Joint Maximum Likelihood Research Articles

Estimation and Hypothesis Testing in Finite Mixture Models

SUMMARY Finite mixture models are a useful class of models for application to data. When sample sizes are not large and the number of underlying densities is in question, likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter, λ, and the parameter of the underlying densities, θ, are problematical. Our approach places a prior distribution on λ and estimates θ by maximizing the likelihood of the data given θ with λ integrated out. Advantages of this approach, computational issues using the EM algorithm and directions for further work are discussed. The technique is applied to two examples.

MAXIMUM LIKELIHOOD AND BAYESIAN PARAMETER ESTIMATION IN ITEM RESPONSE THEORY*

ABSTRACTAdvantages and disadvantages of joint maximum likelihood, marginal maximum likelihood, and Bayesian methods of parameter estimation in item response theory are discussed and compared.

A general latent trait model for response processes

The purpose of the current paper is to propose a general multicomponent latent trait model (GLTM) for response processes. The proposed model combines the linear logistic latent trait (LLTM) with the multicomponent latent trait model (MLTM). As with both LLTM and MLTM, the general multicomponent latent trait model can be used to (1) test hypotheses about the theoretical variables that underlie response difficulty and (2) estimate parameters that describe test items by basic substantive properties. However, GLTM contains both component outcomes and complexity factors in a single model and may be applied to data that neither LLTM nor MLTM can handle. Joint maximum likelihood estimators are presented for the parameters of GLTM and an application to cognitive test items is described.

項目反応モデルにおける母数の同時推定

This article was concerned with Bayesian estimation of parameters in three parameter logistic models. Although a considerable number of estimation procedures based on likelihood (e. g. joint maximum likelihood estimation, marginal likelihood estimation, conditional maximum likelihood estimation) have been developed. only one research (Swaminathan and Gifford. 1980) used Bayesian approach for an estimation of parameters. The proposed method in this study differed from Swaminathan et al's method: (1) our methood was a full realization of the general Bayesian hierarchical model (Lindley and Smith, 1972), that is, our method employed exchangeable prior for all parameters; and (2) our method made use of an efficient quasi-Newton method to obtain the mode of the posterior distribution. We used the mode of joint posterior distribution of the parameters as Baysian estimates. Prior distributions employed at the first stage were as follows; 1) standard normal distribution N (O, 1) for ability parametersθ 2) chi distribution or truncated normal distribution N (μ a, σ 2a) for discrimination parameters a, 3) normal distribution N (μ b, σ b2) for difficulty parameters b, 4) truncated normal distribution N (μ c, σ c2) for pseudo chance-level parameters c. At the second stage, aporopriate distribution were assumed for hyper parameters (e. g. inverse chi-square distribution for σ a2, σ b2 or σ c2).Two programmes by the name of SIMDATA and BAYIRT were made. SIMDATA provided artificial data based on the random numbers generated according to three parameter logistic models. BAYIRT was to obtain modal estimates of parameters by making use of the result that the modal estimate of θ i. e. θ -value of the mode of the joint distribution corresponded to the mode of the conditional distribution of θ evaluated at the modal estimates of a, b and c, and vice versa. BAYIRT also provided MLE's if desired.We obtained Bayesian estimates and MLE's by applying BAYIRT to two kinds of artificial data, and another set of MLE's by applying a well-reputed LOGIST programme, whose algorithm was somewhat different from ours. In order to evaluate accuracy of the estimates, we calculated mean squared errors (MSE) of the estimates, and correlation coefficients between estimates and true values.In terms of MSE, Bayesian estimates were superior to MLE's. In terms of correlation coefficients, Bayesian estimates of parameters θ were superior to MLE's again, and Bayesian estimates of parameters a, b and c proved generally superior to MLE's.(An exception was that LOGIST produces slightly better estimates for parameters a and b. But the estimeters of parameters c were very poor and LOGIST failed to produce good estimates of parameters θ Our overall judgment was that Bayesian estimation procedure could yield better estimates.) Among Bayesian procedures, the estimates with normal distribution as prior of parameters a were found to be generally better than those with chi distribution as prior. Added accuracy of the Bayesian estimates seemed to be due to the collateral information gathered through the use of exchangeable priors.

SAMPLING VARIANCES AND COVARIANCES OF PARAMETER ESTIMATES IN ITEM RESPONSE THEORY

ABSTRACTThis paper develops a possible method for computing the asymptotic sampling variance‐covariance matrix of joint maximum likelihood estimates in item response theory when both item parameters and abilities are unknown. For a set of artificial data, results are compared with empirical values; also with the variance‐covariance matrices found by the usual formulas for the case where the abilities are known, or where the item parameters are known. The results are consistent with the conjecture that the new method is asymptotically correct except for errors due to grouping.

The joint estimation of differential delay, Doppler, and phase (Corresp.)

In radio and sonar applications it sometimes happens that narrow-band signals, originated from a remote source and observed at a pair of receivers, differ by unknown differential phase and Doppler shift in addition to the differential delay corresponding to the range difference. The correspondence presents the joint maximum likelihood (ML) estimate of the differential delay, Doppler, and phase and examines their accuracy by deriving the Cramér-Rao bound. It is shown that the joint ML estimators are the values of the delay and Doppler that maximize the magnitude of a generalized ambiguity function analogous to the one used in radar. It is also shown that for long observation time and high enough signal-to-noise ratio there is no degradation in the accuracy of the time-delay estimator due to the additional phase and Doppler uncertainty and that the differential Doppler is uncorrelated with the differential delay and phase estimators.

Estimation of the parameters of the Markovian representation of the autoregressive-moving average model

Tuan (1978) discussed the approximate maximum likelihood estimation of the parameters of the Markovian representation of the autoregressive-moving average model. He considered an indirect method, using the equivalent quasiautoregressive moving average form, the parameters of which are in one to one correspondence with those of the Markovian representation. A method of finding exact joint maximum likelihood estimates of the parameters and the initial state of the process is described here.

Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4

Abstract Expressions are given for the joint maximum likelihood estimates of the location and scale parameters of a Cauchy distribution based on samples of size 3 and 4.

AN EMPIRICAL MODEL OF PRICE AND OUTPUT BEHAVIOR

This paper proposes a model of short-run price and output behavior and undertakes an initial empirical investigation of the model with data from the manufacturing sector of the U.S. Economy. The model provides a relatively precise specification of the various factors that influence prices and output, and joint maximum likelihood techniques are used to estimate the parameters of the model. The empirical results support the proposition that demand-oriented forces primarily influence output while cost-push forces primarily influence prices and indicate that real interest rates affect both prices and output.

Estimation of Weibull Parameters With Competing-Mode Censoring

Existing results are reviewed for the maximum likelihood (ML) estimation of the parameters of a 2-parameter Weibull life distribution for the case where the data are censored by failures due to an arbitrary number of independent 2-parameter Weibull failure modes. For the case where all distributions have a common but unknown shape parameter the joint ML estimators are derived for i) a general percentile of the j-th distribution, ii) the common shape parameter, and iii) the proportion of failures due to failure mode j. Exact interval estimates of the common shape parameter are constructable in terms of the ML estimates obtained by using i) the data without regard to failure mode, and ii) existing tables of the percentage points of a certain pivotal function. Exact interval estimates for a general percentile of failure-mode-j distribution are calculable when the failure proportion due to failure-mode-j is known; otherwise a joint s-confidence region for the percentile and failure proportion is calculable. It is shown that sudden death endurance test results can be analyzed as a special case of competing-mode censoring. Tabular values for the construction of interval estimates for the 10-th percentile of the failure-mode-j distribution are given for 17 combinations of sample size (from 5 to 30) and number of failures.

Maximum Likelihood Estimation of the Parameters of the Weibull Distribution by Modified Quasilinearization

An iterative procedure is given for joint maximum likelihood estimation of the three parameters of the Weibull population. For this population, the likelihood function is forned and the maximum likelihood equations are obtained. Simultaneous solution of these equations yields joint maximum likelihood estimators for the three parameters. The proposed iterative procedure is a modified quasilinearization algorithm for solving nonlinear equations with bounded variables. Numerical results are given in which the parameters are estimated from real and from simulated life test data drawn from the Weibull population.

On Signal and Noise Level Estimation in a Coherent PCM Channel

Joint maximum likelihood estimators are presented for the signal amplitude and noise power density in a coherent PCM channel with white Gaussian noise and a correlation receiver. The estimates are based upon the correlation coefficient outputs of the receiver. From these estimators, an estimator for the quantity (received signal energy)/bit/,(noise power)/(unit bandwidth) upon which the error probabilities depend, is derived. This estimator is shown to be useful as 1) a point estimator for the signal-to-noise ratio for the higher values of this ratio (about 4 dB or greater), and 2) an easily calculated statistic upon which to base data acceptance or rejection criteria. The acceptance or rejection levels are obtained by the use of confidence interval curves in conjunction with word error probability data.