On the Expansion for Expected Sample Size in Non-Linear Renewal Theory

ABSTRACTType/Token Ratios have been extensively used in child language research as an index of lexical diversity. This paper shows that the measure has frequently failed to discriminate between children at widely different stages of language development, and that the ratio may in fact fall as children get older. It is suggested here that such effects are caused by a negative, though non-linear, relationship between sample size (i.e. number of tokens) and Type/Token Ratio. Effects of open and closed class items are considered and an alternative Verbal Diversity measure is examined. Standardization of the number of tokens before computing Type/Token Ratios is recommended.

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On double stage minimax-shrinkage estimator for generalized Rayleigh model

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v5i1.5553 ◽

2016 ◽

Vol 5 (1) ◽

pp. 39 ◽

Cited By ~ 1

Author(s):

Abbas Najim Salman ◽

Maymona Ameen

Keyword(s):

Sample Size ◽

Shape Parameter ◽

Mean Squared Error ◽

Scale Parameter ◽

Rayleigh Distribution ◽

Shrinkage Estimator ◽

Squared Error ◽

Expected Sample Size ◽

Generalized Rayleigh Distribution ◽

The Mean

This paper is concerned with minimax shrinkage estimator using double stage shrinkage technique for lowering the mean squared error, intended for estimate the shape parameter (a) of Generalized Rayleigh distribution in a region (R) around available prior knowledge (a0) about the actual value (a) as initial estimate in case when the scale parameter (l) is known .In situation where the experimentations are time consuming or very costly, a double stage procedure can be used to reduce the expected sample size needed to obtain the estimator.The proposed estimator is shown to have smaller mean squared error for certain choice of the shrinkage weight factor y(×) and suitable region R.Expressions for Bias, Mean squared error (MSE), Expected sample size [E (n/a, R)], Expected sample size proportion [E(n/a,R)/n], probability for avoiding the second sample and percentage of overall sample saved for the proposed estimator are derived.Numerical results and conclusions for the expressions mentioned above were displayed when the consider estimator are testimator of level of significanceD.Comparisons with the minimax estimator and with the most recent studies were made to shown the effectiveness of the proposed estimator.

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Lower bounds for the expected sample size of sequential procedures for selecting and ranking of binomial and Poisson populations

Lobachevskii Journal of Mathematics ◽

10.1134/s1995080216040119 ◽

2016 ◽

Vol 37 (4) ◽

pp. 455-465 ◽

Cited By ~ 2

Author(s):

I. Kareev

Keyword(s):

Sample Size ◽

Lower Bounds ◽

Expected Sample Size ◽

Sequential Procedures ◽

Poisson Populations

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Limit theorems for first-passage times in linear and non-linear renewal theory

Advances in Applied Probability ◽

10.1017/s000186780002293x ◽

1984 ◽

Vol 16 (04) ◽

pp. 766-803 ◽

Cited By ~ 6

Author(s):

S. P. Lalley

Keyword(s):

Limit Theorems ◽

Large Deviation ◽

Local Limit Theorem ◽

Renewal Theory ◽

Local Limit ◽

Linear Boundary ◽

First Passage ◽

Non Linear ◽

First Time ◽

Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

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Optimized adaptive enrichment designs for three-arm trials: learning which subpopulations benefit from different treatments

Biostatistics ◽

10.1093/biostatistics/kxz030 ◽

2019 ◽

Author(s):

Jon Arni Steingrimsson ◽

Joshua Betz ◽

Tianchen Qian ◽

Michael Rosenblum

Keyword(s):

Sample Size ◽

Disease Severity ◽

Adaptive Design ◽

Type I Error ◽

Type I ◽

Expected Sample Size ◽

Common Control ◽

Enrichment Designs ◽

Adaptive Enrichment Designs ◽

Standard Designs

Summary We consider the problem of designing a confirmatory randomized trial for comparing two treatments versus a common control in two disjoint subpopulations. The subpopulations could be defined in terms of a biomarker or disease severity measured at baseline. The goal is to determine which treatments benefit which subpopulations. We develop a new class of adaptive enrichment designs tailored to solving this problem. Adaptive enrichment designs involve a preplanned rule for modifying enrollment based on accruing data in an ongoing trial. At the interim analysis after each stage, for each subpopulation, the preplanned rule may decide to stop enrollment or to stop randomizing participants to one or more study arms. The motivation for this adaptive feature is that interim data may indicate that a subpopulation, such as those with lower disease severity at baseline, is unlikely to benefit from a particular treatment while uncertainty remains for the other treatment and/or subpopulation. We optimize these adaptive designs to have the minimum expected sample size under power and Type I error constraints. We compare the performance of the optimized adaptive design versus an optimized nonadaptive (single stage) design. Our approach is demonstrated in simulation studies that mimic features of a completed trial of a medical device for treating heart failure. The optimized adaptive design has $25\%$ smaller expected sample size compared to the optimized nonadaptive design; however, the cost is that the optimized adaptive design has $8\%$ greater maximum sample size. Open-source software that implements the trial design optimization is provided, allowing users to investigate the tradeoffs in using the proposed adaptive versus standard designs.

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Comparison of a Two-Stage and Three-Stage Interim-Analysis Procedure

Psychological Reports ◽

10.2466/pr0.1992.71.1.3 ◽

1992 ◽

Vol 71 (1) ◽

pp. 3-14 ◽

Cited By ~ 1

Author(s):

John E. Overall ◽

Robert S. Atlas

Keyword(s):

Sample Size ◽

Interim Analysis ◽

Type I Error ◽

Substantial Reduction ◽

Error Rates ◽

Sampling Plan ◽

Type I ◽

Two Stage ◽

Expected Sample Size ◽

Analysis Strategy

A statistical model for combining p values from multiple tests of significance is used to define rejection and acceptance regions for two-stage and three-stage sampling plans. Type I error rates, power, frequencies of early termination decisions, and expected sample sizes are compared. Both the two-stage and three-stage procedures provide appropriate protection against Type I errors. The two-stage sampling plan with its single interim analysis entails minimal loss in power and provides substantial reduction in expected sample size as compared with a conventional single end-of-study test of significance for which power is in the adequate range. The three-stage sampling plan with its two interim analyses introduces somewhat greater reduction in power, but it compensates with greater reduction in expected sample size. Either interim-analysis strategy is more efficient than a single end-of-study analysis in terms of power per unit of sample size.

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Surface and kinetic effects in gas chromatographic studies of polystyrene

Canadian Journal of Chemistry ◽

10.1139/v76-503 ◽

1976 ◽

Vol 54 (22) ◽

pp. 3496-3507 ◽

Cited By ~ 6

Author(s):

Gregory J. Courval ◽

Derek G. Gray

Keyword(s):

Sample Size ◽

Flow Rate ◽

Retention Volume ◽

Rate Dependence ◽

Retention Data ◽

Scanning Electron Micrographs ◽

Linear Absorption ◽

Solvent Interactions ◽

Non Linear ◽

Polymer Solvent

Considerable variation in the measurement of polymer–solvent interactions using gc retention data may occur due to kinetic factors, surface excess concentrations of probe vapour, and non-linear partition isotherms. The kinetic factors, which appear as a flow rate dependence of the retention volume, are analysed in terms of a previously reported theoretical model for retention on polymeric stationary phases passing through the glass transition. The predicted linear extrapolations to zero flow rate are obtained for the retention of n-tetradecane on polystyrene. The variation of this flow rate dependence with temperature and with the thickness of the stationary phase are also in qualitative agreement with the theory. A simplified model for the effect of loading on the retention diagram is presented. Non-linear absorption and bulk sorption isotherms result in a dependence of retention volume on sample size, necessitating an extrapolation of the measured retention volumes to zero peak height. The temperature variation of the flow rate dependence, the effect of loading, and the effect of sample size on retention volume are all further complicated by uneven distribution of polymer on support. From scanning electron micrographs of the beads it is evident that 'beading up' of the polystyrene on the glass surface may occur at low loadings, resulting in a non-uniform coating with large areas of the beads uncoated. It is concluded that in order to obtain reliable data on polymer–solvent interactions using gas chromatography, all of the above-mentioned factors must be considered.

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