Sleep is essential for our physical and mental health, and disruptions in normal sleep patterns can negatively impact our well-being and performance. Studying sleep patterns can therefore help predict relapses in unhealthy or harmful behaviors. Existing tools used for quantifying sleep duration and patterns include self-report questionnaires and actigraphy data, but these methods do not perform well when the data is sparse. We propose a method that leverages smartphone data to predict bed times and wake times with screen-on and screen-off events. We use a latent variable model for length of time between screen-on events, which is expected to be greater when the individual is asleep than when he or she is awake. On nights when there is an insufficient amount of smartphone data, our method uses information from previous nights to produce estimates of bed times and wake times. The proposed method was validated using a sample of 16 teenagers with borderline personality disorder and sleep estimates from the proposed method were compared to those obtained with actigraphy data and self-report questionnaires.