Abstract
We consider the extreme value problem of the minimum-maximum models for the independent and identically distributed random sequence and stationary random sequence, respectively. By invoking some probability formulas and Taylor’s expansions of the distribution functions, the limiting distributions for these two kinds of sequences are obtained. Moreover, convergence analysis is carried out for those extreme value distributions. Several numerical experiments are conducted to validate our theoretical results.
Highlights
Consider a collection of random variables { Xij } for i = 1, . . . , n, j = 1, . . . , m following a common distribution function F
We present some theoretical results of limiting distributions for the minimum-maximum model (1), which probably give some insights on application
We focus on the problem of obtaining the limiting distributions for the minimum-maximum model (1), as well as the convergence rate of P( Mn ≤ an x + bn ) to its extreme value limit
Summary
We focus on the problem of obtaining the limiting distributions for the minimum-maximum model (1), as well as the convergence rate of P( Mn ≤ an x + bn ) to its extreme value limit. Motivated by [1,4,5], we first provide the methods for selecting the normalized constants an , bn Combining their properties with Taylor’s expansions of the distribution functions, we obtain the limiting distributions for i.i.d. and stationary random sequence, respectively. The following theorem implies that the limiting distribution of (4) with i.i.d. random sequence belongs to the Gumbel class. Suppose { Xij , 1 ≤ i ≤ n, 1 ≤ j ≤ m} are i.i.d. random variables with common distribution function F having a continuous and bounded first derivative F 0 , there exist some normalized constants.
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