Minimum Risk Point Estimation Research Articles

Multi-stage minimum risk point estimation (MRPE) of a function of unknown $$\varvec{\beta }$$ in a Gamma($$\varvec{\alpha ,\beta }$$) model with $$\varvec{\alpha }$$ known: related problems and illustrations with analysis from simulations and cancer data

Multi-stage minimum risk point estimation (MRPE) of a function of unknown $$\varvec{\beta }$$ in a Gamma($$\varvec{\alpha ,\beta }$$) model with $$\varvec{\alpha }$$ known: related problems and illustrations with analysis from simulations and cancer data

Multi-stage Minimum Risk Point Estimation Strategies for Comparing the Locations from Two Negative Exponential Models and Second-Order Asymptotics: Illustrations with Simulated Data and Bone Marrow Transplant Data

Multi-stage Minimum Risk Point Estimation Strategies for Comparing the Locations from Two Negative Exponential Models and Second-Order Asymptotics: Illustrations with Simulated Data and Bone Marrow Transplant Data

Triple Sampling Inference Procedures for the Mean of the Normal Distribution When the Population Coefficient of Variation Is Known

This paper discusses the triple sampling inference procedures for the mean of a symmetric distribution—the normal distribution when the coefficient of variation is known. We use the Searls’ estimator as an initial estimate for the unknown population mean rather than the classical sample mean. In statistics literature, the normal distribution under investigation underlines almost all the natural phenomena with applications in many fields. First, we discuss the minimum risk point estimation problem under a squared error loss function with linear sampling cost. We obtained all asymptotic results that enhanced finding the second-order asymptotic risk and regret. Second, we construct a fixed-width confidence interval for the mean that satisfies at least a predetermined nominal value and find the second-order asymptotic coverage probability. Both estimation problems are performed under a unified optimal framework. The theoretical results reveal that the performance of the triple sampling procedure depends on the numerical value of the coefficient of variation—the smaller the coefficient of variation, the better the performance of the procedure.

Distribution-free minimum risk point estimation of the mean under powered absolute error loss plus cost of sampling: Illustrations with cancer data

We begin with a discussion of results on the asymptotic properties associated with purely sequential minimum risk point estimation (MRPE) methodology for an unknown mean when the distribution is unspecified, under squared error loss (SEL) plus the cost of sampling. We proceed to show that, given certain moment conditions, such properties can be extended to estimation problems under a broader class of loss functions, namely: first order risk-efficiency under powered absolute error loss (PAEL) plus the powered cost of sampling. In addition to purely sequential strategy, we also prove these asymptotic properties for a more operationally convenient, accelerated sequential procedure. We follow-up with extensive data analyses from simulations under mixture distributions. We conclude with illustrations using data from two cancer studies.

An optimal purely sequential strategy with asymptotic second-order properties: Applications from statistical inference and data analysis

We develop a new class of purely sequential methodologies under an assumption that the population distribution belongs to a location-scale family. Both asymptotic first-order and second-order theories are put forward with substantial generality under a big and unified tent that successfully lead to a broad set of illustrations. After we identify an appropriately defined optimal strategy under this unified structure, we introduce applications that handle a variety of interesting inference problems. These are associated with, but not limited to, the following areas: (a) the fixed-width confidence interval (FWCI) estimation, (b) the minimum risk point estimation (MRPE), (c) the fixed-size confidence region (FSCR) estimation, (d) multiple comparisons, and (e) selecting the best normal treatment (StBNT). In illustrations (a)–(d), we have highlighted a number of choices of population distributions. Some illustrations are accompanied with data analyses.

Purely Sequential Minimum Risk Point Estimation (MRPE) for a Survival Function in an Exponential Distribution: Illustration with Remission Times for Bladder Cancer Patients

Purely Sequential Minimum Risk Point Estimation (MRPE) for a Survival Function in an Exponential Distribution: Illustration with Remission Times for Bladder Cancer Patients

Three-stage minimum risk point estimation with termination defined via Gini’s mean difference

In this article, we revisit the classic inference problem of minimum risk point estimation for an unknown normal mean when the variance also remains unknown. We propose an alternative three-stage sampling procedure with termination defined via Gini’s mean difference rather than the traditional sample standard deviation. A number of asymptotic properties are investigated both theoretically and empirically. An extensive set of simulations is conducted to demonstrate the remarkable performance of the new procedure. For practical purposes, we also include illustrations using real data sets on the number of days marigold seeds need to flower.

A new formulation of minimum risk fixed-width confidence interval (MRFWCI) estimation problems for a normal mean with illustrations and simulations: Applications to air quality data

Research on classical fixed-width confidence interval (FWCI) estimation problems for a normal mean when the variance remains unknown have steadily moved along under a zero-one loss function. On the other hand, minimum risk point estimation (MRPE) problems have grown largely under a squared error loss function plus sampling cost. However, the FWCI problems customarily do not take into account any sampling cost in their formulations. This fundamental difference between the two treatments has led the literature on the FWCI and MRPE problems to grow in multiple directions in their own separate ways from one another. In this article, we introduce a new formulation combining both MRPE and FWCI methodologies with desired asymptotic first-order (Theorems 2.1–2.2) and asymptotic second-order characteristics (Theorem 2.3) under a single unified structure, allowing us to develop a genuine minimum risk fixed-width confidence interval (MRFWCI) estimation strategy. Fruitful ideas are proposed by incorporating illustrations from purely sequential and other multistage MRFWCI problems with an explicit presence of a cost function incurred due to sampling. We supplement the general theory and methodology by means of illustrations and analyses from simulated data along with applications to air quality data.

Second-order asymptotics for comparing treatment means from purely sequential estimation strategies under possible outlying observations

We consider two independent populations, and with all four parameters assumed unknown, and In this paper, our goal is to revisit sequential minimum risk point estimation (MRPE) problems for the parameter say) under the squared error loss (SEL) plus linear cost of sampling in our quest for comparing two treatments. We explicitly consider two separate scenarios: (i) but is assumed unknown, and (ii) are unknown and unequal. In defining the relevant purely sequential stopping rules, customarily one incorporates multiples of sample standard deviations to set up requisite boundary crossing conditions. We set out to replace the multiples of sample standard deviations used in defining requisite boundary crossing conditions with Gini’s mean difference (GMD), mean absolute deviation (MAD), along with a number of combinations of the sample standard deviations, the GMD’s, and the MAD’s. We prove that the ensuing body of such newly developed purely sequential MRPE strategies have associated second-order regret expansions and that the GMD-based as well as the MAD-based strategies are better equipped to withstand occurrences of possible outlying observations. Our theory and methodologies are amply supported by both large-scale simulations and the illustrations with real data.

Minimum risk point estimation for a function of a normal mean under weighted power absolute error loss plus cost: First-order and second-order asymptotics

We have designed multistage minimum risk point estimation (MRPE) strategies for a function of an unknown mean in a normal population with its variance unknown. These are developed under a weighted power absolute error loss (PAEL) function plus nonlinear cost of sampling by incorporating purely sequential, accelerated sequential, and three-stage estimation methodologies. Crucial asymptotic first-order and asymptotic second-order properties have been proved thoroughly under all three MRPE strategies. Extensive sets of simulations tend to validate nearly all desirable asymptotic properties for small to medium to large optimal fixed sample sizes.

Purely Sequential Point Estimation of a Function of the Mean in an Exponential Distribution

A purely sequential minimum risk point estimation methodology with an associated stopping time N is designed to come up with a unified strategy in an exponential distribution (Expo). We work under a weighted squared error loss due to the estimator $$g(\overline{X}_{N})$$ of $$g(\beta ),$$ a function of the unknown mean parameter $$\beta (>0)$$ , plus linear cost of sampling from an Expo $$(\beta )$$ population. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, $$\rightarrow 0$$ ) results are laid out including (1) the first- and second-order efficiency properties, (2) the risk efficiency property, and (3) the second-order regret expansion under suitable conditions on g(.).

Minimum risk point estimation of the size of a finite population under mark–recapture strategy

In this article, we address the problem of minimum risk point estimation for the size of a closed population using the weighted squared error loss function with a penalty of cost for each observation under the well-known mark–recapture strategy. In order to achieve minimum risk, a crucial point is to determine the number of items to obtain in the recapture phase. We first develop a purely sequential sampling scheme that will require much less sampling operations without losing accuracy of the estimation. Then, we develop a novel practical accelerated sequential method that will further accelerate the whole recapture process when sampling can be easily done in batches. The stopping rules of both purely sequential and accelerated sequential sampling schemes possess desirable asymptotic properties. All theoretical findings are double validated by extensive data analyses.

Two-stage and sequential procedures for estimation of powers of parameter of a family of distributions

For the generalized positive exponential family of distributions, the problems of confidence interval estimation, bounded risk point estimation, and minimum risk point estimation of the rth power of unknown parameter θ are handled. Two-stage and purely sequential methodologies are developed for the estimation purpose. Sequential procedures are developed based on the maximum likelihood estimator (MLE) of the parameter. The exact operating characteristics of these procedures are given.

Multistage Estimation of the Rayleigh Distribution Variance

In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results.

A double-sequential sampling scheme

In this article, we have developed a new sequential sampling scheme, called the double-sequential sampling scheme, in which sampling is proceeded k-at-a-time () part of way, and then one-at-a-time sequentially. Requiring the same projected sample size as the Anscombe-Chow-Robbins (ACR) scheme, it further helps to save sampling operations. We include the statistical inference problems of fixed-width confidence interval estimation and minimum risk point estimation for a normal mean when the population variance also remains unknown as two illustrations. These are followed by extensive sets of simulations and real-data illustrations using horticulture data, showing great practical applicability of this new sequential sampling scheme.

A broader class of modified two-stage minimum risk point estimation procedures for a normal mean

In this paper, we design an innovative and general class of modified two-stage sampling schemes to enhance double sampling and modified double sampling procedures. Under the squared error loss plus linear cost of sampling, we revisit the classic problem of minimum risk point estimation (MRPE) for an unknown normal mean μ () when the population variance () also remains unknown. With stopping variables constructed based on an arbitrary general estimator Wm for σ, which satisfies a set of certain conditions, our procedures are proved to enjoy asymptotic first- and second-order efficiency as well as asymptotic first-order risk efficiency. For illustrative purposes, we further investigate specific modified two-stage MRPE procedures, where we substitute appropriate multiples of sample standard deviation, Gini’s mean difference (GMD), and mean absolute deviation (MAD) in the place of Wm , respectively. Extensive simulation studies are utilized to validate our theoretical findings. A real-life data set of weight change from female anorexic patients is then analyzed to demonstrate the practical applicability of these modified two-stage MRPE procedures. Comparing them in the case where there exist suspect outliers in the pilot sample, we are empirically confident that the GMD- and MAD-based procedures appear more robust than the sample-standard-deviation-based procedures.

Purely sequential estimation problems for the mean of a normal population by sampling in groups under permutations within each group and illustrations

Purely sequential estimation for unknown mean () in a normal population having an unknown variance () when observations are gathered in groups has been recently discussed in Mukhopadhyay and Wang (2020). In this article, we briefly revisit two fundamental problems on sequential estimation: (i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem. However, we substitute the estimators defining the stopping boundaries with newly constructed unbiased and consistent estimators under permutations within each group. These new estimators incorporated in the definition of the stopping boundaries have led to tighter estimation of requisite optimal fixed sample sizes. We have analyzed the first-order and second-order asymptotic properties under appropriate requirements on the pilot size. Large-scale computer simulations and substantial data analysis have validated such first-order and second-order results. The methodologies are illustrated with the help of time series data on offshore wind energy.

On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) strategies for a normal mean and optimality

We develop a generalized class of purely sequential sampling strategies associated with both fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) problems for the unknown mean $$\mu $$ of a normally distributed population having its variance $$\sigma ^{2}$$ also unknown. Under this newly proposed general class of associated estimation strategies, we develop a variety of asymptotic first-order and asymptotic second-order properties such as asymptotic consistency, first-order efficiency, first-order risk efficiency, second-order efficiency, and second-order regret analysis. Next, we proceed to locate an optimal strategy within our newly built large class of possibilities. Such optimality is defined as having been associated with the minimal second-order asymptotic variance of a stopping time within the general class of proposed strategies. We follow through by exploring both the FWCI and MRPE problems with the help of data analysis from simulations.

Minimum risk point estimation (MRPE) of the mean in an exponential distribution under powered absolute error loss (PAEL) due to estimation plus cost of sampling

We begin with a review of asymptotic properties of a purely sequential minimum risk point estimation (MRPE) methodology for an unknown mean in a one-parameter exponential distribution under a class of generalized loss functions. This class of powered absolute error loss (PAEL) includes both squared error loss (SEL) and absolute error loss (AEL) plus cost of sampling. We prove the asymptotic second-order efficiency property and asymptotic first-order risk efficiency property associated with the purely sequential MRPE problem. For operational convenience, we then move to implement an accelerated sequential MRPE methodology and prove the analogous asymptotic second-order efficiency property and asymptotic first-order risk efficiency property. We follow up with extensive data analysis from simulations and provide illustrations using cancer data.

Purely sequential FWCI and MRPE problems for the mean of a normal population by sampling in groups with illustrations using breast cancer data

Two fundamental problems on purely sequential estimation are revisited—(i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem—in the context of estimating an unknown mean (μ) in a normal population having an unknown variance (). We begin by laying down general frameworks for the second-order asymptotic analyses, in both problems, under sequential sampling of one observation at a time. Then, instead of gathering one observation at a time, we consider sequentially sampling k observations at a time in defining our proposed estimation strategies. We replace the customary sample standard deviation as an estimator for σ with a number of other pertinent estimators to come up with new and more appropriate stopping rules to suit the occasion. We do so because in real life we know that packaged items purchased in bulk often cost less per unit sample than the cost of an individual item. This article builds the whole array of estimation methodologies in order to address both FWCI and MRPE problems with appropriate first-order and second-order asymptotic analyses. These are followed by extensive sets of carefully laid out data analyses assisted via large-scale computer simulations. These are wrapped up with illustrations using breast cancer data.