In this paper, we obtain certain expressions and recurrence relations for two general classes of distributions based on some conditional expectations of k-th lower record values. We consider the necessary and sufficient conditions such that these conditional expectations hold for some distribution functions. Furthermore, an expression of conditional expectation of other general class of distributions through truncated moments of some random variable is considered. Some distributions as examples of these general classes are shown in Tables 1 and 2 accordingly.
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Manuscript submitted on 9-05-2016 |
Original Manuscript | Some Characterization and Relations Based on K-th Lower Record Values |
Record values are found in many real lifetime datasets such as observing new records in k-th highest or k-th lowest values of weather-conditions, temperatures, water-levels, Olympic or world records in sports. In addition, record values are used in theory of reliability. Furthermore, these statistics are related to the occurrence times of some non-homogenous Poisson process used in shock models Kamps [1U. Kamps, "Reliability properties of record values from non-identically distributed random variables", Communication in Statistics, Theory and Methods, vol. 23, pp. 2101-2112, 1994.
[http://dx.doi.org/10.1080/03610929408831373] ]. Chandler [2K.N. Chandler, "The distribution and frequency of record values", Journal of the Royal Statistical Society: Series B, vol. 14, pp. 220-228, 1952.] started the statistical study of record values as a model considering dependence structure for successive extremes in a sequence of independent and identically distributed (iid) random variable. This means that, the life-length distribution of system components may change after the failure of each component. Dziubdziela and Kopocinski [3W. Dziubdziela, and B. Kopociński, "Limiting properties of the k-th record values", Applicationes Mathematicae, vol. 2, pp. 187-190, 1976.] proposed the limiting distribution of k-th record values where k is some positive integer. Many authors have considered characterization of distributions through conditional expectation of record values, for instance, Nagaraja [4H.N. Nagaraja, "Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values", sankhyaseriesa Sankhya: The Indian Journal of Statistics, Series A (1961-2002), vol. 50, pp. 70-73, 1988.], Franco and Ruiz [5M. Franco, and J.M. Ruiz, "On characterization of continuous distributions by conditional expectation of record values", sankhyaseriesa Sankhya: The Indian Journal of Statistics, Series A (1961-2002), vol. 58, pp. 135-141, 1996., 6M. Franco, and J.M. Ruiz, "On characterizations of distributions by expected values of order statistics and record values with gap", Metrika Metrika: International Journal for Theoretical and Applied Statistics, vol. 45, pp. 107-119, 1997.
[http://dx.doi.org/10.1007/BF02717097] ], Khan and Alzaid [7A.H. Khan, and A.A. Alzaid, "Characterization of distributions through linear regression of non-adjacent generalized order statistics", Journal of Applied Statistical Science, vol. 13, pp. 123-136, 2004.], Khan, Faizan, and Haque [8A.H. Khan, M. Faizan, and Z. Haque, "Characterization of continuous distributions through record statistics", Communications of the Korean Mathematical Society, vol. 25, pp. 485-489, 2010.
[http://dx.doi.org/10.4134/CKMS.2010.25.3.485] ], and Lopez-Blazques and Moreno-Rebollo [9F.L. Bláquez, and J.L.M. Rebollo, "A characterization of distributions based on linear regression of order statistics and record values", Sankhyā: The Indian Journal of Statistics, Series A, pp. 311-323, 1997.]. For more information in the theory of records and its distributional properties and some characterizations of k-th record values can be found in, for example, Ahsanullah [10M. Ahsanullah, Record statistics., Nova Science Publishers: Commack, N.Y., 1995., 11M. Ahsanullah, Record values--theory and applications., University Press of America: Dallas, 2004.], Arnold, Balakrishnan, and Nagaraja [12B.C. Arnold, N. Balakrishnan, and H.N. Nagaraja, Records., Wiley: New York, 1998.
[http://dx.doi.org/10.1002/9781118150412] ], Nevzorov [13V.B. Nevzorov, Records: Mathematical theory., American Mathematical Society: Providence, RI, 2001.], Deheuvels [14P. Deheuvels, "The characterization of distributions by order statistics and record values a unified approach", Journal of Applied Probability, vol. 21, pp. 326-334, 1984.], Nagaraja [15H.N. Nagaraja, "Record values and related statistics - a review", Communications in Statistics - Theory and Methods, vol. 17, pp. 2223-2238, 1988.
[http://dx.doi.org/10.1080/03610928808829743] ], Raqab and Awad [16M.Z. Raqab, and A.M. Awad, "Characterizations of the Pareto and related distributions", Metrika Metrika: International Journal for Theoretical and Applied Statistics, vol. 52, pp. 63-67, 2000.
[http://dx.doi.org/10.1007/s001840000061] ] and references therein.
Let {X_{i}, i ≥ 1} be a sequence of independently and identically distributed (iid) random variables with cumulative distribution function (cdf)F(x) and probability density function (pdf)f(x). For a fixed k ≥ 1, the kth lower record value of X'_{i} s is defiend by:
Note that with
are lower record values.
For
we have the following (see Ahsanullah [10M. Ahsanullah, Record statistics., Nova Science Publishers: Commack, N.Y., 1995., 11M. Ahsanullah, Record values--theory and applications., University Press of America: Dallas, 2004.], Arnold, Balakrishnan, and Nagaraja [12B.C. Arnold, N. Balakrishnan, and H.N. Nagaraja, Records., Wiley: New York, 1998.
[http://dx.doi.org/10.1002/9781118150412] ], Nevzorov [13V.B. Nevzorov, Records: Mathematical theory., American Mathematical Society: Providence, RI, 2001.]:
The pdf of Z_{r}^{(k)} and (Z_{r}^{(k)}, Z_{s}^{(k)}) are as follows:
(1) |
(2) |
We shall denote:
(3) |
(4) |
(5) |
Such that g is a continuous, monotonic and differentiable function on (α,β).
In this paper, we present three general classes of distributions whose cdf’s are:
(6) |
(7) |
(8) |
such that g is a continuous, monotonic and differentiable function on (α,β).
We extend using the cdf in (6) some work of Al-Shomrani and Shawky [17A.A. Al-Shomrani, and A.I. Shawky, "Some characterization for record values", Acta Scientiae Et Intellectus, vol. 2, pp. 49-56, 2016.] as shown in Theorem 1 by characterizing this first general form of distributions through conditional expectation of p-th power of difference of functions of two k-th lower record values. Moreover, Theorems 2.5 and 2.6 in Shawky and Abu-Zinadah [18A. Shawky, and H.H. Abu-Zinadah, "General recurrence relations and characterizations of certain distributions based on record values", Journal of Approximation Theory and Applications, vol. 2, pp. 149-159, 2006.] and Theorems 3 and 4 in Shawky and Bakoban [19A.I. Shawky, and R.A. Bakoban, "Conditional expectation of certain distributions of record values", International Journal of Mathematical Analysis, vol. 3, pp. 829-838, 2009.] are generalized as shown in Theorems 2 and 3 using the cdf in (7) by characterizing the second general class of distributions through conditional expectation of k-th lower record values. Lastly, we show that equation (2.1.1) in Hamedani, Javanshiri, Maadooliat, and Yazdani [20G.G. Hamedani, Z. Javanshiri, M. Maadooliat, and A. Yazdani, "Remarks on characterizations of Malinowska and Szynal", AMC Applied Mathematics and Computation, vol. 246, pp. 377-388, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.08.030] ] is a special case of Theorem 4 by using the cdf in (8) as the third general class of distributions based on truncated moments of some random variable. Some distributions as members of these general classes are given as examples in Tables 1 and 2.
Let X be an absolutely continuous random variable with cdf F(x) and pdf f(x) on the support (α,β), F(α) = 0 and F(β) = 1. Then, for two values of r and s, 1 ≤ r < s ≤ n (where as defined above).
(9) |
if and only if:
Where g(x) is a continuous, differentiable and non-decreasing function of x and p is a positive integer.
For proving the necessary part, from (1) and (2), we have for s≥ r+1:
(10) |
Using (6), suppose,
Hence the necessary part is proven.
For the sufficiency part, it is clear from (10) that:
(11) |
Differentiating both sides in (11) with respect to x, we get:
Using (10), we can get that:
and from (9),
which gives:
Therefore, the proof is completed.
When k = 1and c=1, which is the special case of theorem 1 above, has been established in Al-Shomrani and Shawky [17A.A. Al-Shomrani, and A.I. Shawky, "Some characterization for record values", Acta Scientiae Et Intellectus, vol. 2, pp. 49-56, 2016.].
Let X be an absolutely continuous random variables with distribution function F(x) and xϵ (α,β), F(α) = 0 and F(β) = 1, then:
if and only if:
(12) |
Where b, c and d ≠ 0 are finite constants and g(x) is a continuous, monotonic and differentiable of x on the support (α,β).
Proving the necessary part, in view of (3), we can get:
(13) |
Now, from (7), we obtain:
which gives:
(14) |
Substituting (14) into (13), we get:
Therefore, (12) is obtained.
For proving the sufficiency part, from (12) and (13), we obtain:
(15) |
Differentiating both sides in (15) with respect to x, we get:
Hence, the proof is completed.
(1) For k = 1 and d=1 in theorem 2 above, it has been obtained in Shawky and Abu-Zinadah [18A. Shawky, and H.H. Abu-Zinadah, "General recurrence relations and characterizations of certain distributions based on record values", Journal of Approximation Theory and Applications, vol. 2, pp. 149-159, 2006.].
(2) For k = 1 in theorem 2, it has been addressed in Shawky and Bakoban [19A.I. Shawky, and R.A. Bakoban, "Conditional expectation of certain distributions of record values", International Journal of Mathematical Analysis, vol. 3, pp. 829-838, 2009.].
(3) If a = 0 in (11) then .
If F(x) < 1 be any cdf of the continuous random variable X and xϵ (α,β), F(α) = 0 and F(β) = 1, then:
if and only if,
(16) |
Where b, c and d ≠ 0 are finite constants and g(x) is a continuous, monotonic and differentiable of x on the support (α,β).
For the necessary part, in view of (4), it is straightforward to get:
(17) |
Substituting (14) into (17) result in:
this completes the necessary part.
For the sufficiency part, we have from (16) and (17) that:
(18) |
From (16), we get,
(19) |
Differentiating both sides in (18) with respect to y and substituting (19), we have:
hence, the theorem is proved.
(1) Fork=1 and d=1 in theorem 3 above, this case has been addressed in Shawky and Abu-Zinadah [18A. Shawky, and H.H. Abu-Zinadah, "General recurrence relations and characterizations of certain distributions based on record values", Journal of Approximation Theory and Applications, vol. 2, pp. 149-159, 2006.].
(2) Fork=1 in theorem 3, it has been obtained in Shawky and Bakoban [19A.I. Shawky, and R.A. Bakoban, "Conditional expectation of certain distributions of record values", International Journal of Mathematical Analysis, vol. 3, pp. 829-838, 2009.].
Referring to (5) and (8), then:
(20) |
If a ≠ 0 and from (5) and using integration by parts, we get,
(21) |
Now, using (8), we have:
this implies,
(22) |
Substituting (22) into (21), we get:
Let , then
Where
Let u = , then
Therefore, the upper part of (20) is achieved.
If a = 0 and from (21) and (22) then,
Let , then
Thus, the lower part of (20) is achieved.
The RHS of equation (20) where a = 0 is the same as that of equation (2.1.1) in Hamedani, Javanshiri, Maadooliat, and Yazdani [20G.G. Hamedani, Z. Javanshiri, M. Maadooliat, and A. Yazdani, "Remarks on characterizations of Malinowska and Szynal", AMC Applied Mathematics and Computation, vol. 246, pp. 377-388, 2014.
[http://dx.doi.org/10.1016/j.amc.2014.08.030] ].
In this study some characterization results and recurrence relations of certain distributions based on the k-th lower record values for three general classes are obtained. Firstly, we characterize the first general form of distributions through conditional expectation of p-th power of difference of functions of two k-th lower record values as shown in Theorem 1. Secondly, two Theorems 2 and 3 are presented for two recurrence relations of the second general class of distributions through conditional expectation of k-th lower record values. Thirdly, we establish an expression of conditional expectation for the third general class of distributions based on truncated moments of some random variable as shown in Theorem 4. Finally, we show examples of some distributions related to these general classes as in Tables 1 and 2.
The author confirms that this article content has no conflict of interest.
The author thanks the anonymous referees for their valuable suggestions which let to the improvement of the manuscript.
[1] | U. Kamps, "Reliability properties of record values from non-identically distributed random variables", Communication in Statistics, Theory and Methods, vol. 23, pp. 2101-2112, 1994. [http://dx.doi.org/10.1080/03610929408831373] |
[2] | K.N. Chandler, "The distribution and frequency of record values", Journal of the Royal Statistical Society: Series B, vol. 14, pp. 220-228, 1952. |
[3] | W. Dziubdziela, and B. Kopociński, "Limiting properties of the k-th record values", Applicationes Mathematicae, vol. 2, pp. 187-190, 1976. |
[4] | H.N. Nagaraja, "Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values", sankhyaseriesa Sankhya: The Indian Journal of Statistics, Series A (1961-2002), vol. 50, pp. 70-73, 1988. |
[5] | M. Franco, and J.M. Ruiz, "On characterization of continuous distributions by conditional expectation of record values", sankhyaseriesa Sankhya: The Indian Journal of Statistics, Series A (1961-2002), vol. 58, pp. 135-141, 1996. |
[6] | M. Franco, and J.M. Ruiz, "On characterizations of distributions by expected values of order statistics and record values with gap", Metrika Metrika: International Journal for Theoretical and Applied Statistics, vol. 45, pp. 107-119, 1997. [http://dx.doi.org/10.1007/BF02717097] |
[7] | A.H. Khan, and A.A. Alzaid, "Characterization of distributions through linear regression of non-adjacent generalized order statistics", Journal of Applied Statistical Science, vol. 13, pp. 123-136, 2004. |
[8] | A.H. Khan, M. Faizan, and Z. Haque, "Characterization of continuous distributions through record statistics", Communications of the Korean Mathematical Society, vol. 25, pp. 485-489, 2010. [http://dx.doi.org/10.4134/CKMS.2010.25.3.485] |
[9] | F.L. Bláquez, and J.L.M. Rebollo, "A characterization of distributions based on linear regression of order statistics and record values", Sankhyā: The Indian Journal of Statistics, Series A, pp. 311-323, 1997. |
[10] | M. Ahsanullah, Record statistics., Nova Science Publishers: Commack, N.Y., 1995. |
[11] | M. Ahsanullah, Record values--theory and applications., University Press of America: Dallas, 2004. |
[12] | B.C. Arnold, N. Balakrishnan, and H.N. Nagaraja, Records., Wiley: New York, 1998. [http://dx.doi.org/10.1002/9781118150412] |
[13] | V.B. Nevzorov, Records: Mathematical theory., American Mathematical Society: Providence, RI, 2001. |
[14] | P. Deheuvels, "The characterization of distributions by order statistics and record values a unified approach", Journal of Applied Probability, vol. 21, pp. 326-334, 1984. |
[15] | H.N. Nagaraja, "Record values and related statistics - a review", Communications in Statistics - Theory and Methods, vol. 17, pp. 2223-2238, 1988. [http://dx.doi.org/10.1080/03610928808829743] |
[16] | M.Z. Raqab, and A.M. Awad, "Characterizations of the Pareto and related distributions", Metrika Metrika: International Journal for Theoretical and Applied Statistics, vol. 52, pp. 63-67, 2000. [http://dx.doi.org/10.1007/s001840000061] |
[17] | A.A. Al-Shomrani, and A.I. Shawky, "Some characterization for record values", Acta Scientiae Et Intellectus, vol. 2, pp. 49-56, 2016. |
[18] | A. Shawky, and H.H. Abu-Zinadah, "General recurrence relations and characterizations of certain distributions based on record values", Journal of Approximation Theory and Applications, vol. 2, pp. 149-159, 2006. |
[19] | A.I. Shawky, and R.A. Bakoban, "Conditional expectation of certain distributions of record values", International Journal of Mathematical Analysis, vol. 3, pp. 829-838, 2009. |
[20] | G.G. Hamedani, Z. Javanshiri, M. Maadooliat, and A. Yazdani, "Remarks on characterizations of Malinowska and Szynal", AMC Applied Mathematics and Computation, vol. 246, pp. 377-388, 2014. [http://dx.doi.org/10.1016/j.amc.2014.08.030] |