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We consider a semidirect product of two locally compact groups S and T, with S Abelian, denoted by SσT. An action of SσT on S is introduced to make S a homogeneous space of SσT. Then we define a unitary representation from SσT into the unitary group of L2(S) which is our main tool for defining the continuous wavelet transform on L2(S). Also the main properties of the transform are discussed. We prove the Plancherel and inversion formulas and reproducing kernel’s formula for this transform. This is finally specialized to the case of the continuous wavelet transform on L2(Rd).

In the present work, some characterization results are established based on the number of observations near the order statistics. Under some conditions, it is shown that the parent distribution can be uniquely determined by the moments of the number of observations in a random sample that fall within a left-hand or right-hand neighborhood of a specific order statistic. It is proved that the underlying distribution F belongs to the class of symmetric distributions if and only if the first moment of the number of observations in the right neighborhood of the kth order statistic and in the left neighborhood of the(n − k + 1)th order statistic from a sample of size n are equal. Also, characterizations of the exponential distribution are presented.

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