Least Favorable Configuration Research Articles

Bonferroni-type Plug-in Procedure Controlling Generalized Familywise Error Rate

Consider the multiple hypotheses testing problem controlling the generalized familywise error rate k-FWER, the probability of at least k false rejections. We propose a plug-in procedure based on the estimation of the number of true null hypotheses. Under the independence assumption of the p-values corresponding to the true null hypotheses, we first introduce the least favorable configuration (LFC) of k-FWER for Bonferroni-type plug-in procedure, then we construct a plug-in k-FWER-controlled procedure based on LFC. For dependent p-values, we establish the asymptotic k-FWER control under some mild conditions. Simulation studies suggest great improvement over generalized Bonferroni test and generalized Holm test.

The Slippage Configuration Is Always the Least Favorable Configuration for Two Alternatives

Performance guarantees for multinomial selection procedures are usually derived by finding the least favorable configuration (LFC)—the one for which the probability of correct selection is minimum outside the indifference zone—and then evaluating the procedure on that configuration. The slippage configuration has been proved to be the LFC for several procedures and has been conjectured to be the worst for some other procedures. The principal result of this article unifies and extends all previous results for two alternatives: the slippage configuration is the worst for all procedures that have a finite expected number of trials and always select the alternative with more successes. A generalization of the key inequality in the proof to an arbitrary number of alternatives is conjectured.

Plug-in tests for nonequivalence of means of independent normal populations.

In this paper, we consider the problem of testing nonequivalence of several independent normal population means. It is a well-known problem to test the equality of several means using the analysis of variance (ANOVA). Instead of determining the equality, one may consider more flexible homogeneity, which allows a predetermined level of difference. This problem is known as testing nonequivalence of populations. We propose the plug-in statistics for two different measures of variability: the sum of the absolute deviations and the maximum of the absolute deviations. For each test, the least favorable configuration (LFC) to ensure the maximum rejection probability under the null hypothesis is investigated. Furthermore, we demonstrate the numerical studies based on both simulation and real data to evaluate the plug-in tests and compare these with the range test.

An enhanced lognormal selection procedure

John and Chen (IEEE Trans Reliab 55(1):135---148, 2006) propose an exact two-stage solution based on the Least Favorable Configuration (LFC) to the ranking and selection problem of determining the best system from k lognormal populations. Lognormal density is commonly used to model certain lifetimes in reliability and survival analysis. It is known that selection procedures that are developed based on the LFC are conservative and become inefficient when the number of systems is large. We propose to take into account the differences of sample values within the parameter of interest when determining sample sizes, which can significantly increase the efficiency of the selection procedures. We also sequentialize the selection procedure and provide a procedure for estimating the constant needed to apply the solution. An experimental performance evaluation demonstrates the validity and efficiency of the enhanced lognormal selection procedure.

On testing the equivalence of treatments using the measure of range

A studentized range test is proposed to test the hypothesis of average equivalence of treatments in terms of the distance between means against the alternative hypothesis of inequivalence. A least favorable configuration (LFC) of means to guarantee the maximum level at a null hypothesis and a LFC of means to guarantee the minimum power at an alternative hypothesis are obtained. The level and power of the test are fully independent of the unknown means and variances. For a given level and a given power, the critical value and the required sample size for an experiment can be simultaneously determined, and the tables of critical values and sample sizes are provided for practitioners. A real numerical example to demonstrate the use of the test procedure is provided. In situations where the common population variance is unknown and the equivalence is the actual distance between means without standardization, a two-stage sampling procedure can be employed to find these solutions. It proves to be a quite feasible solution for practitioners.

Comparing multiple treatments to both positive and negative controls

In the past, most comparison to control problems have dealt with comparing k test treatments to either positive or negative controls. Dasgupta et al. [2006. Using numerical methods to find the least favorable configuration when comparing k test treatments to both positive and negative controls. Journal of Statistical Computation and Simulation 76, 251–265] enumerate situations where it is imperative to compare several test treatments to both a negative as well as a positive control simultaneously. Specifically, the aim is to see if the test treatments are worse than the negative control, or if they are better than the positive control when the two controls are sufficiently apart. To find critical regions for this problem, one needs to find the least favorable configuration (LFC) under the composite null. In their paper, Dasgupta et al. [2006. Using numerical methods to find the least favorable configuration when comparing k test treatments to both positive and negative controls. Journal of Statistical Computation and Simulation 76, 251–265] came up with a numerical technique to find the LFC. In this paper we verify their result analytically. Via Monte Carlo simulation we compare the proposed method to the logical single step alternatives: Dunnett's [1955. A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association 50, 1096–1121] or the Bonferroni correction. The proposed method is superior in terms of both the Type I error and the marginal power.

A Note on a Subset Selection Procedure for the Most Probable Multinomial Event

Panchapakesan (1971) proposed and investigated a subset selection procedure for selecting the most probable cell in a multinomial distribution on k (≥2) cells. He showed that the least favorable configuration (LFC) for the probability of a correct selection (PCS) is that of equal cell-probabilities. He showed that this result holds exactly for k = 2 and asymptotically for k ≥ 3. Later, Chen (1986) and Liu and Lin (1991) showed that the result holds exactly for k ≥ 3. Their proofs involve differentiation of type-2 Dirichlet integrals with the restriction that the cell-probabilities add up to unity. We now give a fairly simple proof of this result by obtaining the PCS as a single integral involving the gamma distribution. This was the proof behind the claim of the LFC result made without details in the abstract by Panchapakesan (1973). Recommended by N. Mukhopadhyay

Using numerical methods to find the least favorable configuration when comparing k test treatments with both positive and negative controls

Traditionally, comparisons to control problems dealt with comparing k test treatments to either a positive control or a negative control. However, in certain situations, it is necessary to include both types of control in the study. When comparing k test treatments to both positive and negative controls, the null hypothesis is composite and finding the least favorable configuration (LFC) under the null analytically is difficult. We propose a numerical solution for this problem. Using our method, we show that the LFC under the null is when half of the test treatment means are equal to the mean of the negative control and the other half of the test treatment means are equal to the mean of the positive control. We calculate critical points at the LFC for k=2–6. This work was motivated by real life problems that we discuss and illustrate with an example.

An interval estimation for the number of signals

We propose a multi-step procedure for constructing a confidence interval for the number of signals present. The proposed procedure uses the ratios of a sample eigen-value and the sum of different sample eigen-values sequentially to determine the upper and lower limits for the confidence interval. A preference zone in the parameter space of the population eigen-values is defined to separate the signals and the noise. We derive the probability of a correct estimation, P (CE), and the least favorable configuration (LFC) asymptotically under the preference zone. Some important procedure properties are shown. Under the asymptotic LFC, the P (CE) attains its minimum over the preference zone in the parameter space of all eigen-values. Therefore a minimum sample size can be determined in order to implement our procedure with a guaranteed probability requirement.

Development of a lower confidence limit for the number of signals

We propose a multistep procedure for constructing a lower confidence limit for computing the number of signals present in a vector measurement. We derive the probability of correct estimation P(CE) and the least favorable configuration (LFC) for our procedure. Under the LFC, P(CE) attains its minimum over the parameter space of all eigenvalues. Therefore, to implement our technique, procedure parameters are determined for the LFC for each sample size n so that the probability requirement is reached.

A selection procedure for estimating the number of signal components

This paper proposes a selection procedure to estimate the multiplicity of the smallest eigenvalue of the covariance matrix. The unknown number of signals present in a radar data can be formulated as the difference between the total number of components in the observed multivariate data vector and the multiplicity of the smallest eigenvalue. In the observed multivariate data, the smallest eigenvalues of the sample covariance matrix may in fact be grouped about some nominal value, as opposed to being identically equal. We propose a selection procedure to estimate the multiplicity of the common smallest eigenvalue, which is significantly smaller than the other eigenvalues. We derive the probability of a correct selection, P(CS), and the least favorable configuration (LFC) for our procedures. Under the LFC, the P(CS) attains its minimum over the preference zone of all eigenvalues. Therefore, a minimum sample size can be determined from the P(CS) under the LFC, P(CS|LFC), in order to implement our new procedure with a guaranteed probability requirement. Numerical examples are presented in order to illustrate our proposed procedure.

An integrated selection formulation for the best normal mean: the unequal and unknown variance case

This paper considers an integrated formulation in selecting the best normal mean in the case of unequal and unknown variances. The formulation separates the parameter space into two disjoint parts, the preference zone (PZ) and the indifference zone (IZ). In the PZ we insist on selecting the best for a correct selection (CS1) but in the IZ we define any selected subset to be correct (CS2) if it contains the best population. We find the least favorable configuration (LFC) and the worst configuration (WC) respectively in PZ and IZ. We derive formulas for P(CS1|LFC), P(CS2|WC) and the bounds for the expected sample size E(N). We also give tables for the procedure parameters to implement the proposed procedure. An example is given to illustrate how to apply the procedure and how to use the table.

An integrated formulation for selecting the best normal population: the common and unknown variance case

This paper considers an integrated formulation in selecting the best normal means in the case of common and unknown variance. The formulation separates the parameter space into two disjoint parts, the preference zone (PZ) and the indifference zone (IZ). In the PZ we insist on selecting the best for a correct selection (CSi) but in the IZ we define any selected subet to be correct (CS2) if it contains the best population. We find the least favorable configuration (LFC) and the worst configuration (WC) respectively in PZ and in the entire parameter space. We derive formulas for P(CSi|LFC), P(CS2|WC), and the bounds for the expected sample size E(N). We also provide tables for the procedure parameters to implement the proposed procedure. An example is given to illustrate the procedure and the table.

Lower percentage points of hartley's extremal quotient statistic and their applications

Consider K(>2) independent populations π1,..,π k such that observations obtained from π k are independent and normally distributed with unknown mean µ i and unknown variance θ i i = 1,…,k. In this paper, we provide lower percentage points of Hartley's extremal quotient statistic for testing an interval hypothesisH 0 θ [k] θ [k] > δ vs. H a : θ [k] θ [1] ≤ δ , where δ ≥ 1 is a predetermined constant and θ [k](θ [1]) is the max (min) of the θi,…,θ k . The least favorable configuration (LFC) for the test under H 0 is determined in order to obtain the lower percentage points. These percentage points can also be used to construct an upper confidence bound for θ[k]/θ[1].

On using hartley's statistic to test the hypothesis of not much difference among normal variances

In this paper we investigate the significance level and power of the Hartley's maximum F-ratio test for testing the null hypothesis versus the alternative hypothesis , where and is the maximum (minimum) of the variances of several normal distributions. Under the null hypothesis a least favorable configuration (LFC) for the maximum F-ratio test to be greater than or equal to a critical value is determined. The result is important for the calculation of the critical value of the test statistic to ensure a specified significance level under the hypothesis. Furthermore, this result may be used to construct lower confidence bounds for the ratio . A numerical procedure to implement the testing procedure is provided.

Percentage points of the studentized range test for dispersion of normal means

Consider k (≥ 2) independent populations such that observations obtained from πi are independent and normally distributed with unknown mean μiand a common unknown variance σ2, Let the standardized difference of the means be measured by is the maximum (minimum) of . In this paper, we provide tables for testing an interval hypothesis is a predetermined constant. The null hypothesis thus stipulates that the population means fall into a zone of indifference of size δ (standard units). As a test statistic for this hypothesis the studentized range is proposed and the least favorable configuration (LFC) for the test under H0 is determined. umerical quadrature is employed to obtain percentage points of the test statistic under the LFC for testing the interval hypothesis H0. These tables can also be used to construct a lower confidence bound for d(μ) if δ is not specified.

Percentage points of a studentized range statistic arising from non-identical normal random variables

Assume that there are k(≥2) independent populations π1,…,πk such that the observations obtained from πi:are independent and normally distributed with unknown mean μ1 and a common unknown variance σ2, i = 1, …, k. Let the standardized variation of the means be measured by where μ is the arithmetic mean of μ1:, …, μk:and The interest is to test the equivalence hypothesis against , where δ≥0 is a predetermined constant. A studentized range test is proposed and the least favorable configuration (LFC) under the null hypothesis for the studentized range to be less than or equal to a critical value is determined. Percentage points of the studentized range statistic at this LPC are tabulated and programmed. These percentage points can also be used to construct a lower confidence bound for if δ, and hence the null hypothesis is difficult to specify.

Some theorems, counterexamples, and conjectures in multinomial selection theory

Consider the problem of selecting the tcells of largest probability in a multinomial distribution with unknown cell probabilities P1P2…,Pk where 1 ≤ t< kand k≥ 3. The procedure we are concerned with is a single-stage fixed sample size one which selects the tcells of highest count in the sample, with ties broken by randomization. The preference zone is where δ is a constant in and p[1] ≤ p[2] ≤…≤p[k] denote the ranked multinomial cell probabilities. For a given sample size n the probability vector p in D(t, k, δ) which minimizes the probability of a correct selection over D(t, k, delta) is called a least favorable configuration (LFC) over the preference zone D(t, k,δ). In this paper, we first give some theorems which tell us that a least favorable configuration over the preference zone D(t, k, δ) must be in a certain subset D 0(t, k, δ) of D(t, k, delta). Then we use these theorems to construct examples which disprove the conjecture that the usual slippage Configuration is a least favorable configuration over the preference zone D(t, k, δ). Finally, we make some conjectures of our own which would be very interesting and useful if true.