# Test of hypotheses for linear regression models with non-sample prior information

PhD Thesis

Pratikno, Budi. 2012. Test of hypotheses for linear regression models with non-sample prior information. PhD Thesis Doctor of Philosophy. University of Southern Queensland.
Title Test of hypotheses for linear regression models withnon-sample prior information PhD Thesis Pratikno, Budi Khan, Shahjahan Li, Yan University of Southern Queensland Doctor of Philosophy 207 2012 Classical inferences about population parameters are usually drawn from the sample data alone. This applies to methods used in parameter estimation and hypothesistesting. Inferences about population parameters could be improved using non-sample prior information (NSPI) on the value of another related parameter. However, any NSPI on the value of any parameter is likely to be uncertain (or unsure). The NSPI can be classified as (i) unknown (unspecified), (ii) known (specified), and (iii) uncertainif the suspected value is unsure. For the three different scenarios, three different statistical tests: (i) unrestricted test (UT), (ii) restricted test (RT) and (iii) preliminary test test (PTT) are defined. The current research is to test the intercept parameter(s) when NSPI is available on the slope parameter(s). The test statistics,their sampling distributions, and power functions of the tests are derived. Comparison of power functions of the tests are used to recommend a best test. In this thesis, we test (1) the intercept of the simple regression model (SRM) when there is NSPI on the slope, (2) the intercept vector of the multivariate simple regression model (MSRM) when there is NSPI on the slope vector, (3) a subset of regression parameters of the multiple regression model (MRM) when NSPI is available on another subset of theregression parameters, and (4) the equality of the intercepts for p (≥ 2) lines of the parallel regression model (PRM) when there is NSPI on the slopes.For each of the above four regression models, the following steps are carried out: (1) derived the test statistics of the UT, RT and PTT for both known and unknown variance, (2) derived the sampling distributions of the test statistics of the UT, RT and PTT, (3) derived and compared the power function and the size of the UT, RT and PTT. For known variance, under a sequence of an alternative hypothesis, the sampling distributions of the UT and RT of the simple regression model follows a normal distribution. However, the PTT follows a bivariate normal distribution. For unknownvariance, the sampling distribution of the UT and RT of the simple regression model follows a Student’s t distribution but the PTT follows a correlated bivariate Student’st distribution. For the multivariate simple regression, multiple regression and parallel regression models, the sampling distribution of the UT and RT follows a univariatenoncentral F distribution under the alternative hypothesis. However, the PTT follows a correlated bivariate noncentral F distribution. For the four regression models above, there is a correlation between the UT and PT but there is no such correlation between the RT and PT. To evaluate the power function of the PTT the probability integral of the bivariate normal, bivariate Student’s t and bivariate noncentral F distributions are used. For the computations of the power function of the PTT of the MSRM, MRM and PRM require the cumulative distribution function (cdf) of a correlated bivariate noncentral F (BNCF) distribution. But the correlated BNCF distribution is not available in the literature, and hence we derive the probability density function (pdf) and cdf of the BNCF distribution. The R package is used for all computations and graphical analyses.The statistical criteria that are used to compare the performance of the UT, RT and PTT are the size and power of the tests. A test that minimizes the size and maximizes the power is preferred over any other tests. In reality, the size of a test is fixed, and then the choice of the best test is based on its maximum power. The study shows that the power of the RT is always higher than that of the UTand PTT, and the power of the PTT lies between the power of the RT and UT. The size of the UT is smaller than that of the RT and PTT. Among the three tests, the UT has the lowest power and lowest size. In terms of power it is the worst and in terms of size it is the best. The RT has maximum power and size. The PTT has smaller size than the RT and the RT has larger power than the PTT. The PTT protects against maximum size of the RT and minimum power of the UT. Thus the the PTT attains a reasonable dominance over the UT and RT for all regression models when the suspected value of the slope parameter(s) suggested by the NSPI is not too far away from that under the null hypothesis. simple regression model; sampling distributions; slope parameters; linear regression; non-sample prior information 461399. Theory of computation not elsewhere classified 490399. Numerical and computational mathematics not elsewhere classified Faculty of Sciences
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## Related outputs

##### The power of the tests on the multiple regression model
Pratikno, Budi and Khan, Shahjahan. 2020. "The power of the tests on the multiple regression model." International Journal of Advanced Science and Technology. 29 (6), pp. 3274-3279.
##### Tests on a subset of regression parameters for factorial experimental data with uncertain higher order interactions
Khan, Shahjahan, Pratikno, Budi, Rahman, Shafiqur and Hossain, M. Zakir. 2019. "Tests on a subset of regression parameters for factorial experimental data with uncertain higher order interactions." Journal of Statistical Theory and Applications. 18 (2), pp. 103-112. https://doi.org/10.2991/jsta.d.190514.001
##### Linear model inference with non-sample prior information
Khan, Shahjahan, Pratikno, Budi and Yunus, Rossita M.. 2012. "Linear model inference with non-sample prior information." Ahmad, Munir (ed.) 12th Islamic Countries Conference on Statistical Sciences . Doha, Qatar 19 - 22 Dec 2012 Lahore, Pakistan.
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##### The correlated bivariate noncentral F distribution and its application
Khan, Shahjahan, Pratikno, Budi, Ibrahim, Adriana I. N. and Yunus, Rossita M.. 2016. "The correlated bivariate noncentral F distribution and its application." Communications in Statistics: Simulation and Computation. 45 (10), pp. 3491-3507. https://doi.org/10.1080/03610918.2014.999085
##### Improving statistical inference with uncertain non-sample prior information
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##### Testing equality of two intercepts for the parallel regression model with non-sample prior information
Pratikno, Budi and Khan, Shahjahan. 2017. "Testing equality of two intercepts for the parallel regression model with non-sample prior information." Communications in Statistics: Theory and Methods. 46 (6), pp. 3099-3110. https://doi.org/10.1080/03610926.2015.1054941
##### Testing of intercept when slope is under suspicion
Pratikno, Budi and Khan, Shahjahan. 2012. "Testing of intercept when slope is under suspicion." Ahmad, Munir (ed.) 11th Islamic Countries Conference on Statistical Sciences. Lahore, Pakistan 19 - 22 Dec 2011 Lahore, Pakistan.
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