Parameter Estimation Of Poisson Distribution Under Q-Symmetric Entropy Loss Function

The poisson distribution is a widely applied discrete distribution and can serve as a model for a number of different types of experiments. For example ,if we are modeling a phenomenon in which we are waiting for an occurrence (such as waiting for a bus .waiting for customers to arrive in a bank ), the number of occurrences in agiven time interval can sometimes be modeled by the Poisson distribution.Theorem 2.1 Assume X = {X₁,X₂, … ,X_n),under the los function L(λ,δ) = ,for an prior distribution ,the Bayes estimation of the parameter A is δ_B(X) = [E(λ^q|X)/E{λ^-q|X)]^1/(2q), and if exists δ' ,its Bayes risk satisfies R(δ')) <∞ .then the Bayes estimation is unique.If the prior distribution of the parameter A is Gmma distribution ,which is conjugate distribution of Poisson distribution ,its parameter is α,β.Then the Bayes estimation of the parameterλis its Bayes risk satisfies R(δ') <∞ .Then the Bayes estimation is unique.Basing on the above discuss, we know that the Bayes estimation has the form under the Gmma distribution. The admissibility and inadmissi-bility of estimator with the form of is related to c and d ,we discuss it when c and d hace different value and we assume thatc* = n^-1,where α ≥ q + 1.Theorem 3.1 when 0 < c < c*,the estimator with the form of is admissibility.Lemma 3.1 Under the loss function with the form of if E|δ|^q < +∞,, when the risk functioni R(λA, δ) is continuous about the parameter λ.Theorem 3.2 whenc = c*, the estimator with the form of c\[$ is admissibility.Theorem 3.3 The estimator with the form of cfil^L^T + d- k)}% is inad-missibility if c and d satisfy one of the following condition ,(l)c > c*, (2)c < 0.Parameter Estimation of Binomial Distribution Under Q-Symmetric Entropy Loss FunctionBinomial distribution ,one of the more useful discrete distribution ,is base on the idea of a Bernoulli trial.For a Binomial population of its density is p(Jfc) = c*0fc(l — 0)n^k, (A = 1,2, ? ? ? ,n,0 < 0 < 1), we give the estimation about the unknown parameter 0,and loss function we study also is the q- symmetric entropy loss function.If the prior distribution of the parameter 6 is Beta distribution ,which is conjugate distribution of Bmomial distribution ,its parameter is a, fr.Then the Bayes estimation of the parameter** SB(X) = rgffBT ??â–  "^Sg1]*, Iff- (E^ + fl + a- l)±,ta Bayes risk satisfies/i((S/) < oo .Then the Bayes estimation is unique.Basing on the above discuss, we know that the Bayes estimation has the form ofcfnjfcLiCT1 + d — fc)]3?, under the Beta distribution. The admissibility and inadmiasi-bility of estimator with the form ofcffljjlj (T + d — k)\ * is related to c and d ,we discuss it when c and d hace different value and we assume thatc* = [n8_x +o ? ? ? a-H|81 ₁]^, wherel,n2 > q.Theorem 3.1 when 0 < c < c*,the estimator with the form of c^^{T + d-fis admissibility.Lemma 3.1 Under the loss function with the form of (f)? + ($)? - 2, {q > 0), if E\6\g < +00, ï¿¡m? < +00, when the risk function.R(0, S) is continuous about the parameter 0.Theorem 3.2 whenc = c*, the estimator with the form of cffj^j (T+d— k)\ T< is admissibility.Theorem 3.3 The estimator with the form of dï¿¡\%Lx{T + d - k)}% is inad-missibility if c and d satisfy one of the following condition ,(l)c > c*, (2)c < 0.Parameter Estimation of Geometric Distribution Under Q-Symmetric Entropy Loss FunctionThe Geometric distribution is the simplest of the waiting time distributions and is a special case of the negative binomial diatribution .For a Geometric population of its density is f(x) = 0(1 - 0)1"1, (0 < 6 < l,x = 1,2, ? ? ? ),the parameter 6 is unknown, here .we give the estimation under q- symmetric entropy loss function.If the prior distribution of the parameter $ is Beta distribution ,which is conjugate distribution of Geometric distribution ,\ts parameter is a, b.Then the Bayes estimation of the parameter^ 6B(X) =Bayes risk satisfiesi2(ï¿¡') < oc .Then the Bayes estimation is unique.Basing on the above discuss, we know that the Bayes estimation has the form ofc[Yl^=1(T + d — fc)]"5* t under the Beta distribution. The admissibility and inadmissibU-ity of estimator with the form ofc[fl^.1(r + d — k)]^*i is related to c and d ,we discuss it when c and d hace different value and we assume thatc* = [ni*+a ''' a+n^+q-il ' wnere a > q+ l,n2 > q.Theorem 3.1 when 0 < c < c*,the estimator with the form of cjlitli(T + d-k)} 2? is admissibility.Lemma 3.1 Under the loss function with the form of (f )9 + (|)' - 2, (q > 0), if E\6\q < +oo, Erh < +cc, when the risk function.R(0,8) is continuous about the parametere.Theorem 3.2 whenc = c*, the estimator with the form of c[[\lq=1(T+d-k)]^{^}is admissibility.Theorem 3.3 The estimator with the form of dJJ^L^T + d- k)]^& is inad-missibility if c and d satisfy one of the following condition ,(l)c > c*, (2)c < 0.