Standard Distance in Univariate and Multivariate Analysis

We propose to use the term standard distance for the quantity in univariate analysis and show that it can be easily generalized to the multivariate situation, where it coincides with the square root of the Mahalanobis distance between two samples.     

Aggregation of Variables in Least-Squares Regression

I consider the properties of the estimator when the true model is y = β₁ x ₁ + β₂ x ₂ + u, but the restriction β₁ = β₂ = β is incorrectly imposed. I show that the probability limit of is a weighted sum of β₁ and β₂; the weights sum to 1 but do not necessarily lie in the unit interval, so plim need not be bounded by β₁ and β₂. Sufficient conditions for such bounding are derived. Certain changes in the moments of x ₁ and x ₂ have “perverse” effects on the weights. I illustrate the consequences of inappropriate aggregation of variables with an empirical example of the effect of research and development investment on productivity.     

The Gauss—Markov Theorem and Random Regressors

In the standard linear regression model with independent, homoscedastic errors, the Gauss—Markov theorem asserts that = (X'X)^-1(X'y) is the best linear unbiased estimator of β and, furthermore, that is the best linear unbiased estimator of c'β for all p × 1 vectors c. In the corresponding random regressor model, X is a random sample of size n from a p-variate distribution. If attention is restricted to linear estimators of c'β that are conditionally unbiased, given X, the Gauss—Markov theorem applies. If, however, the estimator is required only to be unconditionally unbiased, the Gauss—Markov theorem may or may not hold, depending on what is known about the distribution of X. The results generalize to the case in which X is a random sample without replacement from a finite population.     

On Francis Galton's Difference Problem

Francis Galton proposed to split the money available for the first two prizes in a competition according to some ratio X, depending on the marks of the three best competitors, but invariant under change of location or scale of the marks. Assuming normality, Galton found that EX is about .75 and empirically he observed that X is nearly uniformly distributed between and 1. Our main purpose is to show that Galton was indeed right for a wide class of underlying distributions. As the number of competitors tends to ∞, the ratio X tends (in distribution) to a uniform random variable.     

Use of Dummy Variables in Testing for Equality between Sets of Coefficients in Two Linear Regressions: A Note

Elementary inductive proofs are presented for the binomial approximation to the hypergeometric distribution, the density of an order statistic, and the distribution of when X ₁, ···, X _n are a sample from N (μ, 1).     

Sample Size Requirements for the Back-of-the-Envelope Binomial Confidence Interval

The simplest approximate confidence interval for the binomial parameter p, based on x successes in n trials, is

where c is a suitable percentile of the normal distribution. Because I ₀ is so useful in introductory teaching and for back-of-the-envelope calculation, it is desirable to have guidelines for deciding when it provides a good answer. (It is clearly unwise to use I ₀ when x is too near 0 or n.) This article proposes such guidelines, based on the criterion that I ₀ should differ from the exact Clopper-Pearson confidence interval by an amount that is small compared to the length of the interval.     

How to Approximate a Histogram by a Normal Density

Which normal density curve best approximates the sample histogram? The answer suggested here is the normal curve that minimizes the integrated squared deviation between the histogram and the normal curve. A simple computational procedure is described to produce this best-fitting normal density. A few examples are presented to illustrate that this normal curve does indeed provide a visually satisfying fit, one that is better than the traditional , s answer. Some further aspects of this procedure are investigated. In particular it is shown that there is a satisfactory answer that is independent of the bar width of the histogram. It is also noted that this graphical procedure provides highly robust estimates of the sample mean and standard deviation. We demonstrate our technique by using data including Newcomb's data of passage time of light and Fisher's iris data.     

On a new type of sufficiency in undominated models

If a family of measures is dominated by a σ-finite measure, then the classical Fisher formulation of sufficiency and the pairwise sufficiency coincide. However, in undominated models, these definitions are essentially different. In particular, the Fisher formulation lacks the basic property: if a σ-algebra is sufficient, then also any larger σ-algebra ? is sufficient. We introduce a new definition of sufficiency, based on the concept of randomization, which has the property described above. We show that in the undominated case, our definition implies pairwise sufficiency and that the converse does not hold. If we assume that the underlying measurable space is a standard Borel space, then Fisher sufficiency implies our new formulation, and the converse implication does not hold.