Empirical Likelihood Inference For Quantile Differences And Means With Missing Data

In practical applications, we sometimes need to compare the quantile differences of two nonparametric populations and make hypothesis testing on means of populations. However, due to all kinds of reasons, we often obtain samples with missing data (incomplete sample) in reality. In this situation, the usual inference procedures cannot be applied directly, so a common method to handle samples with missing data is to impute a value for each missing value and then obtain a'complete sample'. After that, we use usual statistical methods to make inference based on the'complete sample'. There are two chapters in this paper, empirical likelihood confidence intervals of two populations quantile differences and one populations mean are discussed, respectively.First, for the situation of two populations quantile differences, considering the following simple random sample with missing data:( x_i ,δ_xi), i = 1, ,m and( y_j ,δ_yj), j = 1,…,n.We assume that x , y are missing completely at random(MCAR),namely, P (δ_x= 1| x )= P₁(constant) and P (δ_y= 1| y )= P₂(constant). In this paper, we also assume that ( x ,δ_x)and ( y ,δ_y)are independent, we gives empirical likelihood confidence intervals for the differences of two populations quantile differences. Assuming x₁ ,…, x_m i. i . d ., distribution function is F ( x ); Assuming y₁ ,…, y_ni. i . d ., distribution function is G ( y ) , F ( x ) and G ( y ) are unknown . For fixed q , 0 < q< 1 , We assume the q - thquantile of F and G is unique, and letθis q - thquantile of F andθ+Δisq - thquantile of G , that is,θ= F ^-1( q)andθ₀ +Δ= G^-1( q). We obtain'complete sample'by hot deck random imputation and empirical likelihood ratio statistic -2 R (Δ,θ_mn)'s asymptotical distribution, which givesΔ's empirical likelihood confidence intervals, which has better precision according to data imitation.Usingθ₀ to denote the true value ofθ, we make some assumptions as follows:(1).θ₀∈ΩandΩis an open interval.(2).Denote, for some integer t₀≥2, suppose that f^(t₀-1) (t )exists and is continuous in a neighborhood ofθ₀, and that g^(t₀-1) (t )exists and is continuous in a neighborhood ofθ₀+Δ, and also f(θ₀) g(θ₀+Δ) > 0.(4).For i = 1,2, K i are bounded and satisfy Lipschitz condition of order 1; K_i² exist and are bounded. Assuming that for some C > 0,(5).There exists We obtain the following empirical likelihood equationλ_j (θ), j= 1,2 are determined by the following two equations:Theorem 1.1 Suppose that assumptions(1)-(5)are satisfied, then there exists a rootθmnof above-mentioned empirical likelihood equation at probability of 1,θ_mn is strong consistency estimation ofθ₀, and that R (Δ,θ)attains its local maximum atθ_mn, as m, n→∞Second, for the situation of one populations mean, considering the following simple random sample with missing data:( x_i ,δ_i), i = 1,…, n; Whereδ_i =In this chapter, we assume that x is missing completely at random (MCAR), P (δ= 1|x )= p(constant). We obtain'complete sample'through two different methods of data imputation, and then obtain empirical likelihood confidence intervals for means and compare advantages and disadvantages of two different methods of imputation through imitation. Assuming x₁ ,…x_m i. i . d ., distribution function is G_θ( x); G_θ( x) is known butθis unknown. We obtain'complete sample'by (I) (II) two different methods of imputation. Data imputation(I)is obtaining'complete sample'through hot deck random imputation : (II) is making maximum likelihood estimate ofθby use of nomissing data , (θ|^) denotes the estimate value ofθ,missing data are imputed at random from G_θ|^(x) , and obtain'complete sample'after imputation . So we obtain empirical likelihood ratio statistic -2R(Δ,θ_mn)'s asymptotical distribution which gives empirical likelihood confidence intervals of means.Usingθ₀ to denote the true value ofθ, We make some assumptions as follows:(1).θ₀∈ΩandΩis an open interval.(2). The distribution of G_θ( x) have common support, that is, the set A = { x : gθ( x) > 0}is independent ofθ.(3). For every x∈A,the density g_θ( x) is differentiable three times with respect toθ. (7). Ex =μ,0 < Var ( x)<∞, and E|x|³<∞. Through procedures, we obtain empirical likelihood ratio statistic whereλ_n =λ_n (μ) is determined by the following equationTheorem 2.1 Suppose that assumptions (1) (7) are satisfied. Incomplete sample in (I) imputation, we have where a₁ = 1- p + p^-1Theorem 2.2 Suppose that assumptions (1)-(7) are satisfied. Incomplete sample in (II) imputation, we have Var ( x )is population variance.