Most of the results in this thesis are obtained for the logit/probit model for binary response data given by Bartholomew (1980), which is sometimes called the two-parameter logistic model. In most the cases the results also hold for other common binary response models. By profiling and an approximation, we investigate the behaviour of the likelihood function, to see if it is suitable for ML estimation. Particular attention is given to the shape of the likelihood around the maximum point in order to see whether the information matrix will give a good guide to the variability of the estimates. The adequacy of the asymptotic variance-covariance matrix is inwestigated through jackknife and bootstrap techniques. We obtain the marginal ML estimators for the Rasch model and compare them with those obtained from conditional ML estimation. We also test the fit of the Rasch model against a logit/probit model with a likelihood ratio test, and investigate the behaviour of the likelihood function for the Rasch model and its bootstrap estimates together with approximate methods. For both fixed and decreasing sample size, we investigate the stability of the discrimination parameter estimates ai, 1 when the number of items is reduced. We study the conditions which give rise to large discrimination parameter estimates. This leads to a method for the generation of a (p+1)th item with any fixed ap+1,1 and ap+1,0. In practice it is importante to measure the latent variable and this is usually done by using the posterior mean or the component scores. We give some theoretical and applied results for the relation between the linearity of the plot of the posterior mean latent variable values, the component scores and the normality of those posterior distributions.

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Nous étudions, dans cette thèse, la construction des fonctions L p-adiques des motifs sur $\Q$ et, plus particulièrement, des formes modulaires.Dans les premiers trois chapitres on étend des constructions de Perrin-Riou pour construire, pour une représentation p-adique de de Rham $V$ du groupe de Galois absolu $\mathscr{G}_\qp$ de $\qp$ (ou, plus généralement, un alpha gamma-module de de Rham sur l'anneau de Robba) et un système compatible d'éléments globaux, une fonction L p-adique. On montre, en utilisant des lois de réciprocité montrées par Perrin-Riou, Colmez, Cherbonnier-Colmez, Berger et Nakamura, que ces fonctions interpolent des valeurs arithmétiques intéressantes aux caractères localement algébriques.Dans les derniers trois chapitres, on se spécialise au cas de dimension $2$. On démontre, en s'inspirant des techniques de Nakamura et des nouvelles techniques de changement de poids de Colmez introduites pour l'étude des vecteurs localement algébriques dans la correspondance de Langlands L p-adique pour $\mathrm{GL}_2(\qp)$, une équation fonctionnelle pour notre fonction L p-adique. Comme une application de cette équation fonctionnelle, on fournit les argument manquants dans les travaux de Nakamura, complétant la preuve de la conjecture $\epsilon$ locale de Kato pour les représentations de dimension $2$. Pour le motif associé à une forme modulaire, on utilise tous ces résultats pour interpréter les valeurs interpolées par la fonction L p-adique en termes des valeurs spéciales de la fonction $L$ complexe de cette forme. / This thesis studies the construction of $p$-adic $L$-functions associated to motives over $\Q$ and, in particular, to modular forms.In the first three chapters we generalize some constructions of Perrin-Riou in order to construct, for any $p$-adic de Rham representation $V$ of the absolute Galois group $\mathscr{G}_\qp$ of $\qp$ (or, more generally, any de Rham $(\varphi, \Gamma)$-module over the Robba ring) and any compatible system of global elements, a $p$-adic $L$-function. We show, by the use of some reciprocity laws proved by Perrin-Riou, Colmez, Cherbonnier-Colmez, Berger and Nakamura, that these functions interpolate interesting arithmetic values at locally algebraic characters.The last three chapters deal with the particular case of dimension $2$. We show, inspired by some techniques of Nakamura and certain weight change techniques introduced by Colmez for the study of locally algebraic vectors in the $p$-adic Langlads correspondence for $\mathrm{GL}_2(\qp)$, that our $p$-adic $L$-function satisfies a functional equation. As an application of our functional equation, we fulfil the missing arguments in the work of Nakamura, providing a complete proof of Kato's local $\epsilon$-conjecture for $2$-dimensional representations. For the motive associated to a modular form, we use these results to interpret the interpolated values of the $p$-adic $L$-function in terms of special values of the complex $L$-function of the form.