Showing papers on "Likelihood principle published in 1971"

The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters

Abstract: Raiffa and Schlaifer's theory of conjugate prior distributions is here applied to Jeffrey's theory of tests for a sharp hypothesis, for simple normal sampling, for model I analysis of variance, and for univariate and multivariate Behrens-Fisher probelms. Leonard J. Savage's Bayesianization of Jeffrey's theory is given with new generalizations. A new conjugate prior family for normal sampling which allows prior independence of unknown mena and variance is given.

Likelihood ratio and associated test criteria

Abstract: SUMMARY A more accurate approximation is derived for the power function of the likelihood ratio criterion for testing a simple null hypothesis against a class of composite alternative hypotheses. Two rival test statistics suggested by Rao (1965) and asymptotically equivalent to the likelihood ratio procedure are examined and approximations are obtained to their distributions under the null and alternative hypotheses. Comparisons between the power functions of all three statistics are made.

Interval Estimation from the Likelihood Function

Abstract: During the last few years, several authors have devoted attention to both old and new methods of making inferences from likelihood functions. In this paper we study the properties of interval estimates which are obtained by drawing a horizontal line across the graph of the likelihood function. An exact evaluation is reported for one case of positive and one of negative binomial sampling, with parameter 0. The attained confidence coefficient cx(O) is graphed over 0? 0 < 1 for these two cases. A brief look is taken at intervals obtained when the random variable is continuous.

The Asymptotic Distribution of Conditional Likelihood Ratio Tests

Abstract: A conditional likelihood ratio test for testing a hypothesis concerning structural parameters in the presence of infinitely many incidental parameters is suggested. It is shown that the usual X2 approximation to the log-likelihood ratio fails to work in many situations involving incidental parameters. In contrast it is shown that the X2 approximation can be used in large samples under fairly general assumptions if we use conditional likelihood ratio tests instead. The relationship between the theory of UMPU-test and conditional likelihood ratio tests is discussed, and some examples are given to show that the conditional likelihood ratio approach covers more cases than the UMPU approach.

An Estimation Problem in Life-Testing

Abstract: This paper considers the maximum likelihood estimator (m.l.e.) of the scale parameter of the exponential distribution in a life-testing situation, where the stopping rule is a prestated time. The life-test is examined periodically and the number of items that have failed since the previous inspection are counted. Approximations to the m.l.e. are considered along with their properties.

Combining Component and System Information

Abstract: The situation in which results are available on both components and simple systems consisting of m identical, but independent components in series or in parallel is considered. The cases considered are attribute testing both at single and multiple stress levels and life testing. The method of maximum likelihood is used to obtain estimates; and approximate tests of hypotheses and confidence intervals are obtained using the asymptotic variances of the maximum likelihood estimates.

Some models for individual-group comparisons and group behavior

Abstract: In problem solving situations, it has been suggested the superiority of groups over individuals is due simply to the fact that groups consist of several individuals. In this paper, the Lorge and Solomon [1955] approach to such situations is reexamined using the method of maximum likelihood. Extensions to trichotomous response situations are also presented, and the resulting models are applied to data gathered by Staub [1970]. The partitioning of the likelihood ratio goodness-of-fit statistic is then discussed in order that the effects of additional variables on the response variate can be assessed. Finally, the small sample behavior of the likelihood ratio statistic is examined.

Maximum likelihood estimators for non-1-1 functions

Abstract: Dudewicz (1969) discussed the problem of providing maximum likelihood estimators (MLE’s) for ranked means, noted the Berk-Zehna-MLE, and studied another method of providingMLE’s for ranked means Below we formulate this latter method as a general inference principle and study it in some specific cases

Sequential Discrimination with Likelihood Ratios

Abstract: Let $X_1, X_2, \cdots$ be independent identically distributed random variables. You observe the $X$'s sequentially, knowing that their distribution is one of countably many different probabilities. Within an arbitrary error level, can you decide which one? This is the general problem of sequential discrimination. Freedman [4] showed that the discriminability of a family $\Theta$ is equivalent to a seemingly weaker condition. Namely, for any error level $\alpha$ and any particular $\theta \in \Theta$ there is a uniformly powerful fixed sample size test of $\{\theta\}$ versus $\Theta-\{\theta\}$ with error level uniformly as small as $\alpha$. The proof is constructive. Given the fixed sample size tests there is a recipe for manufacturing a sequential procedure to decide among all the members of $\Theta$. The fixed sample size tests are, however, still required. Hoeffding and Wolfowitz [5] considered this problem at length. LeCam and Schwartz [7] also touched upon it briefly. Both papers considered separations in various topologies and structures. Here we return to the original problem and ask whether likelihood ratios can be sensibly used. A rule is easy to specify0. For each $\theta \in \Theta$ you pick a number greater than one. Now watch $X_1, X_2, \cdots$. At each step compute the likelihood ratio for every pair of probabilities. Eventually it may happen that for some $\theta$ all the ratios with $\theta$ in the numerator are as big as the pre-assigned number. If so, stop and declare $\theta$ to be the true distribution. This rule is the extension to the countable case of the general sequential probability ratio test proposed by Barnard [2] and detailed by Armitage [1]. It does require the computation of all the likelihood ratios; but since $\Theta$ is countable, there is always at least one base for calculating densities. Any one will do. Likelihood ratio procedures have the advantage of being easy to formulate. Also, the comparison of densities seems to be a reasonably natural technique. However, it does not always work. An example will illustrate that it may fail spectacularly. When do likelihood ratio procedures work? The principal result is a characterization of families which are likelihood ratio discriminable. Check each probability $\theta$ separately. There may be a number $K(\theta)$ bigger than one which will be eventually exceeded simultaneously by all the ratios with $\theta$ in the numerator. If not, likelihood ratios will not work. If so, then the values $K(\theta)$ may be chosen so as to limit the error to any desired level. Despite their failures, likelihood ratio procedures may work when other natural conditions fail. Freedman showed that if each $\theta \in \Theta$ is isolated in the topology of setwise convergence, then $\Theta$ is disciminable. The converse is false. There is a family which is likelihood ratio discriminable, but which has one element in the setwise closure of all the others. This is a direct consequence of a recent theorem by LeCam [8]. Finally, with many familiar families likelihood ratio procedures have finite expected stopping time. Cases vary, however, and there is a discriminable family which has infinite expected stopping time for sampling under one of its elements.

Optimum Nonparametric Inference of the Partial Regression Coefficients

Abstract: SummaryIN this paper the problem of nonparametric inference about the regression vector in a linear regression in a (k + 1) variate population has been considered. It is assumed that the conditional density function of Y given (X1X2, ..., Xk) = (x1 , x2, ..., xk) is f(y—β0 —β1x1—...—βkxk)where the form of f is unknown and (β1, β2, ..., βk) is the regression vector (in the linear regression of Yon X1, X2, ..., Xk) which is to be estimated. Without loss of generality we assume β0 to be zero. It is also assumed that X1, X2, ..., Xk are bounded random variables. In the present study nonparametric estimates of the density function are obtained by the so-called kernel method. This gives rise to the concept of an empirical likelihood function. Motivated by the likelihood principle we then obtain an estimate of the regression vector, proceeding formally by maximizing the empirical likelihood function. For technical reasons, the tail observations have been treated in a different way from other observations. In fac...

Maximum likelihood analysis of balanced incomplete block models

Abstract: Maximum likelihood analysis of balanced incomplete blocks, and amall-sample properties of maximum likelihood estimates