## August 31, 2020

### Books to Read While the Algae Grow in Your Fur, August 2020

Attention conservation notice: I have no taste, and no qualifications to opine on the economics of the Internet or its implications.

I omit a number of books I read this month and didn't care for, but where my attempts at critique seem mean-spirited even to myself, and, worse, unlikely to inform anyone else. $\newcommand{\Expect}{\mathbb{E}\left[ #1 \right]} \DeclareMathOperator*{\argmin}{argmin}$

Christopher C. Heyde, Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation [SpringerLink]
The most basic sort of quasi-likelihood estimator is for regression problems. It requires us to know the model's prediction for the conditional mean of $Y$ given $X=x$, say $m(x;\theta)$, and the conditional variance of $Y$, say $v(x;\theta)$. It then enjoins us to minimize variance-weighted squared errors: $\hat{\theta} = \argmin_{\theta}{\frac{1}{n}\sum_{i=1}^{n}{\frac{\left(y_i - m(x_i;\theta)\right)^2}{v(x_i;\theta)}}}$ Equivalently, we solve the estimating equation $\frac{1}{n}\sum_{i=1}^{n}{\frac{\nabla_{\theta} m(x_i;\theta)}{v(x_i;\theta)} \left(y_i - m(x_i;\theta)\right)} = 0$ (I've left in the $\frac{1}{n}$ on the left-hand side to make it more evident that this last expression ought to converge to zero at, but only at, the right value of $\theta$.)
This is what we'd do if we thought $Y|X=x$ had a Gaussian distribution $\mathcal{N}(m(x;\theta), v(x,\theta))$; the objective function above would then be (proportional to) the log-likelihood. But there are many situations where a quasi-likelihood estimate works well, even if the real distribution isn't Gaussian. If we're dealing with linear regression functions, for instance, the Gauss-Markov theorem tells us that weighted least squares is the minimum-variance linear estimator, Gaussian distribution or no Gaussian distribution.
Heyde's book is about a broad family of quasi-likelihood estimators for lots of different stochastic processes. The basic idea is to find functionals of these processes which involve both the data and the parameters, which will have expected value zero at, but only at, the right parameters. (As with $Y_i - m(X_i;\theta)$ in the regression example.) More exactly, Heyde enjoins us to look for functionals which will be martingale difference sequences. We then form linear combinations of these functionals and solve for the parameter which sets the combination to zero. (That is, we solve the estimating equation.) The best weights in this linear combination (generally) reflect the variation in the martingale increments. This is an extremely flexible set-up which nonetheless lets Heyde prove some pretty useful results about the properties of his estimators, for a wide range of parametric and non-parametric problems involving stochastic processes.
Recommended for readers who have both a sound grasp on likelihood-based estimation theory for IID data, and some knowledge of stochastic differential equations and martingale theory. But good for those of us with those peculiar deformations. §
(I have been reading this, off and on, since 2002, but I have a rule about not recommending books until I've read them completely...)
Mike Carey and Elena Casagrande, Suicide Risk (vols. 1, 2, 3, 4, 5, 6)
High quality comic book mind candy. It's somewhat reminiscent, in style and theme, of classic Roger Zelazny (especially Creatures of Light and Darkness and Lord of Light), to the point where I'd be surprised if there wasn't direct influence. This is, to be clear, a good thing. §
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