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This paper concerns a new Normal approximation to the beta distribution and its relatives, in particular, the binomial, Pascal, negative binomial, F, t, Poisson, gamma, and chi square distributions. The approximate Normal deviates are expressible in terms of algebraic functions and logarithms, but for desk calculation it is preferable in most cases to use an equivalent expression in terms of a function specially tabulated here. Graphs of the error arc provided. They show that the approximation is good even in the extreme tails except for beta distributions which are J or U shaped or nearly so, and they permit correction to obtain still more accuracy. For use beyond the range of the graphs, some standard recursive relations and some classical continued fractions are listed, with some properties of the latter which seem to be partly new. Various Normal approximations are compared, with further graphs. The new approximation appears far more accurate than the others. Everything an ordinary user of the approximation might want to know is included in this paper. The theory behind the approximation and most proofs are postponed to a second paper immediately following this one. [ABSTRACT FROM AUTHOR]

Several approximations to the binomial distribution were compared in 1956 by Raft [9], including the normal, arcsine, normal Gram-Charlier, Camp-Paulson, Poisson, and Poisson Gram-Charlier approximations. The closeness of these approximations is judged by the maximal error which can arise in evaluating any sum of consecutive binomial terms. Borges [3] investigated a general class of approximations (which includes some of the above) and showed that a certain beta-transformation of a binomially distributed variable is the unique member of this class which is approximately normal with error 0(l/n). In 1963, Bol'shev [2] gave two new Poisson-type approximations. In this paper, some of Raft's tables are extended and tables for the error of these newer approximations are added. As a consequence of the numerical results obtained, the Poisson Gram-Charlier approximation is recommended for small values of p while the Camp-Paulson and Borges' approximations should generally be preferred for large p. [ABSTRACT FROM AUTHOR]

Let X[sub 1], ... X[sub r], be independent B(n,p) random variables. In this paper n is estimated with known p. The MLE is developed and shown to be consistent in a relative sense and asymptotically normal. Motivated by Y[sub i] = X[sub i]/q we consider Y[sub 1], ... ,Y[sub r] independent N(mu, mu) and examine three estimates of mu; (i) the MVUE, (ii) the MLE, and (iii) ?. These are shown to be asymptotically equivalent. [ABSTRACT FROM AUTHOR]

In this paper, the modified method of moments is applied to obtain an estimator of the parameter of a doubly truncated binomial distribution. The estimator is obtained in terms of the first three sample moments. Further, the efficiency of this estimator relative to the maximum likelihood estimator is also given here. [ABSTRACT FROM AUTHOR]

In the classification of the elements in a binomial population, there is in many cases a chance of misclassifying an item. When an item in a sample from a binomial population is misclassified, it causes a bias, independent of the standard error, in the estimate of the population parameter P. This paper analyzes this bias when each of two interviewers independently classify the items in a single sample. Upper and lower bounds for the bias are derived. [ABSTRACT FROM AUTHOR]

Specifically, in Section 1, we expand upon Li and Hu's (LH) and Ji and Ji's (JJ) suggestions for future computational, theoretical and usage improvements for specifically the eSVD. Given that the eSVD has been shown to be suitable in analyzing single-cell data, LH and JJ state that a natural next-step is to enable the eSVD to handle many thousands of cells and genes simultaneously. While the above two directions were about the eSVD in isolation, LH mentions that future work could investigate how the eSVD should be used for downstream clustering analysis (or other downstream tasks in general). [Extracted from the article]

Exact distributions of run statistics are traditionally obtained using combinatorial methods, which, under certain situations, become very tedious. Run distributions of multiple object systems, although appearing frequently in applications from various fields, such as computational biology, are not commonly used, due in part to the lack of easy-to-use formulas. In this article, a method for evaluating partition functions of lattice models in the field of statistical mechanics is used to develop a systematic method to study various run statistics in multiple object systems. By using particular generating functions for the specified situation under study, many new distributions can be obtained in a unified and coherent way. The method makes it possible to manipulate formulas of run statistics by using binomial identities to obtain more general, yet simpler formulas. To illustrate the applications of the general method, the distributions of the total number of runs and the longest runs are investigated. Novel and general explicit formulas are derived for the distribution and moments of the total number of runs, and simple explicit formulas are derived for the distributions of the longest runs. In addition, some classical run statistics are recovered and generalized in the same unified way. As examples of applications to biological sequence analysis, the run statistics developed using the general method are applied to several protein sequences to examine their global and local features. [ABSTRACT FROM AUTHOR]

This article presents the views of author on a paper by Emanuel Parzen which focused on statistical data analysis. There is much more in the written article than Professor Parzen could readily discuss. In the introduction, exploratory data analysis seems to be equated to "confirmatory non parametric statistical data analysis." That equivalence might be worth some further discussion, as one might wonder how much there is to confirm. It was concluded tentatively that with some conviction, that the probability distribution was tri-modal. It is a small further step to calculate and plot confidence intervals for the histogram ordinates based on binomial distributions.

Explores methodologies for the estimation of selected binomial information of the true batting abilities of baseball players. Review of related literature; Data source; Likelihood calculations.

Statistical approaches to overdispersion, correlated errors, shrinkage estimation, and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model (GLMM). Given an unobserved vector of random effects, observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function, known prior weights and a scale factor. The random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components. For problems involving time series, spatial aggregation and smoothing, the dispersion may be specified in terms of a rank deficient inverse covariance matrix. Approximation of the marginal quasi-likelihood using Laplace's method leads eventually to estimating equations based on penalized quasilikelihood or PQL for the mean parameters and pseudo-likelihood for the variances. Implementation involves repeated calls to normal theory procedures for REML estimation in variance components problems. By means of informal mathematical arguments, simulations and a series of worked examples, we conclude that PQL is of practical value for approximate inference on parameters and realizations of random effects in the hierarchical model. The applications cover overdispersion in binomial proportions of seed germination: longitudinal analysis of attack rates in epilepsy patients: smoothing of birth cohort effects in an age-cohort model of breast cancer incidence: evaluation of curvature of birth cohort effects in a case-control study of childhood cancer and obstetric radiation: spatial aggregation of lip cancer rates in Scottish counties: and the success of salamander matings in a complicated experiment involving crossing of male and female effects. PQL tends to underestimate somewhat the variance components and (in absolute value) fixed effects when applied to clustered binary data, but the situation improves rapidly for binomial observations having denominators greater than one. [ABSTRACT FROM AUTHOR]

Psychologists often want to detect category structure in subjects' free recall protocols. Although runs tests based on the binomial distribution are commonly used to detect nonrandomness within a sequence, many research situations require tests based on the multinomial distribution. We propose a test of randomness versus clustering based on the number of runs in multinomial data. We illustrate its use with data from a mass communication experiment using a constrained free recall procedure. [ABSTRACT FROM AUTHOR]

Results are developed for play-the-winner sampling for choosing the better of two binomial populations when the maximum number of tests is specified. It is shown that for a fixed number of tests the probability of correct selection with alternating assignment exceeds that for play- the-winner sampling. However, when the probability of correct selection is fixed, neither sampling method is uniformly better than the other as measured by the expected number of tests on the poorer population, or on the total expected number of tests. [ABSTRACT FROM AUTHOR]