It has long been known that Buffon's needle experiments can be used to estimate p. Three main factors influence these experiments: grid shape, grid density, and needle length. In statistical literature, several experiments depending on these factors have been designed to increase the efficiency of the estimators of p and to use all the information as fully as possible. We wrote the package BuffonNeedle to carry out the most common forms of Buffon's needle experiments. In this article we review statistical aspects of the experiments, introduce the package BuffonNeedle, discuss the crossing probabilities and asymptotic variances of the estimators, and describe how to calculate them using Mathematica. ‡ Introduction Buffon's needle problem is one of the oldest problems in the theory of geometric probability. It was first introduced and solved by Buffon [1] in 1777. As is well known, it involves dropping a needle of length l at random on a plane grid of parallel lines of width d > l units apart and determining the probability of the needle crossing one of the lines. The desired probability is directly related to the value of p, which can then be estimated by Monte Carlo experiments. This point is one of the major aspects of its appeal. When p is treated as an unknown parameter, Buffon's needle experiments can be seen as valuable tools in applying the concepts of statistical estimation theory, such as efficiency, completeness, and sufficiency. For instance, in order to obtain better estimators of p, Kendall and Moran [2] and Diaconis [3] examine several aspects of the problem with a long needle (l > d). Morton [4] and Solomon [5] provide the general extension of the problem. Perlman and Wishura [6] investigate a number of statistical estimation procedures for p for the single, double, and triple grids. In their study, they show that moving from single to double to triple grid, the asymptotic variances of the estimators get smaller and hence more efficient estimators can be obtained. Wood and Robertson [7] introduce the concept of grid density and provide an alternative idea. They show that Buffon's original single grid is actually the most efficient if the needle length is held constant (at the distance between lines on the single grid) and the grids are chosen to have equal grid density (i.e., equal length of maximizing the information in Buffon's experiments.

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In this article, we examine F , Wald, LR, and LM test statistics in the linear regression model using vector geometry. These four statistics are expressed as a function of one random variable -the angle between the vectors of unrestricted and restricted residuals. The exact and nominal sampling distributions of this angle are derived to illuminate some facts about the four statistics. Alternatively, we offer that the angle itself can be used as a test statistic. A Mathematica program is also written to carry out the approach.