There have been many studies done over the years comparing birth safety in the hospital and out of the hospital. Some of these studies are stellar, some are greatly flawed. It is important that studies look at low risk populations planning an out of hospital birth with a trained midwife and compare these births with the same low risk populations in hospital. In some studies, unplanned homebirths, births assisted by untrained birth attendants or no birth attendants and babies born prior to 37 weeks or with birth defects not compatible with life have been included in the data, this gives a higher perinatal mortality rate than is accurate. Here are links to two of the best studies I am aware of. The second one looks at birth center births attended by trained midwives. Midwives carry the same equipment to a home delivery as can be found in a birthing center so it is acceptable to compare the results to births at home with trained midwives (usually licensed or certified professional midwives or certified nurse midwives).
1. Outcomes of planned home births with certified professional midwives: large prospective study in North America. BMJ 2005;330:1416
2. Outcomes of Care in Birth Centers: Demonstration of a Durable Model Stapleton, S.R., Osborne, C. and Illuzzi, J. (2013). Journal of Midwifery & Women’s Health, 58: 3-14, doi: 10.1111/jmwh.12003
3. Outcomes of Care for 16,924 Planned home Births in the United States: The Midwives Alliance of north America Statistics project, 2004-2009