Evidence from both psychology and economics indicates that individuals give statements that appear to overestimate their ability compared to that of others. We test three theories that predict such relative overconfidence. The first theory argues that overconfidence can be generated by Bayesian updating from a common prior and truthful statements if individuals do not know their true type. The second theory suggests that self-image concerns asymmetrically affect the choice to receive new information about one's abilities, and this asymmetry can produce overconfidence. The third theory is that overconfidence is induced by the desire to send positive signals to others about one's own skill; this suggests either a bias in judgement, strategic lying, or both. We formulate this theory precisely. Using a large data set of relative ability judgements about two cognitive tests, we reject the restrictions imposed by the Bayesian model and also reject a key prediction of the self-image models that individuals with optimistic beliefs will be less likely to search for further information about their skill because this information might shatter their self-image. We provide evidence that personality traits strongly affect relative ability judgements in a pattern that is consistent with the third theory of social signalling. Our results together suggest that overconfidence in statements is more likely to be induced by social concerns than by either of the other two factors.