Sequential Estimation Procedure Research Articles

Sequential sampling procedures for query size estimation

We provide a procedure, based on random sampling, for estimation of the size of a query result. The procedure is sequential in that sampling terminates after a random number of steps according to a stopping rule that depends upon the observations obtained so far. Enough observations are obtained so that, with a pre-specified probability, the estimate differs from the true size of the query result by no more than a prespecified amount. Unlike previous sequential estimation procedures for queries, our procedure is asymptotically efficient and requires no ad hoc pilot sample or a a priori assumptions about data characteristics. In addition to establishing the asymptotic properties of the estimation procedure, we provide techniques for reducing undercoverage at small sample sizes and show that the sampling cost of the procedure can be reduced through stratified sampling techniques.

Empirical bayes sequential estimation of the mean

We consider the empirical Bayes problem where the component problem is the sequential estimation of the mean of a distribution with squared error decision loss plus a sampling cost. An empirical Bayes sequential estimation procedure is exhibited which is asymptotically optimal. Asymptotic efficiency of the empirical Bayes stopping time sequence is also established. The performance of the proposed empirical Bayes procedure is studied with the help of a Monte Carlo study.

A new sequential design based on the robbins-monro procedure

A sequential estimation procedure based on the Robbins-Monro process is proposed. Through observations of a random variableY(x), estimates of a regression functionM(x) are obtained, whereM(x)=E[Y(x)]. The estimates of the roots ofM(x) are shown to be strongly convergent and asymptotically normal with minimum variance.

A new strategy of method comparison in clinical chemistry with a improved sequential estimation procedure and diagnostic validation

A new strategy of method comparison in clinical chemistry with a improved sequential estimation procedure and diagnostic validation

Asymptotic Theory of Sequential Estimation: Differential Geometrical Approach

Sequential estimation continues observations until the observed sample satisfies a prescribed criterion. Its properties are superior on the average to those of nonsequential estimation in which the number of observations is fixed a priori. A higher-order asymptotic theory of sequential estimation is given in the framework of geometry of multidimensional curved exponential families. This gives a design principle of the second-order efficient sequential estimation procedure. It is also shown that a sequential estimation can be designed to have a covariance stabilizing effect at the same time.

Educational Attainment and the Returns to Education for Australian Youth: Evidence of Self-Selection?

Abstracts Debates concerning the re-introduction of tertiary fees in Australia have made rates of return on education an active topic for investigation. This paper provides estimates of the returns to education that allow for worker heterogeneity and individual self selection in the education process. Using data drawn from the Australian Longitudinal Survey, a sequential estimation procedure is used that enables us to study the interaction between educational attainment and eamings determination. We find that traditional OLS measures of rates of return to education are understated and hence, counsel caution in the interpretation and use of the estimated returns provided by some authors.

A sequential estimation procedure for m-dimensional gaussian processes with independent inerements

The purpose of the paper is to derive a sequential estimation procedure based on a certain discrete stopping time for the exponential class of processes with independent increments. The statistic which has been used for the construciton of a finite stopping time is Jeffreys divergence of two consecutive values of the maximum likelihood estimate.

On two Sequential Procedures for Estimating the Parameter of a Uniform Distribution

Mukhopadhyay (1987) considered sequential procedure for the point estimation of the parameter θ in a U(0, θ) distribution under quadratic Joss. Given a random sample X1, ..., Xn of size n, he used Tn = max ( X1 , ... , Xn) to estimate θ In the present note, a sequential procedure based of UMVUE of θ is proposed. The asymptotic properties of the proposed procedure are established and its advantages over Mukhopadhyay's procedure are discussed.

Sequential estimarion of functionals of the survival curve under random censorship

Sequential point and interval estimation procedures for functionals of the survival curve F of the form ∫ψdF and ∫Fdψ with censored data are considered. The sequential point estimatior, procedure is shown to be risk efficient and asymptotically normal as c, cost per unit observation, tends to zero. For the interval estimation problem, the sequential procedure is shown to be consistent and efficient as the width of the confidence interval decreases to zero. In both estimation problems, the asymptotic distribution of the underlying stopping rule is obtained.

Third order asymptotic efficiency of the sequential maximum likelihood estimation procedure

Under suitable regularity conditions, the third order asymptotic bounds for distributions of regular estimators are obtained. It is shown that the modified maximum likelihood estimation procedure combined with appropriate stopping rule is uniformly third order asymptotically efficient in the sense that its asymptotic distribution attains the bound uniformly in stopping rules up to the third order.

A Sequential Estimation Approach to Cloud‐Clearing For Satellite Temperature Sounding

AbstractTemperature sounding of the troposphere using satellite‐borne infrared radiometers is complicated by the effects of clouds on the measured radiances. If accurate products are to be obtained these effects must first be detected and, if possible, corrections must be made for them in the retrieval procedures. This is normally done by converting the measured radiances to the ‘clear‐column’ values which would be observed from the same atmospheric profile in the absence of cloud—a process known as ‘cloud‐clearing’.In this paper a review of cloud‐clearing methods is presented. Then a new approach to cloud‐clearing, based on the principles of optimal estimation, is developed and applied to data from HIRS (the High‐resolution Infrared Radiation Sounder) on the TIROS‐N/NOAA satellite series.Preliminary estimates of clear‐column radiances and their expected errors are obtained at each HIRS spot by one of a number of methods, depending on the cloud characteristics. the properties of horizontal consistency expected in the clear‐column radiance field are then employed to improve the initial estimates using a sequential estimation procedure. This scheme is intended for implementation in the Local Area Sounding System of the Meteorological Office, which provides satellite soundings of high horizontal resolution for use in operational weather forecasting. Details of the new scheme, and of the old scheme which it replaces, are given. the improvements in clear‐column radiances are demonstrated by comparing the products of the old and new schemes with clear‐column radiances derived from coincident data of AVHRR (the Advanced Very High Resolution Radiometer).

A Note on Minimum Risk Point Estimation of the Shape Parameter of a Pareto Distribution

The minimum risk point estimation problem is considered for the shape parameter of a Pareto distribution where the Joss function is taken as squared error plus the linear cost of sampling. A suitable purely sequential procedure is proposed for this problem and the asrmptotic behavior of the “regret” function proposed by Robbins (1959) and many other characteristics are examined. An extensive numerical study is presented in order to look into moderate sample behaviors of the proposed sequential estimation procedure. The procedure is found to be very satisfactory.

An Analysis of International Exchange Rates Using Multivariate DLM's

New models for multiple time series are introduced and illustrated in an application to international currency exchange rate data. The models, based on matrix-variate normal extensions of the dynamic linear model (DLM), provide a tractable, sequential procedure for estimation of unknown covariance structure between series. A principal components analysis is carried out providing a basis for easy model assessment. A practically important elaboration of the model incorporates time- variation in covariance matrices.

Evaluation of statistical precision and design of efficient sampling for the population estimation based on frequency of occurrence

SummaryThe binomial sampling to estimate population density of an organism based simply upon the frequency of its occurrence among sampled quadrats is a labour‐saving technique which is potentially useful for small animals like insects and has actually been applied occasionally to studies of their populations. The present study provides a theoretical basis for this convenient technique, which makes it statistically reliable and tolerable for consistent use in intensive as well as preliminary population censuses. Firs, the magnitude of sampling error in relation to sample size is formulated mathematically for the estimate to be obtained by this indirect method of census, using either of the two popular models relating frequency of occurrence (p) to mean density (m), i.e. the negative binomial model, p=1−(1+m/k)−k, and the empirical model, p=1−exp(−amb). Then, the equations to calculate sample size and census cost that are necessary to attain a given desired level of precision in the estimation are derived for both models. A notable feature of the relationship of necessary sample size (or census cost) to mean density in the frequency method, in constrast to that in the ordinary census, is that it shows a concave curve which tends to rise sharply not only towards lower but also towards higher levels of density. These theoretical results make it also possible to design sequential estimation procedures based on this convenient census technique, which may enable us with the least necessary cost to get a series of population estimates with the desired precision level. Examples are presented to explain how to apply these programs to acutal censuses in the field.

A survey of literature on quantal response curves with a view toward application to the problem of selecting the curve with the smallest q-quantile (ediooq)

A survey of the literature is presented on various statistical inference problems and procedures associated with the following experimental setting: Any level of dose of some treatmenc can be administered to each ekperimental unit, and the observed response on each unit is binary (success or failure) with the unknown probability being a nondecreasing function of the dose-level. This function is referred to as a quantal response curve. Both parametric and nonparametrie models for a quantal response curve are considered. In each case both single-stage and sequential estimation procedures are reviewed in detail. This survey was motivated by the following statistical selection problem: Suppose that there are available several treatments, and that each one has associated with it a completely unknown quantal response curve. For each curve define its qth quantile (EDlOOq) as the dose-level which corresponds to a success probability of q where qe (0,1) is specified. For any prespecified q we wish to...

Sequential Estimation of Machinability Parameters for Adaptive Optimization of Machinability Data Base Systems

Mathematical model type machinability data base systems require suitable model building procedures to estimate the model parameters. The estimation procedure should be capable of using subjective prior information about the models and must also be capable of adapting the model parameters to the particular machining environment for which the data are needed. In this paper, the sequential Maximum A Posteriori (MAP) estimation procedure is proposed as the mathematical tool for performing these functions. Mathematical details of this estimation procedure are presented. The advantages of this method over conventional regression analysis are discussed based on the analysis of an experimental tool life data set. Details regarding the selection of the various initial values needed for starting the sequential procedure are presented. The use of prior information about the models in order to improve the parameter estimates is investigated. The adaptive capability of the procedure is analyzed using simulated tool life data. The results of this analysis indicate that the proposed sequential estimation procedure is a valuable tool for estimating machinability parameters and for the adaptive optimization of machinability data base systems.

On asymptotically Risk-effivient sequential versions of generalized U-statistics

Sequential point estimation of means of generalized U-statistics is considered. Based on a well defined stopping rule, the proposed sequential (estimation) procedure is shown to be asymptotically (first order) risk-efficient. Asymptotic distributions of sequential generalized U-statistics and the allied stopping times are also studied.

Second Order Approximation to the Risk of a Sequential Procedure

Given $X_1, X_2, \cdots$, i.i.d. with mean $\mu$ and variance $\sigma^2$, suppose that at stage $n$ one wishes to estimate $\mu$ by the sample mean $\bar{X}_n$, subject to the loss function $L_n = A(\bar{X}_n - \mu)^2 + n, A > 0$. If $\sigma$ is known, the optimal fixed sample size $n_0 \approx A^{1/2}\sigma$ can be used, with corresponding risk $R_{n_0}$, but if $\sigma$ is unknown there is no fixed sample size procedure that will achieve the risk $R_{n_0}$. For the sequential estimation procedure with stopping rule $T = \inf\{n \geq n_A: n^{-1} \sum^n_1 (X_i - \bar{X}_n)^2 \leq A^{-1} n^2\}$, the second order approximation of Woodroofe (1977) to the risk $R_T$ for normal $X_i$ is extended to the distribution-free case. Specifically, if the $X_i$ have finite moments of order greater than eight and are non-lattice, under certain conditions on the delay $n_A$ it is shown that the regret $R_T - R_{n_0} = c + o(1)$ as $A \rightarrow \infty$, where $c$ depends on the first four moments of the distribution of the $X_i$. For the lattice case, bounds of the form $c_1 + o(1) \leq R_T - R_{n_0} \leq c_2 + o(1)$ are obtained, where the $c_j$ are $c \pm 3$. It follows from these approximations that the regret can take arbitrarily large negative values as the distribution of the $X_i$ varies, in contrast to previous results for normal and gamma cases.

Adaptive sequential segmentation of piecewise stationary time series

The problem of adaptive segmentation of time series with abrupt changes in the spectral characteristics is addressed. Such time series have been encountered in various fields of time series analysis such as speech processing, biomedical signal processing, image analysis and failure detection. Mathematically, these time series often can be modeled by zero mean gaussian distributed autoregressive (AR) processes, where the parameters of the process, including the gain factor, remain constant for certain time intervals and then jump abruptly to new values. Identification of such processes requires adaptive segmentation: the times of parameter jumps have to be estimated thoroughly to constitute boundaries of “homogeneous” segments which can be described by stationary AR processes. In this paper, a new effective method for sequential adaptive segmentation is proposed, which is based on parallel application of two sequential parameter estimation procedures. The detection of a parameter change as well as the estimation of the accurate position of a segment boundary is effectively performed by a sequence of suitable generalized likelihood ratio (GLR) tests. Flow charts as well as a block diagram of the algorithm are presented. The adjustment of the three control parameters of the procedure (the AR model order, a threshold for the GLR test and the length of a “test window”) is discussed with respect to various performance features. The results of simulation experiments are presented which demonstrate the good detection properties of the algorithm and in particular an excellent ability to allocate the segment boundaries even within a sequence of short segments. As an application to biomedical signals, the analysis of human electroencephalograms (EEG) is considered and an example is shown.

Sequential and two-stage point estimation for the range in a power family distribution

Sequential and two-stage point estimation procedures for the range of a power family distribution are discussed using a loss function from a fairly general class. Some asymptotic characteristics of the sequential procedure are presented, including approximate expressions for the distribution of the sample size. For the two stage procedure, we present asymptotic properties and exact expressions for the risk of the procedure, and distribution and mean of the stopping rule. The results of numerical simulations are presented to demonstrate some practical merits of our procedures for moderate sample sizes.