Bernoulli trial

Abstract: CONTINUOUS FAILURE TIMES AND THEIR CAUSES Basic Probability Functions Some Small Data Sets Hazard Functions Regression Models PARAMETRIC LIKELIHOOD INFERENCE The Likelihood for Competing Risks Model Checking Inference Some Examples Masked Systems LATENT FAILURE TIMES: PROBABILITY DISTRIBUTIONS Basic Probability Functions Some Examples Marginal vs. Sub-Distributions Independent Risks A Risk-Removal Model LIKELIHOOD FUNCTIONS FOR UNIVARIATE SURVIVAL DATA Discrete and Continuous Failure Times Discrete Failure Times: Estimation Continuous Failure Times: Random Samples Continuous Failure Times: Explanatory Variables Discrete Failure Times Again Time-Dependent Covariates DISCRETE FAILURE TIMES IN COMPETING RISKS Basic Probability Functions Latent Failure Times Some Examples Based on Bernoulli Trials Likelihood Functions HAZARD-BASED METHODS FOR CONTINUOUS FAILURE TIMES Latent Failure Times vs. Hazard Modelling Some Examples of Hazard Modelling Nonparametric Methods for Random Samples Proportional Hazards and Partial Likelihood LATENT FAILURE TIMES: IDENTIFIABILITY CRISES The Cox-Tsiatis Impasse More General Identifiability Results Specified Marginals Discrete Failure Times Regression Case Censoring of Survival Data Parametric Identifiability MARTINGALE COUNTING PROCESSESES IN SURVIVAL DATA Introduction Back to Basics: Probability Spaces and Conditional Expectation Filtrations Martingales Counting Processes Product Integrals Survival Data Non-parametric Estimation Non-parametric Testing Regression Models Epilogue APPENDIX 1: Numerical Maximisation of Likelihood Functions APPENDIX 2: Bayesian Computation Bibliography Index

Abstract: The minimum standard deviations achievable for concurrent estimates of thresholds and psychometric function slopes as well as the optimal target values for adaptive procedures are calculated as functions of stimulus level and track length on the basis of the binomial theory. The optimum pair of targets for a concurrent estimate is found at the correct response probabilities p1 = 0.19 and p2 = 0.81 for the logistic psychometric function. An adaptive procedure that converges at these optimal targets is introduced and tested with Monte Carlo simulations. The efficiency increases rapidly when each subject's response consists of more than one statistically independent Bernoulli trial. Sentence intelligibility tests provide more than one Bernoulli trial per sentence when each word is scored separately. The number of within-sentence trials can be quantified by the j factor [Boothroyd and Nittrouer, J. Acoust. Soc. Am. 84, 101-114 (1988)]. The adaptive procedure was evaluated with 10 normal-hearing and 11 hearing-impaired listeners using two German sentence tests that differ in j factors. The expected advantage of the sentence test with the higher j factor was not observed, possibly due to training effects. Hence, the number of sentences required for a reliable speech reception threshold (approximately 1 dB standard deviation) concurrently with a slope estimate (approximately 20%-30% relative standard deviation) is at least N = 30 if word scoring for short, meaningful sentences (j approximately 2) is performed.

Abstract: The statistics of the number of success runs in a sequence of Bernoulli trials have been used in many statistical areas. For almost a century, even in the simplest case of independent and identically distributed Bernoulli trials, the exact distributions of many run statistics still remain unknown. Departing from the traditional combinatorial approach, in this article we present a simple unified approach for the distribution theory of runs based on a finite Markov chain imbedding technique. Our results cover not only the identical Bernoulli trials, but also the nonidentical Bernoulli trials. As a byproduct, our results also yield the exact distribution of the waiting time for the mth occurrence of a specific run.

Abstract: According to a common belief, the birth of a girl or of a boy are equiprobable events. Let us adopt this as the initial hypothesis, and check how it fits some available data. For example, in the period 1871–1900 there were n = 2, 644, 757 babies born in Switzerland including m = 1, 359, 671 boys and n − m = 1, 285, 086 girls.* How well does this data agree with our hypothesis that the probability of a boy’s birth is 0.5? By calling the last event a ‘success’, let us discuss the data in the framework of n = 2, 644, 757 Bernoulli trials, with unknown success probability p; the corresponding frequency is $$\frac{m}{n} = \frac{{1,359,671}}{{2,644,757}} = 0.5141. $$