Copernican principle

Problem

Sampling numbers from a discrete uniform distribution where is unknown. Let be the largest number of the samples. The problem is estimating .

Solution under the Copernican principle

Basic idea

To explain the basic idea, let , then . Because , we will get .

An another example, , then .

Generalize the idea

The probability of all samples are not greater than ,

where is the largest number in , and is an arbitrary number greater than . Then,

Also, , then,

where

Examples of expectations and variances:

The wolfram code to calculate the expectation and the variance.

Function[{m, k}, N[Sum[i((k/i)^m-(k/(i+1))^m), {i, k, Infinity}], 50]][3, 100]
Function[{m, k}, N[k^m * Sum[1/i^m, {i, k, Infinity} + k - 1], 50]][6, 100]
Function[{m, k}, N[k^m * Zeta[m,k] + k - 1, 50]][9, 100]

en1[m_, k_] := Sum[i((k/i)^m-(k/(i+1))^m), {i, k, Infinity}]

ent1[m_, k_] := Sum[(i^2) * ((k/i)^m-(k/(i+1))^m), {i, k, Infinity}]

varn1[m_, k_] := ent1[m,k] - en1[m,k]^2

varn2[m_, k_] := Sum[((i - en[m, k])^2) * ((k/i)^m-(k/(i+1))^m), {i, k, Infinity}]

en[m_, k_] := k^m * Zeta[m, k] + k - 1

ent[m_, k_] := k^2 + k^m * (-Zeta[m, k+1] + 2 * Zeta[m-1, k+1])

varn[m_, k_] := ent[m,k] - en[m,k]^2

N[en[2, 100], 100]
N[en[3, 100], 100]
N[en[6, 100], 100]
N[en[9, 100], 100]

N[varn[2, 100], 100]
N[varn[3, 100], 100]
N[varn[6, 100], 100]
N[varn[9, 100], 100]