Showing papers on "Conditional probability distribution published in 1971"

Recovery of inter-block information when block sizes are unequal

Abstract: SUMMARY A method is proposed for estimating intra-block and inter-block weights in the analysis of incomplete block designs with block sizes not necessarily equal. The method consists of maximizing the likelihood, not of all the data, but of a set of selected error contrasts. When block sizes are equal results are identical with those obtained by the method of Nelder (1968) for generally balanced designs. Although mainly concerned with incomplete block designs the paper also gives in outline an extension of the modified maximum likelihood procedure to designs with a more complicated block structure. In this paper we consider the estimation of weights to be used in the recovery of interblock information in incomplete block designs with possibly unequal block sizes. The problem can also be thought of as one of estimating constants and components of variance from data arranged in a general two-way classification when the effects of one classification are regarded as fixed and the effects of the second classification are regarded as random. Nelder (1968) described the efficient estimation of weights in generally balanced designs, in which the blocks are usually, although not always, of equal size. Lack of balance resulting from unequal block sizes is, however, common in some experimental work, for example in animal breeding experiments. The maximum likelihood procedure described by Hartley & Rao (1967) can be used but does not give the same estimates as Nelder's method in the balanced case. As will be shown, the two methods in effect use the same weighted sums of squares of residuals but assign different expectations. In the maximum likelihood approach, expectations are taken over a conditional distribution with the treatment effects fixed at their estimated values. In contrast Nelder uses unconditional expectations. The difference between the two methods is analogous to the well-known difference between two methods of estimating the variance o2 of a normal distribution, given a random sample of n values. Both methods use the same total sum of squares of deviations. But

The analysis of several 2× 2 contingency tables

Abstract: SUMMARY Consider data arranged into k 2 x 2 contingency tables. The principal result is the derivation of a statistical test for making an inference on whether each of the k contingency tables has the same relative risk. The test is based on a conditional reference set and can be regarded as an extension of the Fisher-Irwin treatment of a single 2 x 2 contingency table. Both exact and asymptotic procedures are presented. The analysis of k 2 x 2 contingency tables is required in several contexts. The two principal ones are (i) the comparison of binary response random variables, i.e. random variables taking on the values zero or one, for two treatments, over a spectrum of different conditions or populations; and (ii) the comparison of the degree of association among two binary random variables over k different populations. Cochran (1954) has investigated this problem with respect to testing if the success probability for each of two treatments is the same for every contingency table. Cochran's recommendation is that the equality of the two success probabilities should be tested using the total number, summed over all tables, of successes for one of the treatments. Cochran considers the asymptotic distribution of the total number of successes, for one of the treatments, conditional on all marginals being fixed in every table. He recommends this technique whenever the difference between the two populations on a logistic or probit scale is nearly constant for each contingency table. The constant logistic difference is equivalent to the relative risk being equal for each table. Mantel & Haenlszel (1959), in an important paper discussing retrospective studies, have also proposed an asymptotic method for analysing several 2 x 2 contingency tables. Their worlk on this problem was evidently done independently of Cochran, for their method is exactly the same as Cochran's except for a modification dealing with the correction factor associated with a finite population. Birch (1964) and Cox (1966) clarified the problem by showing, that under the assumption of constant logistic differences for each table, same relative risk, the conditional distribution of the total number of successes, for one of the treatments, leads to a uniformly most powerful unbiased test. Birch and Cox also derived the exact probability distribution of this conditional random variable under the given model. In this paper, we investigate the more general situation where the difference between the logits in each table is not necessarily constant. Procedures are derived for making an inference with regard to the hypothesis of constant logistic differences. Both the exact and asymptotic distributions are derived for the null and nonnull cases. This problem has been discussed by several investigators. A constant logistic difference corresponds to no interaction between the treatments and the k populations. The case k = 2 corresponds to one in which Bartlett (1935) has derived both an exact and an asymptotic procedure. Norton (1945)

Probabilistic methods of signal and system analysis

Abstract: Preface 1. Introduction To Probability 1-1 Engineering Applications Of Probability 1-2 Random Experiments And Events 1-3 Definitions Of Probability 1-4 The Relative-Frequency Approach 1-5 Elementary Set Theory 1-6 The Axiomatic Approach 1-7 Conditional Probability 1-8 Independence 1-9 Combined Experiments 1-10 Bemoulli Trials 1-11 Applications Of Bemoulli Trials 2. Random Variables 2-1 Concept Of A Random Variable 2-2 Distribution Functions 2-3 Density Functions 2-4 Mean Values And Moments 2-5 The Gaussian Random Variable 2-6 Density Functions Related To Gaussian 2-7 Other Probability Density Functions 2-8 Conditional Probability Distribution And Density Functions 2-9 Examples And Applications 3. Several Random Variables 3-1 Two Random Variables 3-2 Conditional Probability-Revisited 3-3 Statistical Independence 3-4 Correlation Between Random Variables 3-5 Density Function Of The Sum Of Two Random Variables 3-6 Probability Density Function Of A Function Of Two Random Variables 3-7 The Characteristic Function 4. Elements oOf Statistics 4-1 Introduction 4-2 Sampling Theory- The Sample Mean 4-3 Sampling Theory- The Sample Variance 4-4 Sampling Distributions And Confidence Intervals 4-5 Hypothesis Testing 4-6 Curve Fitting And Linear Regression 4-7 Correlation Between Two Sets of Data 5. Random Processes 5-1 Introduction 5-2 Continuous And Discrete Random Processes 5-3 Deterministic And Nondeterministic Random Processes 5-4 Stationary and Nonstationary Random Processes 5-5 Ergodic And Nonergodic Random Processes 5-6 Measurement Of Process Parameters 5-7 Smoothing Data With A Moving Window Average 6. Correlation Functions 6-1 Introduction 6-2 Example:Autocorrelation Function Of A Binary Profess 6-3 Properties Of Autocorrelation Functions 6-4 Measurement Of Autocorrelation Functions 6-5 Examples Of Autocorrelation Functions 6-6 Crosscorrelation Functions 6-7 Properties Of Crosscorrelation Functions 6-8 Examples And Applications Of Crosscorrelation Functions 6-9 Correlation Matrices For Sampled Functions 7. Spectral Density 7-1 Introduction 7-3 Properties Of Spectral Density 7-4 Spectral Density And The Complex Frequency Plane 7-5 Mean-Square Values From Spectral Density 7-6 Relation Of Spectral Density To The Autocorrelation Function 7-7 White Noise 7-8 Cross-Spectral Density 7-9 Measurement Of Spectral Density 7-10 Periodogram Estimate Of Spectral Density 7-11 Examples And Applications Of Spectral Density 8. Repines Of Linear Systems To Random Inputs 8-1 Introduction 8-2 Analysis In The Time Domain 8-3 Mean And Mean-Swquare Value Of System Output 8-4 Autocorrelation Function Of System Output 8-5 Crosscorrelation Between Input And Output 8-6 Example Of Time-Domain Analysis 8-7 Analysis In The Frequency Domain 8-8 Spectral Density At The System Output 8-9 Cross-Spectral Densities Between Input And Output 8-10 Examples Of Frequency-Domain Analysis 8-11 Numerical Computation Of System Output 9. Optimum Linear Systems 9-1 Introduction 9-2 Criteria Of Optimaility 9-3 Restrictions On The Optimum System 9-4 Optimization By Parameter Adjustment 9-6 Systems That Minimize Mean-Square Error Appendices Appendix A: Mathematical Tables Appendix B: Frequently Encountered Probability Distributions Appendix C: Binomial Coefficients Appendix D: Normal Probability Distribution Function Appendix E: The Q-Function Appendix F: Student's T-Distribution Function Appendix G: Computer Computations Appendix H: Table Of Correlation Function-Spectral Density Pairs Appendix I: Contour Integration

Estimation of the Probability that Y<X

Abstract: The problem of estimating θ = Pr[Y < X] has been considered in the literature in both distribution-free and parametric frameworks. In this article, using a Bayesian approach, we consider the estimation of θ from two approaches. The first, analogous to the classical procedure, is concerned with the problem of parametric estimation. The second, peculiar to the Bayesian approach, is directed to the query, “For two future observations, × and Y, what is the probability (given only the available sample data) that Y is less than X” This probability, termed the predictive probability, is not an estimate but is, in fact, a probability. These two views are related in that this predictive probability is the mean of the posterior distribution of θ. In the following sections, these Bayesian procedures are applied to the case of independent exponentially distributed random variables and to various cases of the normal distribution. The Bayesian estimates thus obtained are compared, whenever possible, with their...

Estimating the Conditional Probability of Misclassification

Abstract: Let x be an observation assumed to be from one of two p-dimensional normal populations II, II2 with unknown mean vectors t , t2 and common known covariance matrix M. The observation is to be classified as coming from one of the populations. Samples S1 , S2 consisting of N1 , N2 observations known to come from HI , II2 respectively are also available, and a classification rule is based on these samples. This paper is concerned with the problem of estimating the conditional probability of misclassifying x given a fixed classification rule. Let xi , x2i (i = 1, ... , N ; j = 1, *. , N2) denote the observations in S1, S2, and let i , 22 denote the sample means. The classification rule will be the commonly used one (see for example Anderson, (1958)) that classifies x as HII if

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.

Modelling Conditional Probability

Abstract: Many studies of the joint frequency of the initial and final conditions of weather elements such as cloud cover, visibility, rainfall or temperature attest to the importance of the initial event as a predictor of the later event. Most efforts have involved the actual collection of the data in contingency tables but there is a strong need for an analytical tool to estimate the conditional probabilities from more readily available climatic frequencies. By assuming the Markov process, and with the help of published tables detailing the bivariate normal distribution, a succinct two-parameter model, using the climatic frequencies in a single equation, has been developed to estimate conditional probabilities, of both frequent and rare events, within a few percentage points. The two parameters have been charted as direct functions of the probability of the initial event and the temporal persistence of the element.

On the identifiability of parameters

Abstract: A precise definition of identifiability of a parameter is given in terms of consistency in probability for the parameter estimate. Under some mild uniformity assumptions on the conditional density parameterized by the unknown parameter, necessary and sufficient conditions for the unknown parameter to be identifiable are established. The assumptions and conditions are expressed in terms of the density of individual observations, conditioned upon all past observations. The results are applied to linear system identification problems.

Admissibility of Certain Location Invariant Multiple Decision Procedures

Abstract: Random variables X, Y1, Y2, -.. are available for observation with X real valued and Y1, Y2, * * * taking values in arbitrary spaces. The distribution of Y= (Y1, Y2, ) is given by Uj (j = 1, * ,r) and the conditional density with respect to Lebesgue measure given Yi = yi (i = 1, , n1) ispjn(x-0, y) wherey = (Yi, Y2, ). The parametersjand 0are unknown. A decision k E {1, -, m} is to be made with loss W(j, k, n, y) when n observations are taken. Following Brown's (1966) methods admissibility is proved for the decision procedure which is Bayes in the class of invariant procedures. The result contains that of Lehmann and Stein (1953).

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...

Evaluation of Conditional Failure Density from Hazard Rate

Abstract: Further to the recent discussion regarding the concepts of hazard rate and conditional failure density, it is shown that the latter may be expressed in terms of the former. Moreover, six functions, namely, failure time distribution function, failure time density, reliability function, hazard rate, conditional failure distribution, and conditional failure density, are shown to be equivalent to the extent that if one of them is known, the other five are completely determined. The results are summarized in a table.