Nonparametric function estimation from inversely sampled record-breaking data

Often, in industrial stress testing, meteorological data analysis, and other similar situations, measurements may be made sequentially and only values smaller than all previous ones are recorded. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. This paper is concerned with nonparametric estimation of the distribution and density functions from such data (successive minima). For a single record-breaking sample, consistent estimation is not possible except in the extreme left tail of the distribution. Hence, replication is required, and for m such independent record-breaking samples, the estimators are shown to be strongly consistent and asymptotically normal as m ∞ →. Computer simulations are used to investigate the effect of the bandwidth on the mean squared errors and biases of the smooth estimators, and are also used to provide a comparison of their performance with the analogous estimators obtained under random sampling for record values.     

On the Evaluation of the Kolmogorov Statistic

At least two computer program packages, SPSS and STRATA, use simulated Bernoulli trials to draw (without replacement) a random sample of records from a finite population of records. Therefore, the size of the sample is a random variable. Two estimators of a population total under this sampling procedure are compared with the usual estimator under simple random sampling. Conditions under which the Bernoulli sampling estimators have almost the same mean squared error as the simple random-sample estimator are illustrated.     

NONPARAMETRIC QUANTILE ESTIMATION FROM RECORD-BREAKING DATA

Sometimes, in industrial quality control experiments and destructive stress testing, only values smaller than all previous ones are observed. Here we consider nonparametric quantile estimation, both the ‘sample quantile function’ and kernel-type estimators, from such record-breaking data. For a single record-breaking sample, consistent estimation is not possible except in the extreme tails of the distribution. Hence replication is required, and for m. such independent record-breaking samples the quantile estimators are shown to be strongly consistent and asymptotically normal as m-→∞. Also, for small m, the mean-squared errors, biases and smoothing parameters (for the smoothed estimators) are investigated through computer simulations.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.     

Inverse Adaptive Cluster Sampling with Unequal Selection Probabilities: Case Studies on Crab Holes and Arsenic Pollution

Adaptive cluster sampling is an efficient method of estimating the parameters of rare and clustered populations. The method mimics how biologists would like to collect data in the field by targeting survey effort to localised areas where the rare population occurs. Another popular sampling design is inverse sampling. Inverse sampling was developed so as to be able to obtain a sample of rare events having a predetermined size. Ideally, in inverse sampling, the resultant sample set will be sufficiently large to ensure reliable estimation of population parameters. In an effort to combine the good properties of these two designs, adaptive cluster sampling and inverse sampling, we introduce inverse adaptive cluster sampling with unequal selection probabilities. We develop an unbiased estimator of the population total that is applicable to data obtained from such designs. We also develop numerical approximations to this estimator. The efficiency of the estimators that we introduce is investigated through simulation studies based on two real populations: crabs in Al Khor, Qatar and arsenic pollution in Kurdistan, Iran. The simulation results show that our estimators are efficient.     

Estimating the population mean in the case of missing data using simple random sampling

In this paper, we suggest three new ratio estimators of the population mean using quartiles of the auxiliary variable when there are missing data from the sample units. The suggested estimators are investigated under the simple random sampling method. We obtain the mean square errors equations for these estimators. The suggested estimators are compared with the sample mean and ratio estimators in the case of missing data. Also, they are compared with estimators in Singh and Horn [Compromised imputation in survey sampling, Metrika 51 (2000), pp. 267–276], Singh and Deo [Imputation by power transformation, Statist. Papers 45 (2003), pp. 555–579], and Kadilar and Cingi [Estimators for the population mean in the case of missing data, Commun. Stat.-Theory Methods, 37 (2008), pp. 2226–2236] and present under which conditions the proposed estimators are more efficient than other estimators. In terms of accuracy and of the coverage of the bootstrap confidence intervals, the suggested estimators performed better than other estimators.     

Smooth nonparametric estimation of thedistribution and density functions from record-breaking data

In some experiments, such as destructive stress testing and industrial quality control experiments, only values smaller than all previous ones are observed. Here, for such record-breaking data, kernel estimation of the cumulative distribution function and smooth density estimation is considered. For a single record-breaking sample, consistent estimation is not possible, and replication is required for global results. For m independent record-breaking samples, the proposed distribution function and density estimators are shown to be strongly consistent and asymptotically normal as m → ∞. Also, for small m, the mean squared errors and biases of the estimators and their smoothing parameters are investigated through computer simulations.     

On inverse sampling without replacement

The inverse sampling (due to Haldane, 1945) was proposed for random sampling with replacement draws when “the numberr of successes in the sample until the process of drawing is stopped” is prefixed. In this paper an extension of that result is proposed when the draws are obtained via random sampling without replacement.     

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.     

Estimation of the means of the bivariate normal using moving extreme ranked set sampling with concomitant variable

The estimation of the means of the bivariate normal distribution, based on a sample obtained using a modification of the moving extreme ranked set sampling technique (MERSS) is considered. The modification involves using a concomitant random variable. Nonparametric-type methods as well as the maximum likelihood estimation are considered. The estimators obtained are compared to their counterparts based on simple random sampling (SRS). It appears that the suggested estimators are more efficient. Also, MERSS with concomitant variable is easier to use in practice than the usual ranked set sampling (RSS) with concomitant variable. The issue of robustness of the procedure is addressed. Real trees data set is used for illustration.     

Estimation of the shape and scale parameters of Pareto distribution using ranked set sampling

In the current paper, the estimation of the shape and location parameters α and c, respectively, of the Pareto distribution will be considered in cases when c is known and when both are unknown. Simple random sampling (SRS) and ranked set sampling (RSS) will be used, and several traditional and ad hoc estimators will be considered. In addition, the estimators of α, when c is known using an RSS version based on the order statistic that maximizes the Fisher information for a fixed set size, will be considered. These estimators will be compared in terms of their biases and mean square errors. The estimators based on RSS can be real competitors against those based on SRS.     

Optimized estimation for population mean using conventional and non-conventional measures under the joint influence of measurement error and non-response

Most of the research work in the theory of survey sampling only deals with the sampling errors under the assumptions: (i) there is a complete response and (ii) recorded information from individuals is correct but in practice it is not always true. Non-sampling errors like non-response and measurement errors (MEs) mostly creep into the survey and become more influential for estimators than sampling errors. Considering this practical situation of non-response and MEs jointly, we proposed an optimum class of estimators for population mean under simple random sampling using conventional and non-conventional measures. Bias and mean square error of the proposed estimators are derived up to first degree of approximation. Moreover, a simulation study is conducted to assess the performance of new estimators which proves that proposed estimators are more efficient than the traditional Hansen and Hurwitz estimator and other competing estimators.     

Unequal probability sampling in fixed area plots of stem volume with and without prior inclusion probabilities

The impact of guessing auxiliary population attributes, as opposed to relying on actual values from a prior survey, was quantified for three unequal probability sampling methods of tree stem volume (biomass). Reasonable prior guesses (no-list sampling) yielded, in five populations and 35 combinations of population size and sample size, results at par with sampling with known auxiliary predictors (list sampling). Realized sample sizes were slightly inflated in no-list sampling with probability proportional to predictions ( PPP ). Mean absolute differences from true totals and root mean square errors in no-list-sampling schemes were only slightly above those achieved with list sampling. Stratified sampling generally outperformed PPP and systematic sampling, yet the latter is recommended due to consistency between observed and expected mean square errors and overall robustness against a systematic bias in no-list settings.     

Use of ranks in probability propoetional to size sampling

Use of ranks in unequal probability sampling is examined for sample selection, stratification as well as determining the strata boundaries. A few sampling schemes are proposed and investigated, For samples of size two, two sampling schemes and their 1PPS versions are discussed, An extension of these schemes to general sample sizes is outlined. Nonnegative unbiased variance estimators are proposed in each case, An empirical comparison is included.     

A modified bootstrap procedure for cluster sampling variance estimation of species richness

Variance estimators for probability sample-based predictions of species richness (S) are typically conditional on the sample (expected variance). In practical applications, sample sizes are typically small, and the variance of input parameters to a richness estimator should not be ignored. We propose a modified bootstrap variance estimator that attempts to capture the sampling variance by generating B replications of the richness prediction from stochastically resampled data of species incidence. The variance estimator is demonstrated for the observed richness (SO), five richness estimators, and with simulated cluster sampling (without replacement) in 11 finite populations of forest tree species. A key feature of the bootstrap procedure is a probabilistic augmentation of a species incidence matrix by the number of species expected to be ‘lost’ in a conventional bootstrap resampling scheme. In Monte-Carlo (MC) simulations, the modified bootstrap procedure performed well in terms of tracking the average MC estimates of richness and standard errors. Bootstrap-based estimates of standard errors were as a rule conservative. Extensions to other sampling designs, estimators of species richness and diversity, and estimates of change are possible.     

A general inverse sampling scheme and its application to adaptive cluster sampling

Not having a variance estimator is a seriously weak point of a sampling design from a practical perspective. This paper provides unbiased variance estimators for several sampling designs based on inverse sampling, both with and without an adaptive component. It proposes a new design, which is called the general inverse sampling design, that avoids sampling an infeasibly large number of units. The paper provide estimators for this design as well as its adaptive modification. A simple artificial example is used to demonstrate the computations. The adaptive and non‐adaptive designs are compared using simulations based on real data sets. The results indicate that, for appropriate populations, the adaptive version can have a substantial variance reduction compared with the non‐adaptive version. Also, adaptive general inverse sampling with a limitation on the initial sample size has a greater variance reduction than without the limitation.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

Finite-population variance estimation under systematic sampling schemes with multiple random starts

Motivated by Sampath [Finite population variance estimation under LSS with multiple random starts, Commun. Statist. – Theory Methods 38 (2009), pp. 3596–3607], in this paper unbiased estimators for population variance have been developed under linear systematic sampling, balanced systematic sampling and modified systematic sampling with multiple random starts. Expressions for variances of the estimators are also developed. Detailed numerical comparative studies have been carried out to study the performances of the estimators under various systematic sampling schemes with multiple random starts and some interesting conclusions have been drawn out of the study.     

Estimations of the parameter of a Dirichlet distribution using residual allocation model representations and sampling properties

Estimating the parameter of a Dirichlet distribution is an interesting question since this distribution arises in many situations of applied probability. Classical procedures are based on sample of Dirichlet distribution. In this paper we exhibit five different estimators from only one observation. They are based either on residual allocation model decompositions or on sampling properties of Dirichlet distributions. Two ways are investigated: the first one uses fragments’ size and the second one uses size-biased permutations of a partition. Numerical computations based on simulations are supplied. The estimators are finally used to estimate birth probabilities per month.