Related papers: Estimating Bernoulli trial probability from a smal…
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…
Sample size derivation is a crucial element of the planning phase of any confirmatory trial. A sample size is typically derived based on constraints on the maximal acceptable type I error rate and a minimal desired power. Here, power…
The last success problem is an optimal stopping problem that aims to maximize the probability of stopping on the last success in a sequence of independent $n$ Bernoulli trials. In the classical setting where complete information about the…
How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We…
Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ \psi_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad…
It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…
Computing the probability of evidence even with known error bounds is NP-hard. In this paper we address this hard problem by settling on an easier problem. We propose an approximation which provides high confidence lower bounds on…
A method for the numerical simulation of signed probability distributions for the case of tossing $1/n$-th of a coin is presented and illustrated by examples.
When a planner must decide whether it has enough evidence to make a decision based on probability, it faces the sample size problem. Current planners using probabilities need not deal with this problem because they do not generate their…
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…
We study the empirical likelihood approach to construct confidence intervals for the optimal value and the optimality gap of a given solution, henceforth quantify the statistical uncertainty of sample average approximation, for optimization…
Conventional Monte Carlo simulations are stochastic in the sense that the acceptance of a trial move is decided by comparing a computed acceptance probability with a random number, uniformly distributed between 0 and 1. Here we consider the…
While there exists a large amount of literature on the general challenges of and best practices for trustworthy online A/B testing, there are limited studies on sample size estimation, which plays a crucial role in trustworthy and efficient…
This paper studies the complexity of estimating Renyi divergences of discrete distributions: $p$ observed from samples and the baseline distribution $q$ known \emph{a priori}. Extending the results of Acharya et al. (SODA'15) on estimating…
In this paper we introduce randomized $t$-type statistics that will be referred to as randomized pivots. We show that these randomized pivots yield central limit theorems with a significantly smaller magnitude of error as compared to that…
We derive adjusted signed likelihood ratio statistics for a general class of extreme value regression models. The adjustments reduce the error in the standard normal approximation to the distribution of the signed likelihood ratio…
We describe briefly in this note a procedure for consistently estimating the marginal likelihood of a statistical model through a sample from the posterior distribution of the model parameters.
We analyse the asymptotic behaviour of the probability of observing the expected number of successes at each stage of a sequence of nested Bernoulli trials. Our motivation is the attempt to give a genuinely frequentist interpretation to the…
Interval estimation of the probability of success in a Binomial model is considered. Zieli\'nski (2018) showed that the confidence interval which uses information about non-homogeneity of the sample is better than the classical one. In the…
The established language for statistical testing --- significance levels, power, and p-values --- is overly complicated and deceptively conclusive. Even teachers of statistics and scientists who use statistics misinterpret the results of…