Stirling's Formula and Its Extensions: Heuristic Approaches

Walsh (1995 Walsh , D. P. ( 1995 ). Equating Poisson and normal probability functions to derive Stirling's formula . Amer. Statist. 49 : 270 – 271 .[Taylor & Francis Online] , [Google Scholar]) introduced a heuristic approach to motivate Stirling's formula by equating a Poisson probability to an analogous value from a normal density function. We explore similar heuristics to derive approximations for various binomial, negative binomial, and multinomial coefficients. Also, using heuristics markedly different from those of Walsh, we develop an approximation of (nk)! for positive integers n (large) and k. These heuristics are then used to validate Stirling's formula for Γ(nα) where α is a positive real number. To derive each of our approximations we use a different probability distribution, and hence each section may serve as pedagogical module.     

Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image

Markov's inequality gives an upper bound on the probability that a nonnegative random variable takes large values. For example, if the random variable is the lifetime of a person or a machine, Markov's inequality says that the probability that an individual survives more than three times the average lifetime in the population of such individuals cannot exceed one-third. Here we give a simple, intuitive geometric interpretation and derivation of Markov's inequality. These results lead to inequalities sharper than Markov's when information about conditional expectations is available, as in reliability theory, demography, and actuarial mathematics. We use these results to sharpen Chebyshev's tail inequality also.     

An application of Lévy's inversion formula for higher order asymptotic expansion

The asymptotic expansions for the distribution of statistics are, in general, given by applying Lévy's inversion formula to the characteristic function. This paper shows an inversion formula for higher order asymptotic expansion of the distribution of a scalar valued function which contains dependent statistics. The usage of the formula is illustrated by derivation of third order asymptotic expansion of the distribution of Hotelling's T²-statistic under the elliptical distribution as an example.     

On the performance of Turing's formula: A simulation study

Turing's formula is an amazing result that allows one to estimate the probability of observing something that has not been observed before. After a brief review of the literature, we perform a simulation study to better understand how well this formula works in a variety of situations. We also compare the performance of Turing's formula with several modifications that have appeared in the literature. We find that these modifications tend to outperform Turing's formula, but usually not by very much. We further find that Turing's formula and its modifications tend to work better for heavy-tailed distributions than for light-tailed ones.     

Confidence Intervals for Survival Probabilities: A Comparison Study

The confidence interval of the Kaplan–Meier estimate of the survival probability at a fixed time point is often constructed by the Greenwood formula. This normal approximation-based method can be looked as a Wald type confidence interval for a binomial proportion, the survival probability, using the “effective” sample size defined by Cutler and Ederer. Wald-type binomial confidence interval has been shown to perform poorly comparing to other methods. We choose three methods of binomial confidence intervals for the construction of confidence interval for survival probability: Wilson's method, Agresti–Coull's method, and higher-order asymptotic likelihood method. The methods of “effective” sample size proposed by Peto et al. and Dorey and Korn are also considered. The Greenwood formula is far from satisfactory, while confidence intervals based on the three methods of binomial proportion using Cutler and Ederer's “effective” sample size have much better performance.     

The Topology of Central Counterparty Clearing Networks and Network Stability

□ This paper derives a measure of central counterparty (CCP) clearing-network risk that is based on the probability that the maximum exposure (the N-th order statistic) of a CCP to an individual general clearing member is large. Our analytical derivation of this probability uses the theory of Laplace asymptotics, which is related to the large deviations theory of rare events. The theory of Laplace asymptotics is an area of applied probability that studies the exponential decay rate of certain probabilities and is often used in the analysis of the tails of probability distributions. We show that the maximum-exposure probability depends on the topology, or structure, of the clearing network. We also derive a CCP's Maximum-Exposure-at-Risk, which provides a metric for evaluating the adequacy of the CCP's and general clearing members’ loss-absorbing financial resources during rare but plausible market conditions. Based on our analysis, we provide insight into how clearing-network structure can affect the maximum-exposure risk of a CCP and, thereby, network stability. We show that the rate function (the exponential decay rate) of the maximum-exposure probability is informative and can be used to compare the relative maximum-exposure risks across different network configurations.     

A Remark on the Alternative Expectation Formula

Students in their first course in probability will often see the expectation formula for nonnegative continuous random variables in terms of the survival function. The traditional approach for deriving this formula (using double integrals) is well-received by students. Some students tend to approach this using integration by parts, but often get stuck. Most standard textbooks do not elaborate on this alternative approach. We present a rigorous derivation here. We hope that students and instructors of the first course in probability will find this short note helpful.     

On derivaton and application of aic as a data-based criterion for histograms

In this paper a derivation of the Akaike's Information Criterion (AIC) is presented to select the number of bins of a histogram given only the data, showing that AIC strikes a balance between the “bias” and “variance” of the histogram estimate. Consistency of the criterion is discussed, an asymptotically optimal histogram bin width for the criterion is derived and its relationship to penalized likelihood methods is shown. A formula relating the optimal number of bins for a sample and a sub-sample obtained from it is derived. A number of numerical examples are presented.     

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.     

The large sample coverage probability of confidence intervals in general regression models after a preliminary hypothesis test

We derive a computationally convenient formula for the large sample coverage probability of a confidence interval for a scalar parameter of interest following a preliminary hypothesis test that a specified vector parameter takes a given value in a general regression model. Previously, this large sample coverage probability could only be estimated by simulation. Our formula only requires the evaluation, by numerical integration, of either a double or a triple integral, irrespective of the dimension of this specified vector parameter. We illustrate the application of this formula to a confidence interval for the odds ratio of myocardial infarction when the exposure is recent oral contraceptive use, following a preliminary test where two specified interactions in a logistic regression model are zero. For this real‐life data, we compare this large sample coverage probability with the actual coverage probability of this confidence interval, obtained by simulation.     

Option pricing for generalized distributions

The Black Scholes formula has been widely used to price financial instruments. The derivation of this formula is based on the assumption of lognormally distributed returns which is often in poor agreement with actual data. An option pricing formula based on the generalized beta of the second kind (GB2) is presented. This formula includes the Black Scholes formula as a special case and accommodates a wide variety of nonlognormally distributed returns. The sensitivity of option values to departures from the skewness and kurtosis associated with the lognormal distribution is investigated.     

A Gaussian alternative to using improper confidence intervals

The problem posed by exact confidence intervals (CIs) which can be either all-inclusive or empty for a nonnegligible set of sample points is known to have no solution within CI theory. Confidence belts causing improper CIs can be modified by using margins of error from the renewed theory of errors initiated by J. W. Tukey—briefly described in the article—for which an extended Fraser's frequency interpretation is given. This approach is consistent with Kolmogorov's axiomatization of probability, in which a probability and an error measure obey the same axioms, although the connotation of the two words is different. An algorithm capable of producing a margin of error for any parameter derived from the five parameters of the bivariate normal distribution is provided. Margins of error correcting Fieller's CIs for a ratio of means are obtained, as are margins of error replacing Jolicoeur's CIs for the slope of the major axis. Margins of error using Dempster's conditioning that can correct optimal, but improper, CIs for the noncentrality parameter of a noncentral chi-square distribution are also given.     

The table auto-regressive moving-average model for (categorical) stationary series: statistical properties (causality; from the all random to the conditional random)

A strictly stationary time series is modelled directly, once the variables' realizations fit into a table: no knowledge of a distribution is required other than the prior discretization. A multiplicative model with combined random ‘Auto-Regressive’ and ‘Moving-Average’ parts is considered for the serial dependence. Based on a multi-sequence of unobserved series that serve as differences and differences of differences from the main building block, a causal version is obtained; a condition that secures an exponential rate of convergence for its expected random coefficients is presented. For the remainder, writing the conditional probability as a function of past conditional probabilities, is within reach: subject to the presence of the moving-average segment in the original equation, what could be a long process of elimination with mathematical arguments concludes with a new derivation that does not support a simplistic linear dependence on the lagged probability values.     

Approximate variances for tolerance-distribution free ld50-estimators supplemented with abbott's correction

This paper is concerned with the 'tolerance-distribution free estimation of a LD50 (LC50) in a toxicity experiment with background mortality. Firstly the observed proportions mortality are adjusted by Abbott's formula. The LD50 is then obtained by inserting the adjusted proportions in the usual formula, but the variance formula is adapted to account for the variability in the background mortality.     

Measuring diversity in biological populations with logarithmic abundance distributions

Fisher's logarithmic distribution for species abundance is derived under a suitable set of conditions. The derivation explains the explicit meaning of the two parameters of the distribution.     

An alternative reduction formula for the central moments of the general bernoulli distribution

This note presents an alternative to Sproule's (1992) reduction formula for the central mcfnerits of the general Bernoulli distribution, This alternative is founded in part on Roranovsky's (1923) formula for the central moments of the conwentional Binomial distribution, The significance of this note for future research is also discussed.     

The Inverse of the Freeman – Tukey Double Arcsine Transformation

A formula for the inverse of the Freeman–Tukey double arcsine transformation is derived. This formula is useful when expressing means of double arcsines as retransformed proportions. When the mean is taken from original proportions involving different n's, it is suggested that the harmonic mean of the n's be used in the inversion formula.     

Test for Trend With a Multinomial Outcome

ABSTRACT

There is no established procedure for testing for trend with nominal outcomes that would provide both a global hypothesis test and outcome-specific inference. We derive a simple formula for such a test using a weighted sum of Cochran–Armitage test statistics evaluating the trend in each outcome separately. The test is shown to be equivalent to the score test for multinomial logistic regression, however, the new formulation enables the derivation of a sample size formula and multiplicity-adjusted inference for individual outcomes. The proposed methods are implemented in the R package multiCA.     

On a gmanova model likelihood ratio test criterion

For the generalized MANOVA (GMANOVA) model of Potthoff and Roy (1964), X = BξA + E, Khatri (1966) derives the likelihood ratio test criterion for test-ing the composite double linear null hypothesis CξV = 0, C,V known. This criterion plays an important role in statistics, and several authors have recently studied its further properties. However, Khatri's (1966) de-reviation of the distribution of this criterion is involved. By noting that the GMANOVA model is re-stricted MANOVA model, this paper presents an alter-native simple derivation of the distribution of this criterion. The derivation is based on the generalized Sverdrup's lemma, Kabe (1965).     

Survival Distributions Satisfying Benford's Law

Hill stated that “An interesting open problem is to determine which common distributions (or mixtures thereof) satisfy Benford's law …”. This article quantifies compliance with Benford's law for several popular survival distributions. The traditional analysis of Benford's law considers its applicability to datasets. This article switches the emphasis to probability distributions that obey Benford's law.