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

Abstract

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.     

Sequential minimum risk point estimation (MRPE) methodology for a normal mean under Linex loss plus sampling cost: First-order and second-order asymptotics

Abstract

We have designed a sequential minimum risk point estimation (MRPE) strategy for the unknown mean of a normal population having its variance unknown too. This is developed under a Linex loss plus linear cost of sampling. A number of important asymptotic first-order and asymptotic second-order properties' characteristics have been developed and proved thoroughly. Extensive sets of simulations tend to validate nearly all of these asymptotic properties for small to medium to large optimal fixed sample sizes.     

Multistage estimation of the difference of locations of two negative exponential populations under a modified Linex loss function: Real data illustrations from cancer studies and reliability analysis

We have designed modified two-stage and purely sequential strategies to estimate the difference of location parameters from two independent negative exponential populations having unknown but proportional scale parameters under a modified Linex loss function. This article extends one-sample methodologies of Mukhopadhyay and Bapat (2016 Mukhopadhyay, N. and Bapat, S. R. (2016). Multistage Point Estimation Methodologies for a Negative Exponential Location under a Modified Linex Loss Function: Illustrations with Infant Mortality and Bone Marrow Data, Sequential Analysis 35: 175–206. http://dx.doi.org/10.1080/07474946.2016.1165532.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Sequential Analysis). Some preliminary results are established along the lines of Mukhopadhyay and Hamdy (1984 Mukhopadhyay, N. and Hamdy, H. I. (1984). On Estimating the Difference of Location Parameters of Two Negative Exponential Distributions, Canadian Journal of Statistics 12: 67–76.[Crossref] , [Google Scholar], Canadian Journal of Statistics) and Mukhopadhyay and Darmanto (1988 Mukhopadhyay, N. and Darmanto, S. (1988). Sequential Estimation of the Difference of Means of Two Negative Exponential Populations, Sequential Analysis 7: 165–190.[Taylor &; Francis Online] , [Google Scholar], Sequential Analysis). We have resorted to Mukhopadhyay and Duggan (1997 Mukhopadhyay, N. and Duggan, W. T. (1997). Can a Two-Stage Procedure Enjoy Second Order Properties? Sankhya, Series A 59: 435–448. [Google Scholar], Sankhya, Series A) in developing asymptotic second-order properties for the modified two-stage methodology and to nonlinear renewal theory of Lai and Siegmund (1977 Lai, T. L. and Siegmund, D. (1977). A Nonlinear Renewal Theory with Applications to Sequential Analysis I, Annals of Statistics 5: 946–954.[Crossref], [Web of Science ®] , [Google Scholar], 1979 Lai, T. L. and Siegmund, D. (1979). A Nonlinear Renewal Theory with Applications to Sequential Analysis II, Annals of Statistics 7: 60–76.[Crossref], [Web of Science ®] , [Google Scholar], Annals of Statistics) and Woodroofe (1977 Woodroofe, M. (1977). Second Order Approximation for Sequential Point and Interval Estimation, Annals of Statistics 5: 984–995.[Crossref], [Web of Science ®] , [Google Scholar], Annals of Statistics) in addressing analogous properties under the purely sequential methodology. Then, we supplement with extensive sets of data analysis via computer simulations validating that both modified two-stage and purely sequential methods perform very well. Both methodologies are also illustrated and implemented using real datasets from cancer studies and reliability analysis.     

Multistage point estimation methodologies for a negative exponential location under a modified linex loss function: Illustrations with infant mortality and bone marrow data

We have designed Stein-type (Stein, 1945 Stein, C. (1945). A Two Sample Test for a Linear Hypothesis Whose Power Is Independent of the Variance, Annals of Mathematical Statistics 16: 243–258.[Crossref] , [Google Scholar], Annals of Mathematical Statistics) two-stage, modified two-stage (Mukhopadhyay and Duggan, 1997 Mukhopadhyay, N. and Duggan, W. T. (1997). Can a Two-Stage Procedure Enjoy Second Order Properties? Sankhya, Series A 59: 435–448. [Google Scholar], Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965 Chow, Y. S. and Robbins, H. (1965). On the Asymptotic Theory of Fixed Width Sequential Confidence Intervals for the Mean, Annals of Mathematical Statistics 36: 457–462.[Crossref] , [Google Scholar], Annals of Mathematical Statistics) to estimate an unknown location parameter of a negative exponential distribution having an unknown scale parameter under a newly defined and modified Linex loss function. We aim at controlling the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, ω, and we emphasize both asymptotic first-order and asymptotic second-order properties for the modified two-stage and purely sequential estimation strategies. In developing asymptotic second-order properties for the modified two-stage methodology, we have heavily relied upon basic ideas rooted in Mukhopadhyay and Duggan (1997 Mukhopadhyay, N. and Duggan, W. T. (1997). Can a Two-Stage Procedure Enjoy Second Order Properties? Sankhya, Series A 59: 435–448. [Google Scholar]). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon nonlinear renewal theory (Lai and Siegmund, 1977 Lai, T. L. and Siegmund, D. (1977). A Nonlinear Renewal Theory with Applications to Sequential Analysis I, Annals of Statistics 5: 946–954.[Crossref], [Web of Science ®] , [Google Scholar], 1979 Lai, T. L. and Siegmund, D. (1979). A Nonlinear Renewal Theory with Applications to Sequential Analysis II, Annals of Statistics 7: 60–76.[Crossref], [Web of Science ®] , [Google Scholar], Annals of Statistics; Woodroofe, 1977 Woodroofe, M. (1977). Second Order Approximation for Sequential Point and Interval Estimation, Annals of Statistics 5: 984–995.[Crossref], [Web of Science ®] , [Google Scholar], Annals of Statistics). Then, we take to extensive data analysis carried out via computer simulations when requisite sample sizes range from small to moderate to large. We find that the Stein-type two-stage estimation methodology oversamples significantly and yet the achieved risk is not close to preset goal ω. On the other hand, both modified two-stage and purely sequential estimation strategies perform remarkably well. We have validated their main theoretical first-order and second-order properties through simulated data. The latter methodologies have been illustrated and implemented using two real data sets from health studies, namely, infant mortality data and bone marrow data.