Frequentist Theory Research Articles

LOWER BAYESIAN CONFIDENCE LIMITS ON THE PROCESS CAPABILITY INDEX Cpm: A COMPARATIVE STUDY

ABSTRACT The main focus of this paper is on a particular type of process capability indices, termed C pm in the quality control community, enabling to consider proximity of the process center to the pre-specified target value as well as process inherent variability altogether whilst investigating process capability. In order to take the probabilistic properties of the point estimator used on C pm into account, the key element of this study is to present a generalized Bayes-type approach that allows for the construction of a lower interval limit for the process measure C pm . In comparison to the classical analogue for C pm (built upon the traditional frequentist theory) entailing a non-central chi-square distribution that the practitioners are generally unfamiliar with, the Bayesian method relates simply to the integration of a gamma distribution. The lower Bayesian interval estimate on C pm is also compared with the other three lower confidence bounds posed in the literature and then various experimental studies under a number of process parameter configurations assumed are conducted to illustrate practical guidance about the types of processes for which the Bayesian-based procedure might be advantageous. Directions of how to choose adequate lower confidence limits on C pm for practical use are reported as well.

Kolmogorov's Contributions to the Foundations of Probability

Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical and philosophical foundations of probability in the twentieth century, and his thinking on the topic is still potent today. In this article we first review the three stages of Kolmogorov's work on the foundations of probability: (1) his formulation of measure-theoretic probability, 1933; (2) his frequentist theory of probability, 1963; and (3) his algorithmic theory of randomness, 1965–1987. We also discuss another approach to the foundations of probability, based on martingales, which Kolmogorov did not consider.

Improved confidence estimators for Fieller's confidence sets

AbstractFieller's confidence set CF for ratios of location parameters, although of great importance in practice, is often cited as an example to criticize frequentist theory. The reason is that the set can consist of the whole parameter space and yet the confidence is γ = 1 – α in any case. In this paper, we study the problem of constructing data‐dependent estimators better than γ+, A reasonable estimator appears to be γ+, which is one if CF is the whole parameter space and γ otherwise. By using an estimated confidence approach and a squared‐error loss, it is shown that γ+ dominates γ. The risk improvement of γ+ over γ can be sizable. Also, by numerically comparing γ+ with a generalized Bayes estimator γL, which is shown to be admissible when one or two ratios are concerned, it is shown that γ+ is nearly admissible. We also conclude that the common practice of reporting 1 – α only when CF is not the whole parameter space is nearly admissible.

Ignorability in general incomplete-data models

SUMMARY Rubin (1976) defined ignorability conditions for frequentist and Bayes/likelihood analyses of data subject to missing observations. More recently, Heitjan & Rubin (1991) and Heitjan (1993) generalised the Rubin model to encompass other forms of incompleteness, establishing ignorability conditions for Bayes/likelihood inferences only. This paper extends the Heitjan-Rubin model by explicitly defining the observed degree of coarseness as a data element. This permits the development of a frequentist theory, including a generalisation of 'missing completely at random', the frequentist ignorability condition for missing data. The model is applied in a number of incomplete-data problems of general interest.

Estimation for Large Successive Samples with Unknown Inclusion Probabilities

We develop a frequentist theory of estimation for the following problem. A statistician is given a sample by successive sampling from a large finite population. Units are listed in the order in which they were observed. With each sampled unit is provided one or more characteristics and the size measure which was used in drawing the successive sample. Based solely on the sample information and with no other information about the sampling frame, the statistician is to estimate population totals for the characteristics of interest. The problem arises in the literature of oil and gas forecasting and of reliability growth.

A Noninformative Bayesian Approach to Interval Estimation in Finite Population Sampling

Abstract A noninformative Bayesian approach to interval estimation in finite population sampling is discussed. Given the sample, this method introduces the Polya distribution as a pseudo posterior distribution over the unobserved members of the population. In many cases this distribution yields interval estimates similar to those of standard frequentist theory. In addition, it can be used in situations where the standard methods are difficult to apply, for example, in producing an interval estimate for the ratio of two medians. We also consider related point estimation problems and observe that estimators derived from the pseudo posterior often perform better than classical alternatives.

A Noninformative Bayesian Approach to Interval Estimation in Finite Population Sampling

Abstract A noninformative Bayesian approach to interval estimation in finite population sampling is discussed. Given the sample, this method introduces the Polya distribution as a pseudo posterior distribution over the unobserved members of the population. In many cases this distribution yields interval estimates similar to those of standard frequentist theory. In addition, it can be used in situations where the standard methods are difficult to apply, for example, in producing an interval estimate for the ratio of two medians. We also consider related point estimation problems and observe that estimators derived from the pseudo posterior often perform better than classical alternatives.

Towards a Frequentist Theory of Upper and Lower Probability

We present elements of a frequentist theory of statistics for concepts of upper and lower (interval-valued) probability (IVP), defined on finite event algebras. We consider IID models for unlinked repetitions of experiments described by IVP and suggest several generalizations of standard notions of independence, asymptotic certainty and estimability. Instability of relative freqencies is favoured under our IID models. Moreover, generalizations of Bernoulli's Theorem give some justification for the estimation of an underlying IVP mechanism from fluctuations of relative frequencies. Our results indicate that an objectivist, frequency- or propensity-oriented, view of probability does not necessitate an additive probability concept, and that IVP models can represent a type of indeterminacy not captured by additive probability.

A "Super-Population Viewpoint" for Finite Population Sampling

Frequently it is reasonable for a sample surveyor to view the finite population of interest as an independent sample of size N from an infinite super-population. This super-population viewpoint is contrasted to the classical frequentist theory of finite population sampling and the classical theory of infinite population sampling. A new technique for making inferences about finite population "parameters' is developed and shown to be applicable for any survey design. Two example applications are given: the estimation of strata- and population means in stratified sampling and the use of the so-called regression estimators for the same purpose.

Statistical Problems in Science. The Symmetric Test of a Composite Hypothesis

Abstract The purpose of the paper is to illustrate a “give and take” type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observations x and the unknown “true” state of nature σ. Quite often, replication of the experiments reveals considerable variation in x, indicating that no mathematical treatment of the problem is possible without the assumption that x is a sample value of a random variable X, treated in terms of frequentist theory of probability. As to the true state of nature, σ, situations vary. Indeed, there are cases where it appears natural to consider that σ is selected at random out of a certain known set Σ, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can “give” something to science. In each example, however, the assumption of the randomness of σ appears extraneous. Section 3 de...

Statistical Problems in Science. The Symmetric Test of a Composite Hypothesis

Abstract The purpose of the paper is to illustrate a “give and take” type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observations x and the unknown “true” state of nature σ. Quite often, replication of the experiments reveals considerable variation in x, indicating that no mathematical treatment of the problem is possible without the assumption that x is a sample value of a random variable X, treated in terms of frequentist theory of probability. As to the true state of nature, σ, situations vary. Indeed, there are cases where it appears natural to consider that σ is selected at random out of a certain known set Σ, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can “give” something to science. In each example, however, the assumption of the randomness of σ appears extraneous. Section 3 describes another example of research in science, again with a non-random Σ, which happened to “give” something to the frequentist theory of statistics. The motivation to reduce the frequency of erroneous conclusions in the general circumstances of the biological study had led to considering the possibility that a certain derivative may fail to exist. This created a problem of testing (Section 4) which does not appear to have been considered. A solution of the problem, in the form of optimal symmetric (Cα) tests, is given in Section 5.