Testing a Pseudo-Random Number Generator Algorithm

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I created a pseudo-random number generator that creates random bits from given numbers. For better visualization, suppose that we have inputs "a", "b", "ab", "abc" etc. For all these inputs, algorithm generates random 16 bits.

"a"   =>  0101101110101011
"b"   =>  1010110110101110
"ab"  =>  0001010111001101
"abc" =>  0101101011011101

(The inputs and outputs above are not quotes from the original algorithm. I don't want you to test these. I just shared it to make it come alive in your mind.)

I'm not sure if it's completely secure and reliable. What are common tests for such algorithms? I need to understand how to test my generator properly to make sure it's safe and works well. Can anyone suggest what tests I should use? Any help would be great, thanks!