Statistics

In this note, we give an explicit expression for the quantile of a mixture of two random variables. We carefully examine all possible cases of discrete and continuous variables with possibly unbounded support. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial information is available (Bernard and Vanduffel (2014)).

Nolan and Temple Lang argue that "the ability to express statistical computations is an essential skill." A key related capacity is the ability to conduct and present data analysis in a way that another person can understand and replicate. The copy-and-paste workflow that is an artifact of antiquated user-interface design makes reproducibility of statistical analysis more difficult, especially as data become increasingly complex and statistical methods become increasingly sophisticated. R Markdown is a new technology that makes creating fully-reproducible statistical analysis simple and painless. It provides a solution suitable not only for cutting edge research, but also for use in an introductory statistics course. We present evidence that R Markdown can be used effectively in introductory statistics courses, and discuss its role in the rapidly-changing world of statistical computation.

Review of Cheryl Misak, Frank Ramsey: A Sheer Excess of Powers (Oxford University Press, 2020).

Randomization procedures are used in legal and statistical applications, aiming to shield important decisions from spurious influences. This article gives an intuitive introduction to randomization and examines some intended consequences of its use related to truthful statistical inference and fair legal judgment. This article also presents an open-code Java implementation for a cryptographically secure, statistically reliable, transparent, traceable, and fully auditable randomization tool.

Innovation in healthcare payment and service delivery utilizes high cost, high risk pilots paired with traditional program evaluations. Decision-makers are unable to reliably forecast the impacts of pilot interventions in this complex system, complicating the feasibility assessment of proposed healthcare models. We developed and validated a Discrete Event Simulation (DES) model of primary care for patients with Diabetes to allow rapid prototyping and assessment of models before pilot implementation. We replicated four outcomes from the Centers for Medicare and Medicaid Services Federally Qualified Health Center Advanced Primary Care Practice pilot. The DES model simulates a synthetic population's healthcare experience, including symptom onset, appointment scheduling, screening, and treatment, as well as the impact of physician training. A network of detailed event modules was developed from peer-reviewed literature. Synthetic patients' attributes modify the probability distributions for event outputs and direct them through an episode of care; attributes are in turn modified by patients' experiences. Our model replicates the direction of the effect of physician training on the selected outcomes, and the strength of the effect increases with the number of trainings. The simulated effect strength replicates the pilot results for eye exams and nephropathy screening, but over-estimates results for HbA1c and LDL screening. Our model will improve decision-makers' abilities to assess the feasibility of pilot success, with reproducible, literature-based systems models. Our model identifies intervention and healthcare system components to which outcomes are sensitive, so these aspects can be monitored and controlled during pilot implementation. More work is needed to improve replication of HbA1c and LDL screening, and to elaborate sub-models related to intervention components.

This note is an extended read of my read of Laplace's book Theorie Analytique des Probabilites, when considered from a Bayesian viewpoint but without historical nor comparative pretentions. A deeper analysis is provided in Dale (1999).

We discuss the role that the null hypothesis should play in the construction of a test statistic used to make a decision about that hypothesis. To construct the test statistic for a point null hypothesis about a binomial proportion, a common recommendation is to act as if the null hypothesis is true. We argue that, on the surface, the one-sample t-test of a point null hypothesis about a Gaussian population mean does not appear to follow the recommendation. We show how simple algebraic manipulations of the usual t-statistic lead to an equivalent test procedure consistent with the recommendation, we provide geometric intuition regarding this equivalence, and we consider extensions to testing nested hypotheses in Gaussian linear models. We discuss an application to graphical residual diagnostics where the form of the test statistic makes a practical difference. We argue that these issues should be discussed in advanced undergraduate and graduate courses.

A simple, intuitive approach to the assessment of probabilistic inferences is introduced. The Shannon information metrics are translated to the probability domain. The translation shows that the negative logarithmic score and the geometric mean are equivalent measures of the accuracy of a probabilistic inference. Thus there is both a quantitative reduction in perplexity, which is the inverse of the geometric mean of the probabilities, as good inference algorithms reduce the uncertainty and a qualitative reduction due to the increased clarity between the original set of probabilistic forecasts and their central tendency, the geometric mean. Further insight is provided by showing that the Rényi and Tsallis entropy functions translated to the probability domain are both the weighted generalized mean of the distribution. The generalized mean of probabilistic forecasts forms a spectrum of performance metrics referred to as a Risk Profile. The arithmetic mean is used to measure the decisiveness, while the -2/3 mean is used to measure the robustness.

This paper proposes a signature-based approach for solving redundancy allocation problems when component lifetimes are not only heterogeneous but also dependent. The two common schemes for allocations, that is active and standby redundancies, are considered. If the component lifetimes are independent, the proposed approach leads to simple manipulations. Various illustrative examples are also analysed. This method can be implemented for practical complex engineering systems.

It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds.