What is the ruling for missing obligatory prayers for a period without a valid excuse? Is it obligatory to make up the missed obligatory prayers when one realizes that one has committed a sin by willfully neglecting them?

Establishing prayers is the second pillar of Islam after the declaration of faith in God and in His Messenger (peace and blessings be upon him). God Almighty says, "Establish obligatory prayers. Verily, such prayers are enjoined on the believers at stated times."
The five daily prayers form the most important devotional rite in a Muslim’s life. The Prophet (peace and blessings be upon him) said, “Prayer is a pillar of the religion [Islam]. Whoever establishes it, establishes religion and whoever destroys it (i.e. neglects it), destroys religion.”
It is obligatory to make up the missed obligatory prayers based on the words of the Prophet (peace and blessings be upon him) who said, "Whoever forgets to offer an obligatory prayer, let him perform it when he remembers it. There is no other expiation for it."
Prayers missed due to menstruation or postnatal bleeding are not to be made up. It was reported that a woman asked ‘Aishah (may God be pleased with her), "Why does a woman make up her missed fast days but not prayers?” ‘Aishah replied, “Are you strict like the people from Harura` (i.e., the Khawarij who settled in Harura`)?" The woman replied, 'No, I am just asking." Lady ‘Aishah said, "We were ordered (at the time of the Prophet) to make up our missed fast days but not the prayers we missed due to menstruation or postnatal bleeding."

The ruling
A Muslim can estimate the period during which he missed obligatory prayers based on what he is certain about. Then he is to spend an equal period making up the missed prayers by offering a missed prayer with each current one of its like. He can leave off performing the nawafil (voluntary prayers) during this time since making up missed prayers takes precedence over the nawafil.
And God Almighty knows best.