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Lévy‐Driven Ornstein‐Uhlenbeck (or CAR(1)) processes have been introduced in the literature as a model for stochastic volatility. A general formula to recover the unobserved driving process from a continuously observed CAR(1) was developed. When the CAR(1) process is observed at discrete times, the driving process must be approximated. Approximated increments of the driving process are used to test the hypothesis that the CAR(1) belongs to a specified class of Lévy processes. Two goodness‐of‐fit tests are proposed. The performance of the tests is illustrated through simulation. The Canadian Journal of Statistics 46: 355–376; 2018 © 2018 Statistical Society of Canada

This paper deals with a new Bayesian approach to the standard onesample z-and t-tests. More specifically, let x1, . . . , xn be an independent random sample from a normal distribution with mean µ and variance σ 2 .The goal is to test the null hypothesis H0 : µ = µ1 against all possible alternatives. The approach is based on using the well-known formula of the Kullbak-Leibler divergence between two normal distributions (sampling and hypothesized distributions selected in an appropriate way). The change of the distance from a priori to a posteriori is compared through the relative belief ratio (a measure of evidence). Eliciting the prior, checking for prior-data conflict and bias are also considered. Many theoretical On one-sample Bayesian tests 2 properties of the procedure have been developed. Besides it's simplicity, and unlike the classical approach, the new approach possesses attractive and distinctive features such as giving evidence in favor of the null hypothesis. It also avoids several undesirable paradoxes, such as Lindley's paradox that may be encountered by some existing Bayesian methods.The use of the approach has been illustrated through several examples.

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