Sequential Fixed-width Confidence Interval Research Articles

Bounded Risk Two-stage Estimation Procedure for a U(aθ, bθ) Distribution

In the literature, an extensive work on sequential fixed-width confidence interval for the parameter of U( q, m q) model, where m > 1 is known, is available. In this article, we propose a two-stage sampling procedure for estimating the parameter q of U( aq, bq) distribution, where a < b are positive and known. Here, the risk of an estimator [Formula: see text] of q is less than a pre-assigned number w (>0), that is, [Formula: see text], 0 < A < ∞ is known. We determine the parameter Bk of stopping variable so that the risk is uniformly bounded by a pre-assigned value w. We have also tabulated the values of the expected stopping time and its standard deviation (SD).

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.

Asymptotic Normality of Sequential Stopping Times with Applications: Confidence Intervals for an Exponential Mean

In sequential methodologies, finally accrued data customarily look like [Formula: see text] where [Formula: see text] is the total number of observations collected through termination. Under mild regulatory conditions, a standardized version of [Formula: see text] follows an asymptotic normal distribution (Ghosh–Mukhopadhyay theorem) which we highlight with a number of illustrations from the recent literature for completeness. Then, we emphasize the role of such asymptotic normality results along with second-order approximations for stopping times in the construction of sequential fixed-width confidence intervals for the mean in an exponential distribution. Two kinds of confidence intervals are developed: (a) one centred at the randomly stopped sample mean [Formula: see text] and (b) the two other centred at appropriate constructs using the stopping variable [Formula: see text] alone. Ample comparisons among all three proposed methodologies are summarized via simulations. We emphasize our finding that the two fixed-width confidence intervals centred at appropriate constructs using the stopping variable [Formula: see text] alone perform as well or better than the customary one centred at the randomly stopped sample mean.

Sequential fixed-width confidence interval for the rth power of the exponential scale parameter: Two-stage and sequential sampling procedures

In this paper, we study two-stage and sequential sampling procedures for estimating the rth power of the exponential scale parameter, i.e., βr, where the fixed-width confidence interval is considered. The exact operating characteristics of sequential procedures for fixed-width, (), confidence interval estimation is given. We show that the results of Zacks (2009), for two-stage and sequential procedures, can be deduced from our general results.

A robust sequential fixed-width confidence interval for count data based on Bhattacharyya-Hellinger distance estimator

ABSTRACTFor count data having an unknown mass function g0, we use the minimum Bhattacharyya-Hellinger distance (MBHD) estimator and a stopping rule to construct a sequential fixed-width confidence interval for a functional T(g0) = θ0, where fθ0 is the best-fitting parametric model achieving the MBHD between g0 and any member of a parametric class of mass functions. We establish the asymptotic consistency and efficiency properties of the sequential confidence interval and the expected sample size, respectively, as the half-width d → 0. When the count data come from a gross-error contamination model gα, L = (1 − α)fθ + αδ{L} for α ∈ (0, 1), where a parametric model fθ is mixed with a point mass δ{L} located at a value L, we reparameterize L = Ld such that Ld → ∞ as d → 0 and theoretically show that the expected sample size is affected by α, whereas the coverage probability of the sequential confidence interval depends on the rate of [T(gα, Ld) − θ]/d, as d → 0. Our reparameterization fully exploits the MBHD estimator's inherent ability to progressively ignore increasing values of L, providing an asymptotically consistent sequential fixed-width interval estimator of θ. When fθ is Poisson (θ), simulations are conducted to corroborate our theoretical results and to contrast the performance of the MBHD with that of the maximum likelihood estimator (MLE) of θ. A real data on Death Notice, modeled as a negative binomial, are analyzed to compare and contrast the performance of our sequential MBHD procedure with that of the MLE.

A Purely Sequential Estimation Procedure for the Parameter of U(θ,mθ) Distribution

Abtsrcat In the literature, an extensive work on sequential fixed width confidence interval (CI) for the parameter of U(0, θ) model is available. In this article we propose a fixed width (1-) level sequential CI for the parameter θ of U(θ, mθ) distribution, where m > 1 is known, based on the large-sample approximation of coverage probability.

Two-Stage Estimation Procedure for the Parameter of U (θ,mθ) Distribution

Abtsrcat In the literature, an extensive work on sequential fixed width confidence interval (CI) for the parameter of U(0, θ) model is available. In this article we propose a two-stage fixed width (1 − α) level sequential CI for the parameter θ of U( θ, mθ) distribution, where m > 1 is known, based on the large-sample approximation of coverage probability.

Fixed Width Confidence Interval for the Parameter of U(θ, mθ) Distribution

In the literature, sequential fixed width confidence intervals for the parameter θ of U(0, θ) distribution have been obtained. Here, the upper limit depends on the parameter. In this article, we propose a sequential fixed width (1 − α) level confidence interval for θ of U(θ, mθ) distribution, m (>1) a known fixed constant.

Sequential Estimation Procedures for End Points of Support in a Non-Regular Distribution

In this article, we consider sequential estimation of the end points of the support based on the extreme values when the underlying distribution has a bound support. Some sequential fixed-width confidence intervals are proposed. Stopping rules based on the range are proposed and the estimation procedures based on them are shown to be asymptotically efficient. The results of numerical simulations are presented. Moreover, the sequential point estimation problem is considered under squared loss plus cost of sampling.

Sequential Nonparametric Fixed-Width Confidence Intervals for Conditional Quantiles

Let {(X i , Y i ): i ≥ 1} be a sequence of bivariate r.v.'s from a continuous distribution H with marginals F and G, respectively, and let G x (·) denote the conditional distribution of Y 1 given X 1 = x, x ∈ Λ(F), the support of F. In this paper sequential fixed-width confidence interval procedures of length (at most) 2d for the conditional quantile q x (λ) = inf {y: G x (y) ≥ λ}, 0 < λ <1, are studied based on sample conditional quantiles: where denotes a ‘smoothed’ or ‘unsmoothed’ rank nearest neighbor or Nadaraya-Watson–type kernel estimator of G x (·). Asymptotics of these sequential confidence interval procedures including their consistency and (relative) efficiency properties are studied, as d → 0, on the lines of Chow and Robbins (1965), Geertsema (1970), and Stute (1983). The relative efficiencies and deficiencies of these procedures with respect to each other along with some supportive Monte Carlo results are also presented.

Constructing the best linear combination of diagnostic markers via sequential sampling

We study the best linear combination of markers in terms of the area under the receiver operating characteristic curve, since no single marker is perfect for classification purposes. The sequential fixed-width confidence interval estimate method is applied. We show that the proposed procedure is efficient in terms of the total sample size, with an optimal ratio of cases to controls, and is asymptotically consistent. The performance of our method is illustrated by synthesized data and a real example.

The Exact Distributions of the Stopping Times and Their Functionals in Two-Stage and Sequential Fixed-Width Confidence Intervals of the Exponential Parameter

The asymptotic properties of two-stage and sequential procedures for fixed-width, 2δ, confidence interval estimation of the mean β of an exponential distribution are well known. In the present paper we derive the exact operating characteristics of these procedures for any given δ > 0. The methodology is similar to that of Zacks and Mukhopadhyay (2005); (2006). The results are, however, more explicit.

On the convergence rate of sequential fixed-width confidence intervals for normal parameters

We consider the convergence rate of sequential fixed-width confidence intervals for θ = a μ + b σ under a normal N ( μ , σ 2 ) model with μ and σ 2 both unknown. We use the fully sequential procedure proposed by [Takada, Y., 1997. Fixed-width confidence intervals for a function of normal parameters. Sequential Anal. 16, 107–117] to construct sequential confidence intervals for θ and investigate the convergence rate of the coverage probability as the width of the confidence interval approaches zero. We also derive a second-order asymptotic expansion of the average sample size.

Fixed-width confidence interval based on a minimum Hellinger distance estimator

In the context of discrete data, a sequential fixed-width confidence interval for an unknown parameter in a parametric model is constructed using a minimum Hellinger distance estimator (MHD) as the center of the interval. It is shown that our sequential procedure is asymptotically consistent and efficient, when the assumed parametric model is correct. These results, in addition to being exactly same as those obtained by Khan [1969, A general method of determining fixed-width confidence intervals. Ann. Math. Statist. 40, 704–709] and Yu [1989, On fixed-width confidence intervals associated with maximum likelihood estimation. J. Theoret. Probab. 2, 193–199] using a maximum likelihood estimator (MLE), offer an alternative which has several in-built robustness properties. Monte Carlo simulations show that the performance of our sequential procedure based on MHD, measured in terms of average sample size and the coverage probability, are as good as those based on MLE, when the assumed Poisson model is correct. However, when the samples come from a gross-error contaminated Poisson model, our numerical results show that the deviation from the Poisson model assumption severely affects the performance of the sequential procedure based on MLE, while the procedure based on MHD continues to perform well, thus exhibiting robustness of MHD against gross-error contaminations even for random sample sizes.

Sequential Estimation for a Functional of the Spectral Density of a Gaussian Stationary Process

Integral functional of the spectral density of stationary process is an important index in time series analysis. In this paper we consider the problem of sequential point and fixed-width confidence interval estimation of an integral functional of the spectral density for Gaussian stationary process. The proposed sequential point estimator is based on the integral functional replaced by the periodogram in place of the spectral density. Then it is shown to be asymptotically risk efficient as the cost per observation tends to zero. Next we provide a sequential interval estimator, which is asymptotically efficient as the width of the interval tends to zero. Finally some numerical studies will be given.

Sequential Fixed-Width Confidence Interval for the Product of Two Means

For the product of two population means, the problem of constructing a fixed-width confidence interval with preassigned coverage probability is considered. It is shown that the optimal sample sizes which minimize the total sample size and at the same time guarantee a fixed-width confidence interval of desired coverage depend on the unknown parameters. In order to overcome this, a fully sequential procedure consisting of a sampling scheme and a stopping rule are proposed. It is then shown that the sequential confidence interval is asymptotically consistent and the stopping rule is asymptotically efficient, as the width goes to zero. Furthermore, a second order result for the difference between the expected stopping time and the (total) optimal fixed sample size is established. The theoretical results are supported by appropriate simulations.

The Convergence Rate of Fixed-Width Sequential Confidence Intervals for a Parameter of an Exponential Distribution

In this paper we consider sequential fixed-width confidence interval estimation for a parameter τ = aµ + bσ with a and b being given constants when the location parameter µ and the scale parameter σ of the negative exponential distribution are unknown. We investigate the rate of convergence of the coverage probability for fixed-width sequential confidence intervals of τ.

On sequential fixed-width confidence intervals for the mean and second-order expansions of the associated coverage probabilities

In order to construct fixed-width (2d) confidence intervals for the mean of an unknown distribution function F, a new purely sequential sam- pling strategy is proposed first. The approach is quite different from the more traditional methodology of Chow and Robbins (1965, Ann. Math. Statist., 36, 457-462). However, for this new procedure, the coverage probability is shown (Theorem 2.1) to be at least (1 - a) + Ad 2 + o(d 2) as d --* 0 where (1 - (~) is the preassigned level of confidence and A is an appropriate functional of F, un- der some regularity conditions on F. The rates of convergence of the coverage probability to (1 - a) obtained by Csenki (1980, Scand. Aetuar. J., 107-111) and Mukhopadhyay (1981, Comm. Statist. Theory Methods, 10, 2231-2244) were merely O(dl/~-q), with 0 < q < 1/2, under the Chow-Robbins stopping time T*. It is to be noted that such considerable sharpening of the rate of con- vergence of the coverage probability is achieved even though the new stopping variable is Op (T*). An accelerated version of the stopping rule is also provided together with the analogous second-order characteristics. In the end, an exam- ple is given for the mean estimation problem of an exponential distribution.

Sequential point and interval estimation of scale parameter of exponential distribution

Sequential fixed‐width confidence intervals are obtained for the scale parameter σ when the location parameter θ of the negative exponential distribution is unknown. Exact expressions for the stopping time and the confidence coefficient associated with the sequential fixed‐width interval are derived. Also derived is the exact expression for the stopping time of sequential point estimation with quadratic loss and linear cost. These are numerically evaluated for certain nominal confidence coefficients, widths of the interval and cost functions, and are compared with the second order asymptotic expressions.

Recursive estimation of intensity function of a Poisson random field

In the paper asymptotic unbiasedness, strong consistency and asymptotic normality of the recursive kernel estimator of an intensity function of a Poisson random field are proved. Three classes of stopping times are defined, which permit constructing an asymptotic sequential fixed-width confidence interval for the unknown intensity function at a fixed point and asymptotic sequential bounds based on the integrated squared error.