Efficient cycle time-throughput curve generation using a fixed sample size procedure

Let x 1, x 2, … x n, … be a sequence of independent random variables with a common density function P(x = 1) = p, P(x = 0) = 1 – p, 0 < p < 1. This paper considers the non-randomized sequential procedures δ's for estimating p and the following three problems on choice of δ. (i) Choose δ to minimize E v (Z δ – P)2 subject to E p N δ ≤ m where m ≥ 1 and Z δ is an unbiased estimate of p; (ii) Choose δ to minimize Ep N δ subject to E p (Z δ – p)2 ≤ α where α is a real positive number; (iii) choose δ to minimize C Ep N δ + E p (Z δ – p)2 where C is the cost of an observation. In each case the minimization is to be done uniformly in p if possible; otherwise the supremum over p of the risk in question is to be minimized. A procedure is constructed for problem (i) when m is not an integer. A fixed sample size procedure is shown to be admissible and minimax for problem (ii). A procedure is constructed which is asymptotically uniformly better than the fixed sample size for problem (ii). Furthermore, for problem (iii) some optimum procedures are constructed.

In this article, two-stage and fixed sample size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. Two different approaches for estimation are considered, namely, fixed-width and bounded-width interval estimation. In the fixed-width confidence interval estimation problem, a two-stage procedure is developed and the exact distribution of the corresponding stopping variable and the exact coverage probability of the interval estimator are derived. Asymptotic optimality properties such as asymptotic first-order efficiency and asymptotic consistency properties of the two-stage procedure are established. Bounded-width confidence intervals are obtained by applying fixed-accuracy estimation methodologies of Mukhopadhyay and Banerjee (2014, 2015a,b) and Banerjee and Mukhopadhyay (2016) using different transformations on ρ. Both fixed sample size and two-stage sampling methodologies for bounded-width confidence interval estimation of are developed incorporating such transformations. Finally, performances of all the procedures are compared via extensive sets of simulation studies.

The problem of choosing the best of k populations when observations are non-negative and small observations are preferred is considered using a truncated play-the-winner sampling scheme. Upper bounds on the expected sample sizes are derived. The procedure is compared with a fixed sample size procedure on the basis of expected sample size and of risk. It is noted that the probability of correct selection (P.C.S.) is the same for the two procedures so that existing tables may be used.

In this paper, our aim is to compare the scale parameters of two independent Pareto distributions using the hypothesis testing approach. We develop testing procedures for the ratio of scale parameters of two independent Pareto populations with unknown but proportional shape parameters and establish that such a comparison is not possible with any fixed sample size procedure. Hence, we implement a practical accelerated sequential sampling design to obtain the required sample sizes while controlling both type I and type II error probabilities. The first and second-order asymptotic properties of the proposed procedure are studied in detail. A large-scale simulation analysis is carried out to validate the theoretical findings under different parameter configurations. In addition to the extensive simulations, we provide two real-data illustrations (incomes of two occupations and rice production of two regions) to complement the practical utility of the proposed study.

Computational comparison for ML estimator of quadratic functions of the Bernoulli parameter in IS and FSS methods

To explore the relationship between sample size in the groups and statistical power of ANOVA and Kruskal-Wallis H test with an imbalanced design. The sample sizes of the two tests were estimated by SAS program with given parameter settings, and Monte Carlo simulation was used to examine the changes in power when the total sample size varied or remained fixed. In ANOVA, when the total sample size was fixed, increasing the sample size in the group with a larger mean square error improved the statistical power, but an excessively large difference in the sample sizes between groups led to reduced power. When the total sample size was not fixed, a larger mean square error in the group with increased sample size was associated with a greater increase of the statistical power. In Kruskal-wallis H test, when the total sample size was fixed, increasing the sample size in groups with large mean square errors increased the statistical power irrespective of the sample size difference between the groups; when total sample size was not fixed, a larger mean square error in the group with increased sample size resulted in an increased statistical power, and the increment was similar to that for a fixed total sample size. The relationship between statistical power and sample size in groups is affected by the mean square error, and increasing the sample size in a group with a large mean square error increases the statistical power. In Kruskal-Wallis H test, increasing the sample size in a group with a large mean square error is more cost- effective than increasing the total sample size to improve the statistical power.

When recruiting participants to a new study developing a clinical prediction model (CPM), sample size calculations are typically conducted before data collection based on sensible assumptions. This leads to a fixed sample size, but if the assumptions are inaccurate, the actual sample size required to develop a reliable model may be higher or even lower. To safeguard against this, adaptive sample size approaches have been proposed, based on sequential evaluation of (changes in) a model's predictive performance. The objective of the study was to illustrate and extend sequential sample size calculations for CPM development by (i) proposing stopping rules for prospective data collection based on minimizing uncertainty (instability) and misclassification of individual-level predictions and (ii) showcasing how it safeguards against inaccurate fixed sample size calculations. Using the sequential approach repeats the predefined model development strategy every time a chosen number (eg, 100) of participants are recruited and adequately followed up. At each stage, CPM performance is evaluated using bootstrapping, leading to prediction and classification stability statistics and plots, alongside optimism-adjusted measures of calibration and discrimination. Learning curves display the trend of results against sample size and recruitment is stopped when a chosen stopping rule is met. Our approach is illustrated for model development of acute kidney injury using (penalized) logistic regression CPMs. Before recruitment based on perceived sensible assumptions, the fixed sample size calculation suggests recruiting 342 patients to minimize overfitting; however, during data collection, the sequential approach reveals that a much larger sample size of 1100 is required to minimize overfitting (targeting a bootstrap-corrected calibration slope ≥0.9). If the stopping rule criteria also target small uncertainty and misclassification probability of individual predictions, the sequential approach suggests an even larger sample size of about 1800. For CPM development studies involving prospective data collection, a sequential sample size approach allows users to dynamically monitor individual-level prediction and classification instability. This helps determine when enough participants have been recruited and safeguards against using inaccurate assumptions in a sample size calculation before data collection. Engagement with patients and other stakeholders is crucial to identify sensible context-specific stopping rules for robust individual predictions.

The multistage Wiener filter (MWF) is an iterative Schur complement procedure for estimating a scalar or a vector of random variables in the linear minimum mean square error sense. When a complementary orthogonal projector is used as a blocking matrix, intermediate vectors in the development of the MWF have singular covariance matrices, and the rationale of the MWF is undermined. For this situation, we validate the structure of the MWF by showing that it can be developed as an iterative generalized Schur complement procedure which relies on the Moore-Penrose generalized inverse. The validated filter does not require any generalized inverse, and it can be implemented as usual by using scalar Wiener filters when estimating a scalar random variable. Next, we show, under a condition which is commonly met in practice, that the MWF is a least squares reduced rank estimate of the Wiener filter from the Krylov subspace associated with the Wiener-Hopf equation. The MWF is then compared with Brillinger's reduced rank minimum mean square error filter. The two filters differ in the subspaces from which signal estimation is performed and in the order in which projection onto the respective subspaces and Wiener estimation are performed. A forward recursion for the mean square estimation error in each stage of the MWF is provided, and the performance of the MWF is demonstrated by a numerical example.

6576 Background: The aim of phase II trials is to determine if a new treatment is promising for further testing in confirmatory clinical trials. Most phase II clinical trials are designed as single-arm trials using a binary outcome with or without interim monitoring for early stopping. In this context, we propose a Bayesian adaptive design denoted as PSSD, predictive sample size selection design (Statistics in Medicine 2012;31:4243-4254). Methods: The design allows for sample size selection followed by any planned interim analyses for early stopping of a trial, together with sample size determination before starting the trial. In the PSSD, we determined the sample size using the predictive probability criterion with two kinds of prior distributions, that is, an ‘analysis prior’ used to compute posterior probabilities and a ‘design prior’ used to obtain prior predictive distributions. In the sample size determination, we provide two sample sizes, that is, N and Nmax, using two types of design priors. At each interim analysis, we calculate the predictive probability of achieving a successful result at the end of the trial using analysis prior in order to stop the trial in case of low or high efficacy, and we select an optimal sample size, that is, either N or Nmax as needed, on the basis of the predictive probabilities. Results: We investigated the operating characteristics through simulation studies, and the PSSD retrospectively applies to a lung cancer clinical trial. As the number of interim looks increases, the probability of type I errors slightly decreases, and that of type II errors increases. The type I error probabilities of the probabilities of the proposed PSSD are almost similar to those of the non-adaptive design. The type II error probabilities in the PSSD are between those of the two fixed sample size (N or Nmax) designs. Conclusions: From a practical standpoint, the proposed design could be useful in phase II single-arm clinical trials with a binary endpoint. In the near future, this approach will be implemented in actual clinical trials to assess its usefulness and to extend it to more complicated clinical trials.

The deployment of statistical process control (SPC) in service process is a prominent global phenomenon in recent years. In this paper, we proposed the optimal variable sample size and sampling interval (VSSI) mean square error (MSE) control chart to monitor the difference between the process mean and the target value and the process variation shifts, and its performance is measured by the average time to signal (ATS). From data analyses, we found the optimal VSSI MSE chart that performs better than the specific VSSI and optimal variable sampling interval (VSI) and fixed sample size and sampling interval (FSSI) MSE chart. An example was given to illustrate this new proposed approach.

A study of predictive coding of component TV signals for compression to transmit at rates that give good picture quality is reported. Designing predictors for minimum mean square prediction error is effective, both for luminance and chrominance signals. Predictors designed in this manner were found to be as good as or better than all other predictors tested. Quantizers designed for minimum mean square system error were primarily used in this study. A programmable testbed was built to permit quick comparisons. Predictors were compared on the testbed using a 2-b quantizer in order to have adequate system error to display visible differences. For simple intrafield coding, component signals compressed for transmission at 30 Mb/s gave about equivalent quality to an NTSC composite signal coded directly for transmission at 45 Mb/s. With 45-Mb/s transmission, component signal coding showed no detectable degradation on close inspection of real pictures. Maximum chrominance changes on test patterns yield visually detectable chrominance slope overload.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The development of an efficient ground sampling strategy is critical to assess uncertainties associated with moderate- or coarse-resolution remote-sensing products. This work presents a comparison of estimating spatial means from fine spatial resolution images using spatial random sampling (SRS), Block Kriging (BK), and Means of Surface with Nonhomogeneity (MSN) at 1 km2 spatial scale. Towards this goal, we focus on the sampling strategies for ground data measurements and provide an assessment of the MODIS LAI product validated by the spatial means estimated by the above-mentioned three methods. The results of this study indicate that: (1) for its effective stratification strategies and its criteria of minimum mean square estimation error, MSN demonstrates the lowest mean squared estimation error for estimating the means of stratified nonhomogeneous surface; (2) BK is efficient in estimating the means of homogeneous surfaces without bias and with minimum mean squared estimation errors. The MODIS LAI product is assessed using the means estimated by SRS, BK, and MSN based on Landsat 8 OLI and SPOT HRV fine-resolution LAI maps. For heterogeneous surfaces, MSN results in low RMSE and high accuracy of MODIS LAI product compared with BK and SRS, whereas for homogeneous surfaces, the statistical parameters outputted by these three methods are similar. These results reveal that MSN is an effective method for estimating the spatial means for heterogeneous surfaces. There are differences in the accuracies of MODIS LAI product assessed by these three methods.

This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared with fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any coordinate-wise sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then log(s)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), where s is the level of sparsity and D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) is the Kullback-Leibler divergence between the underlying distributions. A series of sequential probability ratio tests, which require complete knowledge of the underlying distributions is shown to achieve this bound. Motivated by real-world experiments and recent work in adaptive sensing, we introduce a simple procedure termed sequential thresholding, which can be implemented when the underlying testing problem satisfies a monotone likelihood ratio assumption. Sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than log(s)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), achieving the lower bound. For comparison, we show any nonsequential procedure fails provided the number of measurements grows at a rate less than log(n)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ), where n is the total dimension of the problem.

&lt;p class="MsoNormal" style="margin: 0in 0in 0pt; text-align: justify;"&gt;&lt;span style="font-size: small; font-family: Times New Roman;"&gt;The fixed sample size procedure &lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp;&lt;/span&gt;sometimes fail to deal with the estimation problem in which it&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;is desired to control the combined &lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp;&lt;/span&gt;risk associated with the estimation of parameters of various probability distributions simultaneously. In this paper for a general probabilistic model is proposed an &amp;ldquo;accelerated&amp;rdquo; sequential class to minimize the combined risk for simultaneous estimation of parameters of several probability distributions. The proposed class is shown to provide solutions for many estimation problems under different probabilistic setups for &amp;ldquo; given precision &amp;rdquo; problems. &lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp;&lt;/span&gt;Besides, positive and negative moments for the stopping times are obtained and they are used for deriving the asymptotic expression for the &amp;ldquo;regret&amp;rdquo;&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;associated with&lt;span style="mso-spacerun: yes;"&gt;&amp;nbsp; &lt;/span&gt;the class of &amp;ldquo;accelerated&amp;rdquo; sequential estimation procedure.&lt;/span&gt;&lt;/p&gt;

Wavelet-Based Identification of Linear Discrete-Time Systems