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On the testing hypothesis in uniform family of distributions with nuisance parameter

  • Abbas Eftekharian EMAIL logo and Morad Alizadeh
Published/Copyright: October 4, 2021
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 5

Abstract

The problem of finding optimal tests in the family of uniform distributions is investigated. The general forms of the uniformly most powerful and generalized likelihood ratio tests are derived. Moreover, the problem of finding the uniformly most powerful unbiased test for testing two-sided hypothesis in the presence of nuisance parameter is investigated, and it is shown that such a test is equivalent to the generalized likelihood ratio test for the same problem. The simulation study is performed to evaluate the performance of power function of the tests.

Keywords: Generalized likelihood ratio test; nuisance parameter; one-sided hypothesis; two-sided hypothesis; uniform distribution; uniformly most powerful test
MSC 2010: 62F03

Acknowledgement

The authors express their sincere thanks to the anonymous referees and the associate editor for their useful comments and constructive criticisms on the original version of this manuscript, which led to this considerably improved version.

  1. (Communicated by Gejza Wimmer)

References

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Appendix

Proof of the pdf of Z in Theorem 2.1, i.e., equation (2.7)

Let X1, …, Xn ~ U(a(θ1, b(θ2))). It is clear that Yi=Xi−a(θ1)b(θ2)−a(θ1)∼U(0,1) for any i = 1, …, n. Then, the problem of finding the pdf of Z=X(1)−a(θ1)X(n)−a(θ1) is equivalent to find the pdf of Z=Y(1)Y(n). So, it is enough to obtain the pdf of Z=Y(1)Y(n). To do this, we set W = Y(n) and noting that the Jacobian of transformations z and w is

J = z 1 w 0 = w ,

therefore, the joint pdf of (Z, W) is given by

f Z , W ( z , w ) = w f Y ( 1 ) , Y ( n ) ( z w , w ) = w × n ( n − 1 ) f Y ( z w ) f Y ( z w ) [ F Y ( w ) − F Y ( z w ) ] n − 2 = n ( n − 1 ) w n − 1 ( 1 − z ) n − 2 , 0 < z < w < 1.

Hence, the pdf of Z can be obtained by integrating from the joint pdf of (Z, W) with respect to w as follows

f Z ( z ) = ∫ 0 1 f Z , W ( z , w ) d w = ( n − 1 ) ( 1 − z ) n − 2 , 0 < z < 1.

Received: 2020-03-11
Accepted: 2021-01-11
Published Online: 2021-10-04
Published in Print: 2021-10-26

© 2021 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2021-0054
Keywords for this article
Generalized likelihood ratio test; nuisance parameter; one-sided hypothesis; two-sided hypothesis; uniform distribution; uniformly most powerful test

Articles in the same Issue

  1. Regular papers
  2. Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
  3. Polynomial functions on rings of dual numbers over residue class rings of the integers
  4. Sufficient conditions for p-valent functions
  5. Upper bounds for analytic summand functions and related inequalities
  6. Global structure for a fourth-order boundary value problem with sign-changing weight
  7. On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
  8. Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
  9. Properties of critical and subcritical second order self-adjoint linear equations
  10. Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
  11. Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
  12. Poisson like matrix operator and its application in p-summable space
  13. On the homological and algebraical properties of some Feichtinger algebras
  14. Disjoint topological transitivity for weighted translations generated by group actions
  15. On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
  16. Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
  17. The shifted Gompertz-G family of distributions: Properties and applications
  18. On the testing hypothesis in uniform family of distributions with nuisance parameter
  19. Clarkson inequalities related to convex and concave functions
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