neighborhood of units around a unit that meets a specified condition. However, the edge units produced by sampling neighborhoods have proven to limit the efficiency and applicability of adaptive cluster sampling. We propose a sampling design that is adaptive in the sense that the final sample depends on observed values, but it avoids the use of neighborhoods and the sampling of edge units. Unbiased estimators of population total and its variance are derived using Murthy’s estimator. The modified two-stage sampling design is easy to implement and can be applied to a wider range of populations than adaptive cluster sampling. We evaluate the proposed sampling design by simulating sampling of two real biological populations and an artificial population for which the variable of interest took the value either 0 or 1 (e.g., indicating presence and absence of a rare event). We show that the proposed sampling design is more efficient than conventional sampling in nearly all cases. The approach used to derive estimators (Murthy’s estimator) opens the door for unbiased estimators to be found for similar sequential sampling designs.

A Two-Stage Sequential Sampling Scheme for Warner's Randomized Response Model

In this article we revisit Warner's (Warner, S. L. (1965). Randomized response: a survey technique for elimination evasive answer bias. Journal of the American Statistical Association 60:63–69) randomize response model for estimating the proportion of “sensitive” attributed in a population and propose a two-stage sequential sampling procedure for it. We show that the new procedure can potentially reduce, on the average, the number of interviewees surveyed in the study while allowing instances with smaller error of estimation. The properties and some of the attractive features of the randomized response two-stage sequential sampling estimation procedure is discussed as illustrated.

Two-stage point estimation with a shrinkage stopping rule

Consider the problem of estimating a mean vector in ap-variate normal distribution under two-stage sequential sampling schemes. The paper proposes a stopping rule motivated by the James-Stein shrinkage estimator, and shows that the stopping rule and the corresponding shrinkage estimator asymptotically dominate the usual two-stage procedure under a sequence of local alternatives forp≥3. Also the results of Monte Carlo simulation for average sample sizes and risks of estimators are stated.

Dispersion and Sequential Sampling of Green Cloverworm 1 Eggs in Soybeans 2

The microspatial patterns of green cloverworm eggs were studied within and among soybean plants. The upper leaf surface generally contained significantly more eggs than the lower surface. Few eggs (2.8%) were found on stem or pod surfaces. The distribution of eggs by relative nodal position was influenced by the stage of plant growth. The third node from the terminal was selected as the best relative nodal position from which to sample egg-bearing leaves. Among-plant data were expressed on the basis of three sample-unit sizes. Four dispersion indices were calculated for each data set expressed on the basis of each sample-unit size. The majority of data sets exhibited a random dispersion pattern, although the proportion of data sets categorized as random increased with increasing sample-unit size. Taylor's power function also indicated a random dispersion pattern for all sample-unit sizes. The results of Iwao's regression technique were inconclusive. A two-stage sequential sampling plan is described that estimates egg densities with a predetermined precision level. This plan consists of random samples of leaves, for which variance-to-cost relationships indicated that the optimal number of leaves per sample was four. Cost estimates indicate that this plan is suitable for ecological population studies.