Bivariate Quantile Functions and their Applications to Reliability Modelling


  • Balakrishnapillai Vineshkumar Government Arts College
  • Narayanan Unnikrishnan Nair Cochin University of Science and Technology



Bivariate quantile functions, Hazard and mean residual quantile functions, Bivariate linear hazard (mean residual) quantile function distribution


In this paper we propose a new definition of bivariate quantile function suited for reliability modelling and illustrate its applications. The bivariate hazard and mean residual quantile functions are defined and their properties are studied. Examples of generating new quantile functions and application of the results to model data are provided.


K. ADAMIDIS, S. LOUKAS (1998). A lifetime distribution with decreasing failure rate. Statistics and Probability Letters, 39, pp. 35–42.

F. BELZUNCE, A. CASTANO, A. OLVERA-CERVANTES, A. SUAREZ-LLORENS (2007). Quantile curves and dependence structure of bivariate distributions. Computational Statistics and Data Analysis, 51, pp. 5112–5129.

Y. CAI (2010). Multivariate quantile function models. Statistica Sinica, 20, pp. 481–496.

L. A. CHEN, A. H.WELSH (2002). Distribution function-based bivariate quantiles. Journal of Multivariate Analysis, 83, pp. 208–231.

A. M. FRANCO-PEREIRA, M. SHAKED, R. E. LILLO (2012). The decreasing percentile residual life ageing notion. Statistics, 46, pp. 587–603.

W. G. GILCHRIST (2000). Statistical Modelling with Quantile Functions. Chapman and Hall/CRC Press, Boca Raton.

J. R. M.HOSKING, J. R.WALLIS (1997). Regional Frequency Analysis: An approach based on L-moments. Cambridge University Press, Cambridge.

T. HU, B. E. KHALEDI, M. SHAKED (2003). Multivariate hazard rate ordering. Journal of Multivariate Analysis, 84, pp. 173–189.

N. L. JOHNSON, S. KOTZ (1975). A vector valued multivariate hazard rate. Journal of Multivariate Analysis, 5, pp. 53–66.

S. KAYAL, M. TRIPATHY (2018). A quantile-based tsallis-alpha divergence. Physica A: Statistical Mechanics and its Applications, 492, no. 15, pp. 496–505.

J. P. KLEIN, M. L. MOESCHBERGER (1997). Survival Analysis Techniques for Censored and Truncated Data. Springer, New York.

V. KUMAR, R. RANI (2018). Inference on quantile residual life function under right censored data. Statistica, 78, pp. 105–126.

C. LIN, L. ZHANG, Y. ZHOU (2016). Quantile approach of dynamic generalized entropy (divergence) measure. Journal of Nonparametric Statistics, 28, pp. 617–643.

N. N. MIDHU, P. G. SANKARAN, N. U. NAIR (2013). A class of distributions with linear mean residual quantile function and its generalizations. Statistical Methodology, 15, pp. 1–24.

N. N. MIDHU, P. G. SANKARAN, N. U.NAIR (2014). A class of distributions with linear hazard quantile function. Communications in Statistics: Theory and Methods, 43, pp. 3674–3689.

N. U. NAIR, P. G. SANKARAN (2009). Quantile-based reliability analysis. Communications in Statistics: Theory and Methods, 38, pp. 222–232.

N. U. NAIR, P. G. SANKARAN, N. BALAKRISHNAN (2013). Quantile-based Reliability Analysis. Springer- Science, New York.

N. U.NAIR, B. VINESHKUMAR (2010). L-moments of residual life. Journal of Statistical Planning and Inference, 140, pp. 2618–2631.

N. U. NAIR, B. VINESHKUMAR (2011). Ageing concepts: An approach based on quantile functions. Statistics and Probability Letters, 81, pp. 2016–2025.

F. SADEGHI, F. YOUSEFZADEH, M. CHAHKANDI (2019). Some new stochastic orders based on quantile function. Communications in Statistics- Theory and Methods, 48, no. 2, pp. 942–953.

R. SERFLING (2002). Quantile functions for multivariate analysis: Approaches and applications. Statistica Neerlandica, 56, pp. 214–232.

M. SHAKED, J. G. SHANTHIKUMAR (2007). Stochastic Orders. Springer, New York.

P. SONI, I. DEWAN (2012). Nonparametric estimation of quantile density function. Computational Statistics and Data analysis, 56, pp. 3876–3886.

B. VINESHKUMAR, N. U. NAIR, P. G. SANKARAN (2015). Stochastic orders using quantile-based reliability functions. Journal of the Korean Statistical Society, 44, pp. 221–231.




How to Cite

Vineshkumar, B., & Nair, N. U. (2019). Bivariate Quantile Functions and their Applications to Reliability Modelling. Statistica, 79(1), 3–21.