In these days, online monitoring becomes a common tool for keeping highly reliability of products and systems. The online monitoring information which includes usage history, system conditions, and environmental conditions is reported and stored as big data. On statistical modeling, these variables from the online monitoring are primary candidates for covariates which affect the failure mechanism. There is some literature on modeling by the cumulative exposure model for the products lifetime distribution with covariate effects. The existing literatures require the already known parametric baseline distribution of the cumulative exposure. However such knowledge may be difficult to acquire in advance in some cases. When an incorrect baseline distribution is assumed, it is called misspecification. This paper proposes the strategy which use a likelihood function under a log-normal distribution to estimate parameters which represent covariate effects when the truly underlying baseline distribution is either a Weibull distribution or a log-normal distribution. In this paper, it is derived that the score function of a likelihood function under a log-normal baseline distribution is identified as the approximation for a Weibull cases. Besides, the simulation study and the discussion for the bias of estimation are shown, and this paper clarify the relationship among distribution parameters and the bias of estimation under misspecification.