Publications by Enwu Liu
Calculate R0
\(R_0\), pronounced “R naught, is called the basic reproduction number in Epidemiology, it is defined as the expected number of secondary cases produced by a single case in a susceptible population.\(^1\) \(R_0\) is an important parameter for public health planning such as how quickly people can return to work or when to reopen schools, et a...
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Calculate R0
\(R_0\), pronounced “R naught, is called the basic reproduction number in Epidemiology, it is defined as the expected number of secondary cases produced by a single case in a susceptible popultion.\(^1\) If \(R_0\) is larger than 1 the infection will spread in the population.When \(R_0\) is smaller than 1, the infection will die out. The Wik...
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Markov Chain hitting probability.
For a Markov chain, the hitting probability from state \(i\) to set \(A\) is the probability of ever reaching the Set A, starting from initial state \(i\). The vector of hitting probabilities \(h_A=（h_{iA}:i \in S)\) is the minimal non-negative solution to the following equations \[ h_{iA}=\left\{\begin{matrix} 1 & \text{ for } i \in A\\ \s...
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Conduct natural spline regression 'by hand'
The use of Natural (restricted) spline regression model has been very popular to model non-linear effects of continuous covariates. Statistical software such as R, SAS, STATA and SPSS, et al all can be used to perform the natural spline regression. However, the output results by these software sometimes are quite ‘confusing’ therefore, if ...
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Ralph Smith’s Formal Script Symbol Fonts
\(\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\) Letter Script A \(\mathscr{A}\) B \(\mathscr{B}\) c \(\mathscr{C}\) D \(\mathscr{D}\) E \(\mathscr{E}\) F \(\mathscr{F}\) G \(\mathscr{G}\) H \(\mathscr{H}\) I \(\mathscr{I}\) J \(\mathscr{J}\) K \(\mathscr{K}\) L \(\mathscr{L}\) M \(\mathscr{M}\) N \(\mathscr{N}\) O \(\mathscr{O}\) P \(\mat...
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Notes for Explaining the Gibbs Sampler
Gibbs sampler has been widely used in Bayesian models. In 1992, Georoge Casella and Edward I. George published the paper “Explaining the Gibbs sample”.\(^1\) In the paper they gave several examples of Gibbs sampler and a proof that Gibbs sample sequence will converge to a marginal distribution. However,in the paper they didn’t provide co...
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Center of a Taylor Series
Taylor series is quite important in statistics, such as it can be used to derive the variance of a function of asymptotically normal random variables, derive the likelihood ratio test, the Wald-type test and the score-type test, et al. A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expan...
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Deviance of the Poisson regression
In Poisson regression there are two Deviances. The Null Deviance shows how well the response variable is predicted by a model that includes only the intercept (grand mean) and the Residual Deviance which is −2 times the difference between the log-likelihood evaluated at the maximum likelihood estimate (MLE) and the log-likelihood for a “sa...
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Consistency of MLE
The proof of consistency of MLE is quit abstract and a little bit harder to understand. Here are two methods for proving the consistency of the MLE. I think this two methods are much more easier to understand although not so rigorous. Method 1. Let \(\{X_1,...,X_n\}\) be a sequence of observations, Let \(\hat{\theta_n}\) be the MLE using \(\{X_...
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Product rule of differentiation
The product rule of differentiation will be used when prove the Rao-Cramer lower bound. \[\frac{d(uvw)}{dx}=\frac{du}{dv}vw+u\frac{dv}{dx}w+uv\frac{dw}{dx}\] In general \[\frac{d}{dx}\left[\prod_{i=1}^kf_i(x) \right]=\sum_{i=1}^k\left[(\frac{d}{dx}f_i(x))\prod_{j=1,j \neq i}^k f_j(x)\right]=(\prod_{i=1}^kf_i(x))(\sum_{i=1}^k\frac{f_i'(x)}{f_i...
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