Noninformative priors for the ratio of variabilities in a bivariate normal population

In this paper, we consider some objective priors for the ratio of variabilities in a bivariate normal distribution. We develop the first and second order matching priors and reference priors. We obtain that the second order matching prior matches the alternative coverage probabilities up to the same order. It is also an HPD matching priors. It turns out that the derived reference priors do not satisfy a second order matching criterion. The simulation result shows that the second order matching prior performs better than reference priors based on matching the target coverage probabilities in a frequentist sense. Finally, we show that the second order matching prior and reference priors produce confidence sets with an expected length shorter than the Cox and Reid adjustment.

[1]  Lawrence D. Brown,et al.  INADMISSIBILITY OF THE USUAL ESTIMATORS OF SCALE PARAMETERS IN PROBLEMS WITH UNKNOWN LOCATION AND SCALE PARAMETERS , 1968 .

[2]  D. Dey,et al.  Frequentist validity of posterior quantiles in the presence of a nuisance parameter : higher order asymptotics , 1993 .

[3]  Jayanta K. Ghosh,et al.  On priors providing frequentist validity for Bayesian inference , 1995 .

[4]  On confidence intervals associated with the usual and adjusted likelihoods , 1999 .

[5]  R. Tibshirani.  Noninformative priors for one parameter of many , 1989 .

[6]  Dongchu Sun,et al.  A matching prior based on the modified profile likelihood in a generalized Weibull stress‐strength model , 2013 .

[7]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[8]  A. Gelfand,et al.  On the Estimation of a Variance Ratio. , 1988 .

[9]  G. Iliopoulos.  Decision Theoretic Estimation of the Ratio of Variances in a Bivariate Normal Distribution , 2001 .

[10]  Rahul Mukerjee,et al.  On a property of probability matching priors: matching the alternative coverage probabilities , 1999 .

[11]  B. L. Welch,et al.  On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods , 1963 .

[12]  Laura Ventura,et al.  Prior Distributions From Pseudo-Likelihoods in the Presence of Nuisance Parameters , 2009 .

[13]  Weizhen Wang.  On equivalence of two variances of a bivariate normal vector , 1999 .

[14]  S. Kundu,et al.  DECISION THEORETIC ESTIMATION OF THE VARIANCE RATIO , 1996 .

[15]  W. A. Morgan.  TEST FOR THE SIGNIFICANCE OF THE DIFFERENCE BETWEEN THE TWO VARIANCES IN A SAMPLE FROM A NORMAL BIVARIATE POPULATION , 1939 .

[16]  Malay Ghosh,et al.  ON THE INVARIANCE OF NONINFORMATIVE PRIORS , 1996 .

[17]  M. Madi.  On the invariant estimation of a normal variance ratio , 1995 .

[18]  Thomas J. DiCiccio,et al.  Frequentist and Bayesian Bartlett Correction of Test Statistics Based on Adjusted Profile Likelihoods , 1994 .

[19]  On expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods , 2001 .

[20]  J. Ghosh,et al.  FREQUENTIST VALIDITY OF HIGHEST POSTERIOR DENSITY REGIONS IN THE PRESENCE OF NUISANCE PARAMETERS , 1995 .

[21]  Charles Stein,et al.  On the coverage probability of confidence sets based on a prior distribution , 1985 .

[22]  G. Datta.  On priors providing frequentist validity of Bayesian inference for multiple parametric functions , 1996 .

[23]  R. Mukerjee,et al.  Probability Matching Priors: Higher Order Asymptotics , 2004 .

[24]  S. Kang,et al.  Noninformative priors for the between-group variance in the unbalanced one-way random effects model with heterogeneous error variances , 2019, Journal of Statistical Computation and Simulation.

[25]  C. Stein.  Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean , 1964 .

[26]  M. Ghosh,et al.  Second-order probability matching priors , 1997 .

[27]  B. M. Hill,et al.  Bayesian Inference in Statistical Analysis , 1974 .

[28]  S. Kang,et al.  A matching prior based on the modified profile likelihood for the common mean in multiple log-normal distributions , 2020 .

[29]  James O. Berger,et al.  Estimating a Product of Means: Bayesian Analysis with Reference Priors , 1989 .

[30]  M. Ghosh,et al.  Some new Results on Probability Matching Priors , 2000 .

[31]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[32]  Malay Ghosh,et al.  Some remarks on noninformative priors , 1995 .