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Ordering comparison of zero-truncated poisson random variables with their mixtures.

1. Introduction

Zero-truncated Poisson distribution is used in several applied situations. In marketing literature, e.g., it is used to model the number of units bought by customers (Campo et al, 2003). Typically, the Poisson parameter in such models varies according to some other related factors. Thus, in fact, it is dealt with a mixture of a Poisson distribution, truncated at zero.

Mixtures of distributions are also used in building probability models in biological sciences. For instance, in order to study certain characteristics in natural populations of fish, a random sample might be taken and the characteristic measured for each member of the sample; since the characteristic varies with the age of the fish, the distribution of the characteristic in the total population will be a mixture of the distribution at different ages. For a comprehensive account on mixture of distributions as they occur in diverse fields we refer to Sapatinas (1994).

Several notations other than the means and dispersion of distributions are recently employed to stochastically compare two r.v.'s. Stochastic orderings are useful in several fields such as reliability and risk theory. Misra et al. (2003) considered Poisson and binomial distributions and compared them with their corresponding mixtures, with respect to various stochastic orderings. Motivated by Misra et al's work, Alamatsaz and Abbasi (2008) compared negative binomial distributions with their corresponding mixtures and obtained similar results. Recently, Aghababaei Jazi and Alamatsaz (2010) have studied the stochastic ordering comparison of another known member of the family of generalized power series distributions, i.e., the logarithmic series distribution, with its mixture. In this paper, we consider a zero-truncated Poisson distribution and compare it with its mixture with respect to various stochastic orderings and obtain similar results. Precisely, let X be a r.v. having zero-truncated Poisson distribution with a fixed parameter [theta] [member of] (0, [infinity]), i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

and Y be a zero-truncated Poisson r.v. with a variable parameter [theta], where [theta] is a non-degenerate continuous r.v. having a probability density function g([theta]), [theta] > 0. (Although we have assumed [theta] to be a continuous r.v., all results obtained here would also hold when [theta] is a discrete r.v.). Thus, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Our purpose is to compare r.v.'s X and Y given by (1) and (2), respectively, with respect to various types of stochastic orderings reviewed in section 2.

In section 3, we derive conditions which ensure that a mixed zero-truncated Poisson r.v., with an arbitrary mixing distribution, is larger than the corresponding zero-truncated Poisson r.v. with a fixed parameter in different senses. As a special case, the two distributions are compared when they have equal means.

2. Stochastic Orderings

Here, a brief account of some known stochastic orderings, used in this paper, is provided and their properties are stated. For more details, we refer to, e.g., Muller and Stoyan (2002) and Shaked and Shanthikumar (2007).

Let X and Y be two nonnegative discrete (or continuous) r.v.'s with respective distribution functions F(t) = P(X [less than or equal to] t) and G(t) = P(Y [less than or equal to] t), probability density functions f(t) = P(X = t) and g(t) = P(Y = t), survival functions [bar.F](t) and [bar.G](t), hazard (failure) rate functions [r.sub.X](t) = f(t)/P(X [greater than or equal to] t) and [r.sub.Y](t) = g(t)/P(Y [greater than or equal to] t) and reversed hazard rate functions [[bar.r].sub.X] (t) = f(t)/P(X [less than or equal to] t) and [[bar.r].sub.Y](t) = g(t)/ P(Y [less than or equal to] t).

2.1 Simple stochastic ordering

X is said to be smaller than Y in simple stochastic ordering, denoted by X [[less than or equal to].sub.st] Y, if G(t) [less than or equal to] F(t) for all t [greater than or equal to] 0, or, equivalently, if E([phi](X)) [less than or equal to] E([phi](Y)) for all non-decreasing functions [phi] for which the expectations exist. Thus, in particular, X [[less than or equal to].sub.st] Y implies that E(X) [less than or equal to] E(Y).

2.2 Expectation ordering

X is said to be smaller than Y in expectation ordering, denoted by X [[less than or equal to].sub.E] Y, if E(X) [less than or equal to] E(Y), where expectations are assumed to exist. Thus, by 2.1, simple stochastic ordering implies expectation ordering but the converse may not be true.

2.3 Likelihood ratio ordering

X is said to be smaller than Y in likelihood ratio ordering, denoted by X [[less than or equal to].sub.lr] Y, if f(u)g(v) [greater than or equal to] f (v)g(u), for all u [less than or equal to] v, which is equivalent to h(t) = f(t)/g(t) being non-increasing. Also X [[less than or equal to].sub.lr] Y implies that X [[less than or equal to].sub.st] Y but, in general, the converse may not be true.

2.4 Convex ordering

Y is said to be larger than X in the convex ordering, denoted by X [[less than or equal to].sub.cx] Y, if for every real-valued convex function [phi](.) defined on the real line, E([phi](X)) [less than or equal to] E([phi](Y)).

2.5 Uniformly more variable ordering

X is said to be smaller than Y in uniformly more variable ordering, denoted by X [[less than or equal to].sub.uv] Y, if X and Y have respective supports [R.sub.X] and [R.sub.Y] such that [R.sub.X] [subset or equal to] [R.sub.Y] and the ratio f(t)/g(t) is a unimodal function over [R.sub.Y] but X and Y are not ordered in simple stochastic ordering. For random variables X and Y having a same mean, it is known that X [[less than or equal to].sub.uv] Y implies that X [[less than or equal to].sub.cx] Y.

2.6 (Reversed) Hazard rate ordering

X is said to be smaller than Y in hazard rate ordering, denoted by X [[less than or equal to].sub.hr] Y, if [r.sub.Y](t) [less than or equal to] [r.sub.X](t), for all t [greater than or equal to] 0. Similarly, X is said to be smaller than Y in reversed hazard rate ordering, denoted by X [[less than or equal to].sub.rh] Y, if [[bar.r].sub.X] [less than or equal to] [[bar.r].sub.Y] (t) for all t [greater than or equal to] 0. (Reversed) hazard rate ordering implies simple stochastic ordering but the converse may not be true. Also, X [[less than or equal to].sub.lr] Y is sufficient for X [[less than or equal to].sub.hr] Y and X [[less than or equal to].sub.rh] Y.

2.7 Mean residual life ordering

X is said to be smaller than Y in mean residual life ordering, denoted by X [[less than or equal to].sub.mrl] Y, if [[integral].sup.[infinity].sub.t][bar.F](u)du/[[integral].sup.[infinity].sub.t][bar.G](u)du decreases in t, when defined. A sufficient condition for X [[less than or equal to].sub.mrl] Y is X [[less than or equal to].sub.hr] Y.

2.8 (Factorial) Moments ordering

X is said to be smaller than Y in factorial moments ordering, denoted by X [[less than or equal to].sub.fm] Y, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all i [member of] {1,2...}. Similarly, X is said to be smaller than Y in moments ordering (X [[less than or equal to].sub.moments] Y), if E([X.sup.i]) [less than or equal to] E([Y.sup.i]), for all i [member of] {1,2...}. It is known that simple stochastic ordering implies factorial moments ordering and X [[less than or equal to].sub.fm] Y implies that X [[less than or equal to].sub.moments] Y and specially X [[less than or equal to].sub.E] Y.

2.9 Moment generating function ordering

X is said to be smaller than Y in moment generating function ordering, denoted by X [[less than or equal to].sub.mgf] Y, if E([t.sup.X]) [greater than or equal to] E([t.sup.Y]) for all t [member of] (0,1). Simple stochastic ordering is sufficient for moment generating function ordering.

3. Comparison

Let X and Y be two r.v.'s with pmfs given in (1) and (2), respectively. To prove our main results, we first define:

a(k) = E([[THETA].sup.k+1]/[[e.sup.[THETA]] - 1])/E([[THETA].sup.k]/[[e.sup.[THETA]] - 1]) k = 1,2,... (3)

[d.sub.k](x) = x[(ln(1 + [1/x])).sup.k]/k!, x > 0, k = 1,2,... (4)

s(x) = x/[[e.sup.x] - 1] (5)

t(x) = x[e.sup.x]/[[e.sup.x] - 1] (6)

[[lambda].sub.0] = e([[THETA].sup.2]/[[e.sup.[THETA]] - 1])/E([THETA]/[[e.sup.[THETA]] - 1]), (7)

[[lambda].sup.*.sub.1] = ln(1 + E([e.sup.[THETA]] - 1)), (8)

[[lambda].sub.1] = [s.sup.-1](E(s([THETA))). (9)

Now, we establish the following lemma.

Lemma 3.1. Consider notations (3) to (9). Then, we have

a. a(k) is a non-decreasing function in k = 1,2,...,

b. for each fixed k = 1,2,...., [d.sub.k](x) is a concave function in x > 0,

c. s(x) and t(x) are decreasing and increasing functions in x > 0, respectively,

d. 0 < [[lambda].sup.*.sub.1] [less than or equal to] [[lambda].sub.1].

Proof

(a) We may write for a(k) in (3) that

a(k) = [[integral].sup.[infinity].sub.0][[([theta]]).sup.k+1]/[[e.sup.[theta]] - 1]]g([theta]])d[theta]/ [[integral].sup.[infinity].sub.0][[([theta]]).sup.k]/[[e.sup.[theta]] - 1]]g([theta]])d[theta]=E([Z.sub.k]), k=1,2,...

where [Z.sub.k] is a r.v. having pdf [h.sub.k](z) = [c.sub.k][[z.sup.k]/[[e.sup.z] - 1]]g(z), for all z [member of] (0, [infinity]) with [c.sub.k] as the normalizing constant. Fixing k [member of] {1,2,...}, the ratio [h.sub.k+1](z)/[h.sub.k](z)is obviously a non-increasing function in z [member of] (0, [infinity]), which implies that [Z.sub.k+1] [[greater than or equal to].sub.lr] [Z.sub.k], and thus [Z.sub.k+1] [[greater than or equal to].sub.st] [Z.sub.k]. This, in turn, implies that E([Z.sub.k+1]) [greater than or equal to] E([Z.sub.k]) or, equivalently, a(k+1) [greater than or equal to] a(k). Since k [member of] {1,2,...} was arbitrary, the assertion follows.

(b) It is sufficient to show that [d'.sub.j](x) is a decreasing function, for all x > 0. But, since ln(1 + [1/x]) is a decreasing function, we can see that

[d'.sub.j](x) = [1/j!][(ln(1 + 1/x)).sup.j] + jx(-1/[x.sup.2]) x [(1/[1 + [1/x]])(ln(1 + [1/x])).sup.j-1], j = 0,1,2,...

is a decreasing function for all x > 0.

(c) It is trivially true, because s'(x) < 0 and t'(x) > 0, for all x > 0.

(d) This follows simply because s(x) is a decreasing function and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(The following theorem provides certain restrictions on the parameter [lambda] to ensure that Y is larger than X in various orderings senses.

Theorem 3.1. Let X and Y be r.v.'s having distributions given by (1) and (2), respectively. Then, under the notations (3-9) and Lemma 3.1, we have

a. X [[less than or equal to].sub.lr] Y if, and only if, 0 < [lambda] [less than or equal to] [[lambda].sub.0] = a(1),

b. if X [[less than or equal to].sub.hr] Y then 0 < [lambda] [less than or equal to] [[lambda].sub.1]. Conversely, if 0 < [lambda] [less than or equal to] [[lambda].sup.*.sub.1], then X [[less than or equal to].sub.hr] Y [cross product],

c. X [[less than or equal to].sub.rh] Y if, and only if, 0 < [lambda] [less than or equal to] [[lambda].sub.0] = a(1),

d. if X [[less than or equal to].sub.st] Y then 0 < [lambda] [less than or equal to] [[lambda].sub.1]. Conversely, if 0 < [lambda] [less than or equal to] [[lambda].sup.*.sub.1], then X [[less than or equal to].sub.st] Y [cross product],

e. X [[less than or equal to].sub.E] ([[greater than or equal to].sub.E]) Y if, and only if, [lambda] [less than or equal to] ([greater than or equal to]) [[lambda].sub.2].

Proof.

(a) Consider the likelihood ratio

l(k) = P(Y=k)/P(X=k) = [([e.sup.[lambda]] - 1)/[[lambda].sup.k]]E[[[THETA].sup.k]/[[e.sup.[THETA]] - 1]], k = 1,2,... (10)

Then, by 2.3 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which provides a necessary and sufficient condition for the ratio in (10) to be non-decreasing. Hence, the assertion (a) follows.

(b) First, let X [[less than or equal to].sub.hr] Y. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, since s(x) is a decreasing function, we have [lambda] [less than or equal to] [[lambda].sub.1] = [s.sup.-1] (E(s([THETA]))). Conversely, let 0 < [lambda] [less than or equal to] [[lambda].sup.*.sub.1] where [[lambda].sup.*.sub.1] is as in (8). For k = 1,2,..., consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easily seen that if 0 < [lambda] [less than or equal to] [[lambda].sup.*.sub.1], X ~ TP([lambda]) and [X.sub.1] ~ TP([[lambda].sup.*.sub.1]), then X [[less than or equal to].sub.lr] [X.sub.1]. Hence, by Ross (1983), we have X [[less than or equal to].sub.hr] [X.sub.1]. Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus for all k, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, it is sufficient to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as required.

Note that, by part (d) in Lemma 3.1, we have [[lambda].sup.*.sub.1] [less than or equal to] [[lambda].sub.1]. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as required.

(c) First let X [[less than or equal to].sub.rh] Y. Then

P(X=2)/P(X[less than or equal to]2) [less than or equal to] P(Y=2)/P(Y[less than or equal to]2),

which implies that [lambda] [less than or equal to] [[lambda].sub.0]. Conversely, if 0 < [lambda] [less than or equal to] [[lambda].sub.0] then, by part (a), X [[less than or equal to].sub.lr] Y and consequently X [[less than or equal to].sub.rh] Y.

(d) First, let X [[less than or equal to].sub.st] Y. Then, P(Y[less than or equal to]1) [less than or equal to] P(X[less than or equal to]1) which implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, since s(x) is a decreasing function, we have [lambda] [less than or equal to] [[lambda].sub.1] = [s.sup.- 1](E(s([lambda]))).

Conversely, suppose that 0 < [lambda] [less than or equal to] [[lambda].sup.*.sub.1] where [[lambda].sup.*.sub.1] is as in (8). Then, by part (b) above, X [[less than or equal to].sub.hr] [X.sub.1] which results in X [[less than or equal to].sub.st] Y.

(e) By definition 2.2, since t(x) is an increasing function we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following corollary is valid by Theorem 3.1 and definitions 2.1 to 2.9.

Corollary 3.1.

a. for all 0 < [lambda] [less than or equal to] [[lambda].sub.0] = a(1), X [[less than or equal to].sub.mrl] Y,

b. for all 0 < [lambda] [less than or equal to] [[lambda].sup.*.sub.1], X [[less than or equal to].sub.fm] Y, X [[less than or equal to].sub.mgf] Y and X [[less than or equal to].sub.moments] Y,

c. for all [lambda] > [[lambda].sub.1], Y is not larger than X in simple stochastic and hence not in (reversed) hazard rate orderings,

d. [[lambda].sub.2] [greater than or equal to] [[lambda].sub.1]

Remark 3.1. Theorem 1 provides a condition under which sampling from a zero-truncated Poisson distribution is somehow less favorable than that of its mixture.

In the next theorem, we shall establish that no value of [lambda] > 0 can ensure that Y [[less than or equal to].sub.st] X. Here we also make our comparison in the uniformly more variable ordering sense.

Theorem 3.2. Consider the notations of Theorem 3.1. Then, we have

a. no value of [lambda] > 0 can ensure that Y [[less than or equal to].sub.st] X,

b. if X [[less than or equal to].sub.uv] Y, then [lambda] > [[lambda].sup.*.sub.1]. Also, for [lambda] > [[lambda].sub.1] we have X [[less than or equal to].sub.uv] Y.

Proof. (a) We have

P(X[greater than or equal to]k) = [[infinity].summation over (j=k)][lambda].sup.j]/j!([e.sup.[lambda]] - 1]) k=1,2,... (11)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

From (11), it follows that

P(X[greater than or equal to]k) = [[lambda].sup.k][[infinity].summation over (j=k)][lambda].sup.j-k]/j!([e.sup.j] - 1]) [for all]k = 1,2,... (11)

Now, since 1/j! [less than or equal to] 1/k!(j-k)! for all j = k, k + 1,... and all k = 1,2,..., we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

P(X[greater than or equal to]k) [less than or equal to] [[lambda].sup.k]/k!, [for all]k = 1,2,...

Choose [gamma] > [delta]. Then, from (12), it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus it follows that, for any [gamma] > [lambda], there exist a sufficiently large k (depending on [gamma]) such that P(X[greater than or equal to]k) < P(Y[greater than or equal to]k), which proves the assertion (a).

(b) Let X [[less than or equal to].sub.uv] Y, then P(X=k)/P(Y=k) is unimodal but X and Y are not ordered by the simple stochastic ordering. Thus, from Theorem 3.1(d) and part (a) above, it follows that [lambda] > [[lambda].sup.*.sub.1]. Conversely, suppose that [lambda] > [[lambda].sub.1], then from Theorem 3.1(d) and part (a) above, it is clear that r.v.'s Y and X are not ordered by the simple stochastic ordering. Also, from the arguments used in the proof of Theorem 3.1(a), it follows that P(X=k)/P(Y=k) is unimodal, implying that X [[less than or equal to].sub.uv] Y. []

Remark 3.2. By Theorem 3.1(a), we conclude that Y is not larger than X in likelihood ratio ordering if, and only if, [lambda] [member of] ([[lambda].sub.0],[infinity]). From Theorem 3.2, for all [lambda] [member of] (0,[infinity]), X is not larger than Y in simple stochastic and hence not in likelihood ratio ordering. Therefore, for all [lambda] [member of] ([[lambda].sub.0],[infinity]), neither X is larger than Y in likelihood ratio ordering sense, nor Y is larger than X. []

Finally, as a special case, we compare the r.v.'s X and Y in (1) and (2) when they have a same mean. This is equivalent to the case [lambda] = [[lambda].sub.2].

Theorem 3.3. Suppose that [lambda] = [[lambda].sub.2]. Then,

a. X [[less than or equal to].sub.uv] Y,

b. X [[less than or equal to].sub.cx] Y, which implies that Var(X) [less than or equal to] Var(Y),

c. P(X[greater than or equal to]2) > P(Y[greater than or equal to]2) and

d. neither X is larger than Y in simple stochastic ordering sense, nor Y is larger than X.

Proof. (a) By Theorem 3.2 (b), since [[lambda].sub.2] = [lambda] > [[lambda].sub.1], we have that X [[less than or equal to].sub.uv] Y.

(b) Since X [[less than or equal to].sub.uv] Y and E(X) = E(Y), we have X [[less than or equal to].sub.cx] Y, i.e., E ([phi](X)) [less than or equal to] E([phi](Y)) for any convex function [phi](.), such as [phi](t) = [t.sup.2]. Thus E([X.sup.2]) [less than or equal to] E([Y.sup.2]), which implies that Var(X) [less than or equal to] Var(Y).

(c) Since [[lambda].sub.2] = [lambda] > [[lambda].sub.1], and s(x) is a decreasing function (Lemma 3.1(c)), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which yields P(X=1) [less than or equal to] P(Y=1), or equivalently P(X[greater than or equal to]2) > P(Y[greater than or equal to]2).

(d) By part (a) above, we have X [[less than or equal to].sub.uv] Y. So, Y is not larger than X in simple stochastic ordering sense. Also, by Theorem 3.2(a),X is larger than Y in simple stochastic ordering sense for no value of [lambda] > 0. Therefore, when the zero-truncated Poisson random variable X has the same mean as that of its mixture Y, then there is no simple stochastic ordering between X and Y. []

Remark 3.3. Theorem 3.3 indicates that despite of lack of any simple stochastic ordering, when a zero-truncated Poisson distribution has the same mean as that of its mixture distribution, sampling from the zero-truncated Poisson distribution seems to be more favorable than its mixture. []

Example

In the sampling problem of different species, suppose that X, the number of moths per species caught in a light-trap, is a zero-truncated Poisson r.v., TP([lambda]), with pmf

P(X=k) = [[lambda].sup.k]/k!([e.sup.[lambda]] - 1), k = 1,2,...

It is interesting to specify whether it is better to take [lambda] a random variable with some mixing distribution or to consider [theta] to be a fixed value in (0, [infinity]). For simplicity of computation, assume that [LAMBDA] is an exponential random variable with unit mean. Then, obviously, Y, the number of individuals in different species, is a zero-truncated Poisson mixture r.v. with pmf

P(Y=k) = [[integral].sup.[infinity].sub.0][[lambda].sup.k]/k!([e.sup.[lambda]] - 1)][e.sup.-[theta]]d[theta], k = 1,2,...

By Theorem 3.1, X is smaller than or equal to Y in likelihood ratio (reversed hazard rate) ordering sense if, and only if, 0 < [lambda] [less than or equal to] [[lambda].sub.0] = E([[LAMBDA].sup.2]/([e.sup.[LAMBDA]] - 1))/E([LAMBDA]/([e.sup.[LAMBDA]] - 1)). Since integration methods and numerical analysis for the computation of [[lambda].sub.0] are not applicable, we have used simulation runs of size m =10000000, and calculated [[lambda].sub.0] = 0.626 (it is rounded up to 3 digits). Also, by similar method, we calculated [[lambda].sup.*.sub.1] = 0.821, [[lambda].sub.1] = 0.693 and [[lambda].sub.2] = 1.094 of (8) and Theorem 3.1. So, X is smaller than or equal to Y in simple stochastic (hazard rate) ordering sense, if [lambda] [less than or equal to] [[lambda].sub.1] = 0.693. In addition, X is smaller than or equal to Y in expectation ordering sense if, and only if, [lambda] [less than or equal to] [[lambda].sub.2] = 1.094. Also, for [lambda] > 0.626 we have X [[less than or equal to].sub.uv] Y (Theorem 3.2), specially, for [lambda] = 1.094, we conclude that E(X) = E(Y), X [[less than or equal to].sub.uv] Y, X [[less than or equal to].sub.cx] Y, Var(X) [less than or equal to] Var(Y), Pr(X[greater than or equal to]2) > Pr(Y[greater than or equal to]2) and there is no simple stochastic ordering between X and Y (Theorem 3.3). []

Acknowledgement

The second author wishes to acknowledge the University of Sistan and Baluchestan for their support.

References

[1] Aghababaei Jazi M., and Alamatsaz M.H., 2010, "Ordering comparison of logarithmic series random variables with their mixtures", Communication in Statistics: Theory and Methods, 39,1-12.

[2] Alamatsaz M.H., and Abbasi S., 2008, "Ordering comparison of negative binomial random variables with their mixtures", Statistics and Probability Letters., 78, pp. 2234-2239.

[3] Campo K., Gijsbrechts E., and Nisol P., 2003, "The impact of retailer stockouts on whether, how much, and what to buy", Marketing Research., 20, pp. 273-286.

[4] Misra N., Singh H., and Harner E.J., 2003, "Stochastic comparisons of Poisson and binomial random variables with their mixtures", Statistics and Probability Letters., 65, pp. 279-290.

[5] Muller A., and Stoyan D., 2002, "Comparison methods for stochastic models and risks", Wiley., New York.

[6] Ross S.M., 1983 "Stochastic Processes", Wiley., New York.

[7] Sapatinas T., 1995, "Identifiability of mixtures of power-series distributions and related characterizations", Ann. Inst. Statist, Math., 47, pp. 447-459.

[8] Shaked M., and Shanthikumar J.G., 2007, "Stochastic Orders", Springer, New York.

S. Abbasi (1), M. Aghababaei Jazi (2) and M.H. Alamatsaz (3)

(1) Naghshejahan Institute of Higher Education, Isfahan, Iran

(2) Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

(3) Department of Statistics, University of Isfahan, Isfahan, Iran

E-mail: alamatho@sci.ui.ac.ir
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Author:Abbasi, S.; Jazi, M. Aghababaei; Alamatsaz, M.H.
Publication:Global Journal of Pure and Applied Mathematics
Date:Dec 1, 2010
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