# On Item Parameter Estimation in Certain Latent Trait Models

• J. Pfanzagl
Chapter
Part of the Recent Research in Psychology book series (PSYCHOLOGY)

## Abstract

The paper deals with the estimation of item difficulty parameters (δ1,…, δm) if a subject with ability α produces the response vector (x 1,…, x m) ∈ {0, 1}m with probability $$\prod_{i=1}^{m}H{(\alpha-\delta_{i})}^{x_{i}}(1-H(\alpha-\delta_{i}))^{1-x_{i}}$$.

If H = Ψ (the logistic distribution function), conditional maximum likelihood estimators are consistent under mild conditions on the ability parameters α j , j = 1, 2,…. If the ability parameters are a random sample from an unknown distribution, conditional maximum likelihood estimators are consistent and asymptotically efficient. Marginal maximum likelihood estimators share these properties, since they are asymptotically equivalent to conditional maximum likelihood estimators by a theorem of de Leeuw and Verhelst.

If H = Φ, estimating (δ1,…, δm) is impossible if the distribution of ability parameters is completely unknown, since the item difficulty parameters are not identifiable in this case.

## Keywords

Mixture Model Parametric Family Nuisance Parameter Unknown Distribution Ability Parameter
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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