John W. Pratt

Risk vulnerability and the tempering effect of background risk

Method of operating a warp knitting machine includes, after interruption of a weft running to the weft storage and activation of a machine shut-down device, initially controlling slow-down of the knitting instruments of the machine so that when the machine stops, the weft storage has been emptied of all but a predetermined number of weft lengths, severing the drive connection between the weft storage and the knitting instruments, removing from the weft storage the weft remaining therein, automatically filling the weft storage with weft, and the restoring the drive connection for continuing the knitting operation.

Price Differences in almost Competitive Markets

I. Introduction, 189.—II. Equilibria in models without learning—the case of knowledge, 191.—III. Equilibrium in models with learning, 196.—IV. Empirically observed distributions of prices quoted by different sellers, 204.—V. Qualifications, implications, and conclusions, 205.

On the interpretation and observation of laws

A law with factors x and concomitants z specifies a distribution given z of a potential value Yx that is defined for each x whether or not it is observed. An observed distribution of Y given x and z agrees with the law if and only if, given z, the observed x is independent of Yx or, equivalently, of the joint effect of Ux of excluded variables on Yx. To establish such independence in non-experimental data requires exhaustive exploration of the effects of concomitants, causal and non-causal; R2 and F are irrelevant. We show how the model-free theory applies to linear models, time series, and simultaneous equations, and point out its Bayesian implications.

Concavity of the Log Likelihood

Abstract For a very general regression model with an ordinal dependent variable, the log likelihood is proved concave if the derivative of the underlying response function has concave logarithm. For a binary dependent variable, a weaker condition suffices, namely, that the response function and its complement each have concave logarithm. The normal, logistic, sine, and extreme-value distributions, among others, satisfy the stronger condition, the t (including Cauchy) distributions only the weaker. Some converses and generalizations are also given. The model is that which arises from an ordinary linear regression model with a continuous dependent variable that is partly unobservable, being either grouped into intervals with unknown endpoints, or censored, or, more generally, grouped in some regions, censored in others, and observed exactly elsewhere.

On the Nature and Discovery of Structure

Abstract Extending principles of experimentation, we discuss conditions under which nonexperimental data allow consistent estimation of effects of the kind revealed by experimentation and relevant to decisions. We show how implications of these conditions are often overlooked and how failure to distinguish between “factors” and “concomitants” makes almost anything said about a model ambiguous if not wrong. The effects to be estimated dictate the factors to be included; consistency and efficiency determine the concomitants, whose effects are not to be estimated. Concomitants may affect but must not be affected by the factors. Effects of excluded variables on an included variable may cause inconsistency if the included variable is a factor, can only reduce inconsistency if the included variable is a concomitant. Exclusion of a variable because it is highly correlated with another may sometimes be legitimate if both variables are concomitants, never if either is a factor. A condition for consistent estimatio...

Obustness of Some Procedures for the Two-Sample Location Problem

Abstract The level of ordinary two-sample procedures is not preserved if the two populations differ in dispersion or shape. The effect of such differences, especially differences in dispersion, on the t, median, Mann-Whitney, and normal scores procedures is investigated asymptotically, and tables are given comparing the four procedures.

Aversion to One Risk in the Presence of Others

The more risk-averse of two individuals need not have the smaller certainty equivalent for a risk \~x if another risk or combination of risks w is present. It is shown that he must, however, if either individuals conditional certainty equivalent for x is increasing in w. For independent risks, this condition follows immediately if either individual is decreasingly risk-averse, giving a natural proof of a known result. Another short proof of this result and necessary and sufficient conditions in the independent case are give. For multivariate utilities, the corresponding results do not hold, but it is proved simply that any mixture of decreasingly risk-averse utilities is decreasingly risk-averse. Also touched upon are risk aversions relation to generalized means, concave composition, risk sharing, and interest rates, the application of the results to discounting under uncertainty and selection of investment level, and their connection to singly crossing distributions, noise, and dominance.

Remarks on Zeros and Ties in the Wilcoxon Signed Rank Procedures

Abstract A Wilcoxon one-sample signed rank test may be made when some of the observations are 0 by dropping the 0s before ranking. However, a sample can be not significantly positive while a more negative sample (obtained by decreasing each observation equally), is significantly positive by the ordinary Wilcoxon test. The reverse is also possible. Two-piece confidence regions result. A procedure for avoiding these difficulties is proposed, namely to rank the observations including the 0s, drop the ranks of the 0s, and reject the null hypothesis if the sum of the remaining negative (or positive) ranks falls in the tail of its null distribution (given the number of 0s). If observations are tied in absolute value, their ranks may be averaged before attaching signs. This changes the null distribution. A sample may be significantly positive which is not significant if the observations are increased (unequally), or if the ties are broken in any way. * This research was supported by the United States Navy th...

The Contraction Mapping Approach to the Perron-Frobenius Theory: Why Hilbert's Metric?

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilberts projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...

Paying to improve your chances: Gambling or insurance?

Will a more risk-averse individual spend more or less to improve probabilities, say on marketing efforts that enhance the chance of a sale? For any two payoffs and starting probabilities, the answer is unfortunately indeterminate. However, interpreting gambling as increasing small chances of good outcomes and insurance as reducing small chances of bad outcomes, the more risk-averse individual will pay less (more) to gamble (insure). We find a critical switching probability that depends on the individuals and outcomes involved. If the good outcome is less (more) likely than this critical value, the expenditures represent gambling (insurance).