Journal of Prime Research in Mathematics

# Probability of the moderate deviations for the sum-functions of spacing

Sherzod Mira
GIK Institute of Engineering Sciences and Technology, Topi-23460, N.W.F.P., Pakistan.
Zam Mirakhmedov
GIK Institute of Engineering Sciences and Technology, Topi-23460, N.W.F.P., Pakistan.
Syed Ikram Abbas Tirmizi
GIK Institute of Engineering Sciences and Technology, Topi-23460, N.W.F.P., Pakistan

$$^{1}$$Corresponding Author: shmirakhmedov@yahoo.com,

### Abstract

Let $$0=U_{0,n}\leq U_{1,n}\leq … \leq U_{n,n}=1$$ be an ordered sample from uniform [0,1] distribution $$D_{in}=U_{i,n}-U_{i-1,n}$$, $$i=1,2,..,n$$, $$n=1,2,…$$ , be their spacings,and let $$f_{1n},…,f_{nn}$$ be a set of measurable functions. In this paper theorems on the probabilities of deviations in the moderate zones for $$R_{n}D=f_{1n}(nD_{1n}, … , f_{nn}(nD_{nn}$$ are presented. Application of these results to study an intermediate efficiencies of the tests based on statistic $$R_{n}(D)$$ are also considered.

#### Keywords:

Spacings, uniform distribution, large deviations,goodness-offit,asymptotic efficiencies