Neyman-Pearson Lemma Research Articles

Stability of testing hypotheses

The stability of testing hypotheses is discussed. Differing from the usual tests measured by Neyman-Pearson lemma, the regret and correction of the tests are considered. After the decision is made based on the observationsX 1,X 2, ⋅⋅⋅,X n, one more piece of datumX n+1 is picked and the test is done again in the same way but based onX 1,X 2, ⋅⋅⋅,X n,X n+l There are three situations: (i) The previous decision is right but the new decision is wrong; (ii) the previous decision is wrong but the new decision is right; (iii) both of them are right or both of them are wrong. Of course, it is desired that the probability of the occurrence of (i) is as small as possible and the probability of the occurrence of (ii) is as large as possible. Since the sample size is sometimes not chosen very precisely after the type I error and the type II error are determined in practice, it seems more urgent to consider the above problem. Some optimal plans are also given.

Monotonic expectations, Neyman-Pearson lemma, and some properties of the SPRT

Let f 0( x) and f 1( x) be probability density functions relative to a σ-finite measure μ under respective probability measures P 0 and P 1. Let X be a random variable governed by either P 0 or P 1, and let E i denote the expectation under P i , i = 0, 1. Let ф and ψ ( 0⩽ф⩽1, 0⩽ψ⩽1) be real functions such that (ф − ψ)(λ 1f 1 − λ 0f 0)⩾0 for all x, where λ 0 and λ 1 are positive constants. It is shown in an elementary manner that λ 0E 0(ф − ψ)⩽λ 1E 1(ф − ψ) . This monotonic property immediately gives the Neyman-Pearson lemma and the optimal Bayes test, and hence both share the same intrinsic monotonicity of expectation. This result is generalized in a number of ways which reflect several properties of the Neyman-Pearson test. A sequential extension is also given and this implies the monotonicity of the power function of a certain SPRT. The monotonic property of the average sample number of an optimum sequential test with power one is also discussed.

An application of the neyman-pearson lemma to gaussian processes

Letξ i (t),i=1,2,t ∈ [0, 1], be Gaussian zero mean processes with continous sample paths. Bounds for the probabilitiesβ i =P{ξ i -α i ∈B},i=1,2, are given, where aiε C0, 1., and B is a Borel subset of C[0, 1] Bibliography: 5 titles.

The necessity of strongly subadditive capacities for Neyman-Pearson minimax tests

Generalizing a result of Huber and Strassen and modifying a version due to Bednarski, it is shown that the converse of the Neyman-Pearson lemma for minimax tests holds for arbitrary Polish spaces.

Designs for Group Sequential Phase II Clinical Trials

The main goal of Phase II cancer clinical trials is to identify the therapeutic efficacy of new treatments. For ethical reasons, group sequential procedures, which allow for early stopping when a treatment is either extremely effective or extremely ineffective, have been widely employed in these trials. Although several useful design methods have been discussed in the literature (Fleming, 1982, Biometrics 38, 143-152; Lee, 1979, Cancer Treatment Reports 63, No. 11-12), we are unaware of any results addressing the problem of finding an optimal rule easily by computer. In this paper, using an idea based on the Neyman-Pearson lemma, we propose a method to search over a restricted set of designs and to select the optimal one in this set according to optimality criteria. In all the combinations we have investigated (more than 100) the optimal design produced by our method is the true global optimum. Other applications are discussed.

Characterization of Risk Sets for Simple versus Simple Hypothesis Testing

Abstract Consider testing a simple hypothesis H 0: F = F 0 against a simple alternative H 1: F = F 1 when a random variable X having distribution function F is observed. The risk region is defined to be the set of ordered pairs of error probabilities (E 0[φ(X)], E 1[l - φ(X)]) as φ ranges throughout all tests. This set plays an important role in the Neyman-Pearson lemma, which characterizes the class of most powerful tests of H 0 versus H 1. In a typical introductory course in hypothesis testing, the risk region is shown to be a closed convex subset of the unit square [0, 1] x [0, 1] that is symmetric about (1/2, 1/2) and contains the points (1, 0) and (0, 1). In this article, it is shown that any set with these properties is the risk region for some pair of hypotheses. In fact, one can take F 0 to be the uniform distribution on [0, 1]. This result provides an exercise that illustrates the applicability of some results about convexity and absolute continuity that would be presented in a real analysis cour...

Consecutive Optimizers for a Partitioning Problem with Applications to Optimal Inventory Groupings for Joint Replenishment

We consider several subclasses of the problem of grouping n items (indexed 1, 2, …, n) into m subsets so as to minimize the function g(S1, …, Sm). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g(·) that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S1, …, Sm satisfies a more general condition called “semiconsecutiveness”). Moreover, by restricting attention to “consecutive” (or “semiconsecutive”) partitions, we can solve the partition problem in polynomial time for small values of m. If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman.

A characterization of the demand for land

In a model of an economy where land is modelled explicitly as heterogeneous subsets of the plane and utility takes an integral form, demand can be characterized as a specific map from endowments and prices into subsets of the plane. A modified form of the Neyman-Pearson lemma is used to obtain this result, the modification being necessary because the analog of randomized tests must be excluded from this model.

Binary Experiments, Minimax Tests and 2-Alternating Capacities

The concept of Choquet's 2-alternating capacity is explored from the viewpoint of Le Cam's experiment theory. It is shown that there always exists a least informative binary experiment for two sets of probability measures generated by 2-alternating capacities. This result easily implies the Neyman-Pearson lemma for capacities. Moreover, its proof gives a new method of construction of minimax tests for problems in which hypotheses are generated by 2-alternating capacities. It is also proved that the existence of least informative binary experiments is sufficient for a set of probability measures to be generated by a 2-alternating capacity. This gives a new characterization of 2-alternating capacities, closely related to that of Huber and Strassen.

On solutions of minimax test problems for special capacities

Solutions to minimax test problems between neighbourhoods generated by specially defined capacities are discussed. The capacities are superpositions of probability measures and concave functions, so the paper covers most of the earlier results of Huber and Rieder concerning minimax testing between ɛ-contamination and total variation neighbourhoods. It is shown that the Neyman-Pearson lemma for 2-alternating capacities, proved by Huber and Strassen, can be applied to test problems between noncompact neighbourhoods of probability measures. It turns out that the Radon-Nikodym derivative between the special capacities is usually a nondecreasing function of the truncated likelihood ratio of some probability measures.

On testing the correlation coefficient of a bivariate normal distribution

Here we propose two tests for testing ρ in the bivariate normal population assuming that the ratio of the variances in it is known. The first test (U.M.P.U.) is derived by using the Neyman-Pearson lemma, whereas the second test is obtained through testing the scale parameter of the Cauchy distribution. The powers of the first and second tests are compared with a well-known test, based on the sample correlation coefficient for small and large samples respectively.

Optimality criteria for the selection of a partially sequential indicator set

SUMMARY Wolfe (1977a) introduced the concept of a partially sequential two-sample hypothesis test and studied some of the properties for a general class of such procedures. In this paper we deal with criteria for selecting an indicator set for use in a partially sequential test. In particular, we show how the Neyman-Pearson lemma can be used to generate an asymptotically optimal indicator set, in the sense that it corresponds to an asymptotically most powerful partially sequential test. In addition, we provide an accurate upper bound for the asymptotic expected sample size of the sequentially obtained observations, and obtain a closed form approximation for the maximum number of these observations necessary in order to obtain prespecified asymptotic power bounds against alternatives of interest. Some key word8: Asymptotically most powerful partially sequential test; Asymptotic power bound; Expected sample size; Negative binomial distribution; Optimal indicator set; Partially sequential procedure; Two-sample test.

Zur lösung einer minimax-aufgabe mit anwendung auf ein problem der optimalen steuerung l

With the help of the NEYMAN-PEARSON lemma there are given general assertions about the form of the solutions of the minimax problem (3), (4), (5) with the restrictions (A), (B1), (B2) and the additional assumptions (C), (D). If the additional assumption (E) is fulfilled, too, the solutions can be calculated by means of the VOLTERA integral equations (34), (35). In part II of the paper a class of optimal control problems will be solved by this method.

Bounds for the Power of Likelihood Ratio Tests and Their Asymptotic Properties

Let $P_0$ and $P_1$ be two probability distributions on a measurable space $(\mathscr{H}, \mathscr{B})$. We consider the problem of testing the simple hypothesis $H: P = P_0$ against the simple alternative $K: P = P_1$ on the basis of $n$ independent random variables $X_1, X_2, \cdots, X_n$ with common distribution $P$. The method that is widely used in the literature for investigating large sample properties of optimal tests is the following: according to the Neyman-Pearson lemma an optimal test can be described by means of sums of independent random variables. Theorems from the theory of probabilities of large deviations or ergodic theorems then allow one to obtain results on the asymtotic behavior of the error probabilities of optimal tests. In this paper we use a representation of the power of an optimal test which has its origin in the duality theory of infinite linear programming, in order to derive upper and lower bounds for the power. The bounds holds for any sample size $n$. This is done in Section 1 for two different types of tests, namely for most powerful tests at leve $\alpha_n$ and for tests which minimize a weighted sum of the error probabilities. It turns out that those bounds allow one to derive the well-known asymptotic properties under slightly more general conditions, i.e., $\alpha_n$ is permitted to tend to zero faster than any negative power of $n$ (but not exponentially fast) and the weight $\lambda_n$ to tend exponentially fast to zero. This and another application are discussed in Section 2.

Nonlinear Functional Versions of the Neyman–Pearson Lemma

Versions of the Neyman–Pearson lemma are given (Theorems 1 and 2) which provide sufficiency criteria for constrained extrema of nonlinear functionals with continuous or discrete variation. The use of ratio contours is described as a technique for producing a trial solution to be tested against Theorem 1 or 2. Examples of applications include improvements over previously published methods of solutions of certain problems. Relationship to previous versions of the lemma and to the method of Lagrange multipliers is discussed.

Constrained Discrimination between K Populations

Summary The Bayes rule approach to discrimination, assigning a sample point to the population whose posterior probability is largest, can lead to large misallocation probabilities. There have been several empirical attempts to overcome this difficulty but an objective method is suggested here. This is to find the allocation rule that maximizes the probability of correct allocation subject to the misallocation probabilities satisfying individual upper bounds. Necessary and sufficient conditions for the optimal rule are given and it is shown that a similar approach enables the Neyman–Pearson lemma to be generalized. Analytic methods of finding the optimal rule are discussed but these are of limited application so an approximate, iterative solution is suggested which can be used quite generally.

"Optimal" One-Sample Distribution-Free Tests and Their Two-Sample Extensions

The object of this paper is the development of a theory of optimal one-sample goodness-of-fit tests and of optimal two-sample randomized distribution-free (DF) statistics analogous to the well-known results of Hoeffding (1951), Terry (1952), Lehmann (1953), (1959), Chernoff and Savage (1958), Capon (1961) and others for two-sample nonrandomized rank statistics. For $Y_1, \cdots, Y_n$ a random sample from a population with continuous distribution function $G$, one tests in the one-sample case $H_0 : G = F$ vs. $H_1 : G \neq F$, where $F$ is some known continuous distribution function. From the Neyman-Pearson lemma, distribution-free tests that are most powerful (MP) for any $H$ vs. $K$ satisfying $KH^{-1} = GF^{-1}$, are obtained. From these MP distribution-free tests, one can on paralleling the derivations ([14], [25], [17], [18], [7]) for locally MP tests in the two-sample case obtain locally MP tests in the one-sample case. Further, it is found that the class of alternatives, for which a critical region of the form $\lbrack\sum J\lbrack F(y_i)\rbrack > c\rbrack$ is locally MP, is the class of $G$'s that consists of "contaminated" Koopman-Pitman distributions as given in Section 5. Randomized versions of the two-sample MP and locally MP rank statistics are considered and shown to be asymptotically equivalent to the locally MP rank statistics.

On Fisher's Bound for Asymptotic Variances

On Fisher's Bound for Asymptotic Variances