The error in the normal approximation to the multinomial with an increasing number of classes

In an earlier paper, it was shown that under certain conditions, if the number of classes in a multinomial distribution increases as the number of trials increases, the probabilities assigned to arbitrary regions by the multinomial distribution are close to the probabilities assigned by the distribution of slightly rounded-off normal random variables. A different method of studying the approximation of the multinomial distribution by a normal distribution is to use the multivariate Berry-Esseen bound. In this paper, these two methods are compared, particularly with respect to the class of multinomial distributions for which the bounds on the error remain useful.     

Testing fit with nuisance location and scale parameters

For each n, X₁(n),…, X_n(n) are independent and identically distributed random variables, each with cumulative distribution function F(x) which is known to be absolutely continuous but is otherwise unknown. The problem is to test the hypothesis that \documentclass{article}\pagestyle{empty}\begin{document}$ F(x) = G\left( {{\textstyle{{x - \theta _1 } \over {\theta _2 }}}} \right) $\end{document}, where the cumulative distribution function G_x is completely specified and satisfies certain regularity conditions, and the parameters θ₁, θ₂ are unknown and unspecified, except that the scale parameter θ₂, is positive. Y₁ (n) ≦ Y₂ (n) ≦ … ≦ Y_n (n)are the ordered values of X₁(n),…, X_n(n). A test based on a certain subset of {Y_i(n)} is proposed, is shown to have asymptotically a normal distribution when the hypothesis is true, and is shown to be consistent against all alternatives satisfying a mild regularity condition.     

The normal approximation to the multinomial with an increasing number of classes

For a fixed number of classes and the number of trials increasing, the approach of the multinomial cumulative distribution function to a normal cumulative distribution function is familiar. In this paper we allow the number of classes to increase as the number of trials increases, and show that under certain circumstances the probabilities assigned to arbitrary regions by the multinomial distribution are all close to the probabilities assigned by the distribution of “rounded off” normal random variables. As the number of trials increases, the amount rounded off approaches zero. The result can be used to study the asymptotic distribution of functions of multinomial random variables.     

Tests of independence of continuous random variables which guard against special alternatives

For the problem of testing the independence of a set of continuous random variables, where the test is to be based on n independent and identically distributed observations, tests are constructed that guard against all alternatives but are particularly effective against special alternatives of interest. The tests are based on the numbers of observations falling in the cells of a grid constructed using order statistics.     

The dynamic lot size model with quantity discount: Counterexamples and correction

Federgruen and Lee ([3]) proposed an optimal algorithm for the single-item dynamic lot size model with all-unit discount. In this note we show that their algorithm fails to find the optimal solution for some special cases. We also provide a modification to the algorithm to handle them. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 419–422, 1998     

Scheduling spares with exponential lifetimes in a two-component parallel system

A set of n spare components whose life lengths are exponentially distributed with rates μ₁, …,μ_n are available to keep a two-component parallel system in operation. We derive the optimal order of replacement of failed components in order to maximize the system life length.     

Asymptotic inference about a density function at an end of its range

For each n, X₁(n),…X_n(n) are independent and identically distributed random variables, with common probability density function Where c, θ, α, and r(y) are all unknown. It is shown that we can make asymptotic inferences about c, θ, and α, when r(y) satisfies mild conditions.