Loraine Boettner writes the following in his The Reformed Doctrine of Predestination:
“Most Calvinistic theologians have held that those who die in infancy are saved. The Scriptures seem to teach plainly enough that the children of believers are saved; but they are silent or practically so in regard to those of the heathens. The Westminster Confession does not pass judgment on the children of heathens who die before coming to years of accountability.” (p. 144)
There is no such thing as an “age of accountability” — every person, no matter what age, is accountable.
"Concerning those who die in infancy, Dr. Warfield says:
‘Their destiny is determined irrespective of their choice, by an unconditional decree of God, suspended for its execution on no act of their own; and their salvation is wrought by an unconditional application of the grace of Christ to their souls, through the immediate and irresistible operation of the Holy Spirit prior to and apart from any action of their own proper wills…And if death in infancy does depend on God’s providence, it is assuredly God in His providence who selects this vast multitude to be made participants of His unconditional salvation…This is but to say that they are unconditionally predestinated to salvation from the foundation of the world. If only a single infant dying in irresponsible infancy be saved, the whole Arminian principle is traversed. If all infants dying such are saved, not only the majority of the saved, but doubtless the majority of the human race hitherto, have entered into life by a non-Arminian pathway.’ 
 Two Studies in the History of Doctrine, p. 230.
Certainly there is nothing in the Calvinistic system which would prevent us from believing this; and until it is proven that God could not predestinate to eternal life all those whom He is pleased to call in infancy we may be permitted to hold this view.” (pp. 143-144)
Boettner favorably quotes Warfield denying that the gospel is the power of God to salvation to EVERYONE BELIEVING (Romans 1:16). All of the elect without exception — from the elect infant, to the elect mentally handicapped person — will be regenerated and saved by Christ and will immediately be given knowledge and faith in Him. Men like Warfield and Boettner come to the text of Scripture with their false premise, their false presumption regarding the mental capacity of infants. They assert that infants are not able to believe due to their supposed lack of mental capacity or to the “fact” that infants cannot hear and understand anything while inside their mother’s womb. But Jesus Christ Himself obliterates this pseudo-scientific-psychological nonsense by saying that infants and/or little children ARE ABLE TO RECEIVE the kingdom of God (Luke 18:17). And if that is not enough we have the infant John the Baptist in addition to the passages already mentioned (e.g., Romans 1:16).
More from Boettner:
“The doctrine of infant salvation finds a logical place in the Calvinistic system; for the redemption of the soul is thus infallibly determined irrespective of any faith, repentance or good works, whether actual or foreseen. It does not, however, find a logical place in Arminianism or any other system. Furthermore, it would seem that a system such as Arminianism, which suspends salvation on a personal act of rational choice, would logically demand that those dying in infancy must either be given another period of probation after death, in order that their destiny may be fixed, or that they must be annihilated.” (pp. 144-145)
The blatant denial of the gospel as the power of God to salvation for everyone believing finds a “logical place” in various Calvinistic systems (e.g., Westminster Confession of Faith, The Second Helvetic Confession). For they deny that the gospel of Jesus Christ is the power of God to salvation to EVERYONE believing (Romans 1:16). Presumably to the chagrin of Warfield and Boettner (and all Calvinists of similar belief), their gospel-denying view in the case of the salvation of infants has much in common with Hyper-Calvinism:
“Another common Hyper-Calvinist heresy has to do with the difference between regeneration and conversion. Since regeneration is without means, they argue, then it has no temporal connection whatsoever with repentance and belief of the gospel. Thus, they believe that a person can go for a period of hours, days, weeks, or even years between being regenerated (saved, born again) and being converted (repenting, believing the gospel). They believe that a regenerate person can go for hours, days, weeks, or even years being ignorant of the only ground of salvation that is revealed in the gospel and even being openly hostile to the gospel. They say that this glorifies God’s sovereignty in salvation. They say that to believe that conversion is an immediate and inevitable fruit of regeneration is to believe that God cannot save someone without the means of the gospel, thus denying God’s sovereignty.” (http://www.outsidethecamp.org/hyperheresy.htm)
Here is what the Bible says compared to what is believed by Hyper-Calvinists AND Calvinists like Warfield, Boettner, and adherents to those Reformed Confessions that make exceptions to the Romans 1:16 and Mark 16:16 rule regarding infants:
“For I am not ashamed of the gospel of Christ, for it is [the] power of God to salvation to everyone believing, both to Jew first, and to Greek; for in it the righteousness of God is revealed from faith to faith; even as it has been written, But the just shall live by faith.” (Romans 1:16-17)
Hyper-Calvinists AND the aforementioned Calvinists do NOT believe that the gospel is the power of God to salvation to everyone believing. It is interesting how MULTITUDES of Calvinists who falsely accuse us of Hyper-Calvinism actually have much in common with their Hyper-Calvinist friends.